Leakage Magnetic Field Suppression Using Dual-Transmitter Topology in EV Wireless Charging

Abstract

This paper proposes an active leakage magnetic field (LMF) suppression scheme, which uses the dual-transmitter (DT) topology, for EV wireless charging systems (EVWCS). The two transmitter coils are coplanar, concentric and driven by separate inverters. The LMF components generated by the three coils cancel each other out to reduce the total field strength. This paper gives a detailed theoretical analysis on the operating principles of the proposed scheme. Finite element analysis is used to simulate the LMF distribution patterns. Experimental results show that when there is no coil misalignment, 97% of the LMF strength can be suppressed in a 1kW prototype. These results also show that the impact on efficiency is small. The trade-off between LMF suppression and efficiency is revealed, and a control strategy to balance these two objectives is presented.

I. INTRODUCTION

One of the many challenges that wireless power transfer (WPT) faces is leakage magnetic field (LMF) suppression. Excessive exposure to electromagnetic fields is a threat to human health [1]. In addition, LMFs are a potential source of electromagnetic interference to other sensitive equipment. To suppress the LMF in WPT systems, researchers have proposed many effective measures, which can be divided into three main categories: coil design, passive shielding and counter-LMF generation.

One of the many challenges that wireless power transfer (WPT) faces is leakage magnetic field (LMF) suppression. Excessive exposure to electromagnetic fields is a threat to human health [1]. In addition, LMFs are a potential source of electromagnetic interference to other sensitive equipment. To suppress the LMF in WPT systems, researchers have proposed many effective measures, which can be divided into three main categories: coil design, passive shielding and counterLMF generation. neighboring coils are opposite for laptop charging applications. The DD and DDQ coils introduced in [4] can reduce the LMF in regions close to the coil center. However, in regions further away they generate a stronger LMF than circular coils. A magnetic field shaping technique that uses a TX coil composed of several windings was introduced in [5] for EV wireless charging systems.

Measures in the second category use soft magnetic materials to guide magnetic flux lines or they use the eddy current in metal plates to offset the incident magnetic field. Soft magnetic materials can improve the mutual inductance and lower the LMF strength on its backside. However, in other regions the LMF strength cannot be effectively suppressed, (see Fig. 4 of [6]). Metallic plates can lower the LMF strength in nearly all regions. However, they can also cause eddy current loss. Usually these two measures are combined to achieve better shielding effectiveness and to reduce loss [6]. Due to their simplicity and effectiveness, techniques of this category are the most widely adopted. Theoretical analyses of magnetic field shielding can be found in [7], [8]. Magnetic structure designs and optimization methods are presented in [9], [10]. In addition, metamaterials can also be used to reduce LMF and to improve coupling at the same time [11].

The third category uses extra coil currents to generate counter-LMF. These currents are generated either through magnetic induction or by reverse winding. An active LMF suppression scheme with feedback control was proposed in [12]. However, it was applied for extremely low frequency magnetic field mitigation and is unsuitable for WPT systems whose operating frequencies are much higher. In [13], a combination of active and passive shielding is adopted for power-frequency magnetic field management. In WPT systems, a popular solution is to use a reactive coil. The authors of [14], [15] demonstrated a resonant reactive LMF shielding scheme with automatic tuning control for online electric vehicles. The authors of [16] proposed a reactive shielding scheme with a phase shifter, which solves contradictions between the current phase angle and the amplitude in the reactive coil. The authors of [17] used a hybrid loop array to improve efficiency and reduce LMF strength. A scheme that connects a TX coil in series with a flux cancelling coil was proposed in [18] to cancel the magnetic field on the backside of the TX. In WPT there are no known articles on active LMF suppression schemes that use an independently-driven coil. In addition, existing solutions cannot achieve an extremely high LMF suppression percentage (e.g., >95%).

This paper proposes a dual-transmitter (DT) LMF suppression scheme based on the counter-LMF generation principle. The underlying circuit theories are analyzed in detail, and a strong LMF suppression ability is demonstrated. Finite element analysis is used to simulate the LMF distribution patterns. In terms of efficiency, the DT-EVWCS (EV wireless charging system) is superior to its single-transmitter (ST) counterparts, which are almost the same as the DT system except that they have only one TX. When the two inverters are properly controlled, MOSFET conduction loss can be reduced and DC to DC efficiency can be improved in DT-EVWCS. Due to the trade-off between LMF suppression and efficiency, a control strategy to balance these two objectives has also been proposed.

The rest of this paper is organized as follows. Section II introduces the operating principles of the proposed scheme. Section III analyzes the LMF distribution patterns generated by three coils and proposes a control strategy to balance the LMF suppression and efficiency. Section IV demonstrates some experimental results. Finally, section V concludes the paper.

II. OPERATING PRINCIPLE OF THE DT-EVWCS

In the DT-EVWCS, two concentric and coplanar TXs are driven by separate inverters, as shown in Fig. 1. Coil 1 is the main TX coil, and coil 2 is the auxiliary TX coil. The LCC compensation topology is used in both TXs because of its constant-current-output feature and increased degrees of freedom in terms of parameter optimization [19]. On the receiver (RX) side, series compensation is adopted because of its simplicity. Since compensation circuits are basically bandpass filters, only the fundamental voltage and current components are considered. In this entire paper, the operating frequency is fixed at 85kHz.

Fig. 1. Simplified structure of the DT-EVWCS.

The operating principle of the DT-EVWCS can be briefly explained as follows. Two carefully-designed TX coil current phasors generate a magnetic field, where the required voltage is induced in the RX coil. The LMF generated by the three coils cancel each other out to keep the overall LMF strength below a given limit. In addition, the efficiency can be optimized by properly controlling the two TX coil currents.

A. Interactions Among the Three Coils

In Fig. 2, U₁ and U₂ are the inverter output fundamental voltages; ܴR_L is the equivalent load resistance (ܴR_L = \(\frac{8}{\pi^2}\)R_load, where ܴR_load is the DC load resistance); L₁ and L₂ are the TX coils; L₃ is the RX coil; M₁₂, M₂₃ and M₁₃ are the mutual inductances; C₁, C₂, C₃, C_f1 and C_f2 are the compensation capacitors; and L_f1 and L_f2 are the compensation inductors. The LCC compensation circuit of TX1 comprises L_f1, C_f1, C₁ and L₁, and TX2 has a similar structure. R_inv is the ESR of the MOSFET-based inverter, which is the sum of the MOSFET on-state drain-source resistances on the current conduction path. For the sake of simplicity, the ESRs are omitted in the following analyses except for the efficiency calculation. According to the basic design principle of the LCC compensation topology, L_f1C_f1 = L_f2C_f2 = \(\frac{1}{\omega_0^2}\), and the TX coil currents are controlled by the TX input voltages: I_L1 = \(\frac{U_{1}}{j \omega_{0} L_{f 1}}\), I_L2 = \(\frac{U_{2}}{j \omega_{0} L_{f 2}}\) [19]. On the RX side, L₃C₃ = \(\frac{1}{\omega_0^2}\), where ω₀ is the operating frequency ( 2π × 85000 rad/s). The RX coil current can be obtained as:

\(\boldsymbol{I_{L 3}}=\frac{-j \omega_{0}\left(M_{13} \boldsymbol{I_{L 1}}+M_{23} \boldsymbol{I_{L 2}}\right)}{R_{L}}=\frac{-\left(M_{13} \boldsymbol{U_{1}} / L_{f 1}+M_{23} \boldsymbol{U_{2}} / L_{f 2}\right)}{R_{L}}.\)       (1)

Fig. 2. Simplified schematic of the DT-EVWCS.

To obtain a certain |I_L3|, many combinations of U₁ and U₂ can be used under the condition that |M₁₃ U₁/L_f1 + M₂₃ U₂/L_f2| is equal to a certain value.

TX 1 is driven by three voltages: the inverter output voltage U₁, the induced voltage U₂₁ = -jω₀M₁₂I_L2, and U₃₁ = -jω₀M₁₃I_L3. According to the superposition principle, which is shown in Fig. 3, the output current of inverter 1 can be obtained as:

\(\begin{aligned} I_{L f 1}=& \frac{-j \omega_{0} M_{12} \boldsymbol{I_{L 2}}}{j \omega_{0} L_{f 1}}+\frac{-j \omega_{0} M_{13} \boldsymbol{I_{L 3}}}{j \omega_{0} L_{f 1}}+\frac{\boldsymbol{U_{1}}}{j \omega_{0} L_{f 1}+\left(\frac{1}{j \omega_{0} C_{f 1}} \| j \omega_{0} L_{1}^{\prime}\right)} \\ =& \frac{j M_{12} \boldsymbol{U_{2}}}{\omega_{0} L_{f 1} L_{f 2}}+\frac{M_{13}\left(M_{13} \boldsymbol{U_{1}} / L_{f 1}+M_{23} \boldsymbol{U_{2}} / L_{f 2}\right)}{L_{f 1} R_{L}} \\ &+\frac{\boldsymbol{U_{1}}}{j \omega_{0} L_{f 1}+\left(-j \omega_{0} L_{f 1} \| j \omega_{0} L_{1}^{\prime}\right)} \\ =&\left[\frac{M_{13}^{2}}{L_{f 1}^{2} R_{L}}+\frac{1}{j \omega_{0} L_{f 1}+j \omega_{0} L_{1}^{\prime} /\left(1-\frac{L_{1}^{\prime}}{L_{f 1}}\right)}\right] \boldsymbol{U_{1}} \\ &+\left(\frac{M_{13} M_{23}}{L_{f 1} L_{f 2} R_{L}}+j \frac{M_{12}}{\omega_{0} L_{f 1} L_{f 2}}\right) \boldsymbol{U_{2}} \\ =&\left(\frac{M_{13}^{2}}{L_{f 1}^{2} R_{L}}+j \frac{L_{1}^{\prime}-L_{f 1}}{\omega_{0} L_{f 1}^{2}}\right) \boldsymbol{U_{1}} \\ &+\left(\frac{M_{13} M_{23}}{L_{f 1} L_{f 2} R_{L}}+j \frac{M_{12}}{\omega_{0} L_{f 1} L_{f 2}}\right) \boldsymbol{U_{2}}. \end{aligned}\)       (2a)

Similarly, the output current of inverter 2 is:

\(\begin{aligned} \boldsymbol{I_{L f 2}}=&\left(\frac{M_{23}^{2}}{L_{f 2}^{2} R_{L}}+j \frac{L_{2}^{\prime}-L_{f 2}}{\omega_{0} L_{f 2}^{2}}\right) \boldsymbol{U_{2}} \\ &+\left(\frac{M_{13} M_{23}}{L_{f 1} L_{f 2} R_{L}}+j \frac{M_{12}}{\omega_{0} L_{f 1} L_{f 2}}\right) \boldsymbol{U_{1}}. \end{aligned}\)       (2b)

Here, \(j \omega_{0} L_{1}^{\prime}=j \omega_{0} L_{1}+\frac{1}{j \omega_{0} c_{1}}\), \(j \omega_{0} L_{2}^{\prime}=j \omega_{0} L_{2}+\frac{1}{j \omega_{0} C_{2}}\).

Fig. 3. Superposition of the three components in TX 1.

Let U₁ = |U₁|e^j0 and U₂ = |U₂|e^jθ be the inverter output voltages. Then Equ. (2a) can be changed into:

\(\begin{aligned} I_{L f 1}=&\left(\frac{M_{13}^{2}}{L_{f 1}^{2} R_{L}}+j \frac{L_{1}^{\prime}-L_{f 1}}{\omega_{0} L_{f 1}^{2}}\right)\left|\boldsymbol{{U}_{1}}\right|(\cos 0+j \sin 0) \\ &+\left(\frac{M_{13} M_{23}}{L_{f_{1}} L_{f 2} R_{L}}+j \frac{M_{12}}{\omega_{0} L_{f 1} L_{f 2}}\right)\left|\boldsymbol{{U}_{2}}\right|(\cos \theta+j \sin \theta) \\ &=\left(\frac{M_{13}^{2}\left|\boldsymbol{U_{1}}\right|}{L_{f 1}^{2} R_{L}}+\frac{M_{13} M_{23}\left|\boldsymbol{U_{2}}\right| \cos \theta}{L_{f 1} L_{f 2} R_{L}}-\frac{M_{12}\left|\boldsymbol{U_{2}}\right| \sin \theta}{\omega_{0} L_{f 1} L_{f 2}}\right) \\ &+j\left(\frac{L_{1}^{\prime}-L_{f 1}}{\omega_{0} L_{f 1}^{2}}\left|\boldsymbol{U_{1}}\right|+\frac{M_{12}\left|\boldsymbol{U_{2}}\right| \cos \theta}{\omega_{0} L_{f 1} L_{f 2}}+\frac{M_{13} M_{23}\left|\boldsymbol{U_{2}}\right| \sin \theta}{L_{f 1} L_{f 2} R_{L}}\right). \end{aligned}\)       (3a)

In addition, Equ. (2-b) is equivalent to:

\(\boldsymbol{{I}_{L f 2}}=e^{j \theta} \cdot\left[\begin{array}{c} \left(\frac{M_{23}^{2}\left|\boldsymbol{{U}_{2}}\right|}{L_{f 2}^{2} R_{L}}+\frac{M_{13} M_{23}\left|\boldsymbol{{U}_{1}}\right| \cos \theta}{L_{f 1} L_{f 2} R_{L}}+\frac{M_{12}\left|\boldsymbol{{U}_{1}}\right| \sin \theta}{\omega_{0} L_{f 1} L_{f 2}}\right) \\ +j\left(\frac{L_{2}^{\prime}-L_{f 2}}{\omega_{0} L_{f 2}^{2}}\left|\boldsymbol{{U}_{2}}\right|+\frac{M_{12}\left|\boldsymbol{{U}_{1}}\right| \cos \theta}{\omega_{0} L_{f 1} L_{f 2}}-\frac{M_{13} M_{23}\left|\boldsymbol{{U}_{1}}\right| \sin \theta}{L_{f_{1}} L_{f 2} R_{L}}\right) \end{array}\right].\)       (3b)

Ideally, Eqns. (3a and b) should have zero imaginary parts to maximize the TX input power factors. To realize ZVS, the imaginary parts should be slightly negative, which can be realized by adjusting L_f1, L_f2, C₁ and C₂. A smaller C₁ means a smaller \(L_1^{\prime}\) (\(L_1^{\prime}\) can be negative). Thus, reducing C₁ decreases imag(I_Lf1) and vice versa. Similarly, C₂ can be used to regulate imag(I_Lf2). In addition, L_f1 and L_f2 can also be adjusted. Apparently, the fully resonant LCC compensation topology [19] (\(L_1^{\prime}\) = L_f1, \(L_2^{\prime}\) = L_f2) is not suitable here

B. Power Flow Path Analysis

Referring to Eqns. (3a and b), it is possible to obtain the input active power to each TX and the total input power:

\(\begin{aligned} P_{in1}=& \frac{1}{2} \boldsymbol{U}_{\mathbf{1}} \cdot \boldsymbol{{I}_{L f 1}} \\ =& \frac{M_{13}^{2}}{2 L_{f 1}^{2} R_{L}}\left|\boldsymbol{{U}_{1}}\right|^{2}+\frac{M_{13} M_{23}}{2 L_{f 1} L_{f 2} R_{L}}\left|\boldsymbol{{U}_{1}}\right|\left|\boldsymbol{{U}_{2}}\right| \cos (\theta) \\ &-\frac{M_{12}}{2 \omega L_{f 1} L_{f 2}}\left|\boldsymbol{{U}_{1}}\right|\left|\boldsymbol{{U}_{2}}\right| \sin (\theta) \end{aligned}\)       (4a)

\(\begin{aligned} P_{in2}=& \frac{1}{2} \boldsymbol{U}_{\mathbf{2}} \cdot \boldsymbol{{I}_{L f 2}} \\ =& \frac{M_{23}^{2}}{2 L_{f 2}^{2} R_{L}}\left|\boldsymbol{{U}_{2}}\right|^{2}+\frac{M_{13} M_{23}}{2 L_{f 1} L_{f 2} R_{L}}\left|\boldsymbol{{U}_{1}}\right|\left|\boldsymbol{{U}_{2}}\right| \cos (\theta) \\ &+\frac{M_{12}}{2 \omega L_{f 1} L_{f 2}}\left|\boldsymbol{{U}_{1}}\right|\left|\boldsymbol{{U}_{2}}\right| \sin (\theta) \end{aligned}\)       (4b)

\(\begin{aligned} P_{i n}&=P_{i n 1}+P_{i n 2} & \\ &=\frac{M_{13}^{2}}{2 L_{f 1}^{2} R_{L}}\left|\boldsymbol{U_{1}}\right|^{2}+\frac{M_{23}^{2}}{2 L_{f 2}^{2} R_{L}}\left|\boldsymbol{U_{2}}\right|^{2} \\ &+\frac{M_{13} M_{23}}{L_{f 1} L_{f 2} R_{L}}\left|\boldsymbol{U_{1}}\right|\left|\boldsymbol{U_{2}}\right| \cos (\theta). \end{aligned}\)       (4c)

In the ST-EVWCS, there is only one power flow path from the TX coil to the RX coil. In Equ. (4c), there are direct power flow paths from the TXs to the RX and a cross-coupling term, which shows that P_in is affected by θ.

C. Efficiency Analysis

A passive diode rectifier is adopted in the EVWCS. As a result, power loss on the RX side is determined when the output power and load resistance are fixed. This part focuses on two of the loss terms: the inverter conduction loss and the transmitter loss.

1) Transmitter Efficiency: Transmitter efficiency is defined by \(\eta_{trans}=\frac{P_{load}}{P_{loss, T X} + P_{load}}\), where P_load is the power consumed on the load resistor, and P_loss,TX is the power dissipated on the TX coil ESRs. Strictly speaking, the power losses on the compensation inductors (L_f1 and L_f2) should also be included here. However, they are included in the inverter loss group instead because L_f1 and L_f2 are in series with ܴ௜௡௩ . The power loss on the compensation capacitors is not considered because it is relatively negligible.

Using the law of cosines, the induced voltage in the RX coil is obtained as:

\(\begin{aligned} \left|\boldsymbol{{U}_{3}}\right|^{2}=&\left(\omega_{0} M_{13}\left|\boldsymbol{{I}_{L 1}}\right|\right)^{2}+\left(\omega_{0} M_{23}\left|\boldsymbol{{I}_{L 2}}\right|\right)^{2} \\ &+2 \omega_{0}^{2} M_{13} M_{23}\left|\boldsymbol{{I}_{L 1}}\right|\left|\boldsymbol{{I}_{L 2}}\right| \cos (\theta). \end{aligned}\)       (5)

Thus:

\(\begin{aligned} p=& \frac{P_{load}}{P_{loss, T X}}=\frac{\frac{\left|\boldsymbol{U_{3}}\right|^{2}}{2 R_{L}}}{\frac{1}{2} R_{L 1}\left|\boldsymbol{I_{L 1}}\right|^{2}+\frac{1}{2} R_{L 2}\left|\boldsymbol{I_{L 2}}\right|^{2}} \\ &=\frac{\omega_{0}^{2}}{R_{L}} \cdot \frac{M_{13}^{2}+M_{23}^{2} t^{2}+2 M_{13} M_{23} \cos (\theta) t}{R_{L 1}+R_{L 2} t^{2}}. \end{aligned}\)       (6)

Here t = |I_L2|/|I_L1| is the TX coil current ratio. The derivative of Equ. (6) with respect to t is:

\(\begin{aligned} \frac{d p}{d t} &=\frac{\omega_{0}^{2}}{R_{L}} \cdot \\ & \frac{-2 M_{13} M_{23} \cos (\theta) R_{L 2} t^{2}+2\left(M_{23}^{2} R_{L 1}-M_{13}^{2} R_{L 2}\right) t+2 M_{13} M_{23} \cos (\theta) R_{L 1}}{\left(R_{L 1}+R_{L 2} t^{2}\right)^{2}}. \end{aligned}\)       (7)

As can be seen in Fig. 11, θ>\(\frac{\pi}{2}\) is necessary for effective LMF suppression. Therefore, only this case is considered here. The nominator of the second fraction in Equ. (7) is a quadratic function of t,whose leading coefficient is positive. The function value at t=0 is negative. Therefore, the derivative goes from negative to positive as t increases. Since a larger function value of Equ. (6) means a higher η_trans, the highest η_trans is obtained either at t=0 or t=+∞. In the former case, \(\frac{P_{load}}{P_{loss, T X}}=\frac{\omega_{0}^{2} M_{13}^{2}}{R_{L} R_{L 1}}\); in the latter case, \(\frac{P_{load}}{P_{loss, T X}}=\frac{\omega_{0}^{2} M_{23}^{2}}{R_{L} R_{L 2}}\). These two values correspond to L₁ and L₂ working alone on the TX side. Therefore, the transmitter efficiency of the DT-EVWCS working in the LMF suppression mode should be slightly lower than that of the ST-EVWCS. In this paper, the difference between the ST-EVWCS and the DT-EVWCS is that the former has a single TX (TX1) and this TX uses the fully resonant LCC compensation topology.

2) Inverter Efficiency: Inverter efficiency is defined as \(\eta_{i n v}=\frac{P_{i n}}{P_{l o s s, i n v}+P_{i n}}\), where P_in is defined in Equ. (4c), and P_loss,inv is the conduction loss on ܴR_inv and the ESRs of L_f1 and L_f2 Under the soft-switching (ZVS) condition, there is virtually no overlap of the drain-source voltage and drain current at switching instants. Therefore, switching loss is not considered here because ZVS is pursued in this paper. Assuming that the two inverters are identical, P_loss,inv is obtained as:

\(\begin{aligned} P_{loss, inv}=& \frac{1}{2}\left(R_{inv}+R_{Lf1}\right)\left(\frac{P_{in1}}{\cos a_{1}\left|\boldsymbol{U_{1}}\right|}\right)^{2} \\ &+\frac{1}{2}\left(R_{inv}+R_{L f 2}\right)\left(\frac{P_{in2}}{\cos \alpha_{2}\left|\boldsymbol{U_{2}}\right|}\right)^{2}. \end{aligned}\)       (8)

Here cosα₁ and cosα₂ are the TX input power factors. They should be close to unity to improve inverter efficiency. Obviously, |U₁|and |U₂| can be raised to lower P_loss,inv, which can be realized by increasing L_f1 and L_f2 in the LCC compensation circuits. Nevertheless, the penalty is a lower output power under a limited DC-link voltage (Equ. (4c)).

It is possible for the DT-EVWCS to a achieve higher η_inv than the ST-EVWCS when the inverters used in both systems are identical. This can be roughly explained by the following inequality: \(P_{i n 1}^{2}+P_{i n 2}^{2}<\left(P_{i n 1}+P_{i n 2}\right)^{2}\). When ܴR_Lf1 = ܴR_Lf2, ܿcosα₁ = cosα₂ = 1 and the two inverter output voltages in the DT-EVWCS are equal to those in the ST-EVWCS, the DT-EVWCS has a lower P_loss,inv. Since the inverter output voltages are actually higher in the DT-EVWCS (see part B of section IV), the difference in P_loss,inv. between the two systems is larger than that indicated by this inequality.

D. Principles of LMF Suppression

On a specific measurement point in the intended shielding region (ISR, Fig. 6), the magnetic flux density phasor B_sum is the sum of three components, each contributed by a separate coil. In addition, the amplitude B_sum can be calculated as:

\(\begin{aligned} B_{sum} &=\left|\boldsymbol{I_{L 1}} \overrightarrow{K_{1}}+\boldsymbol{I_{L 2}} \overrightarrow{K_{2}}+\boldsymbol{I_{L 3}} \overrightarrow{K_{3}}\right| \\ &=\left|\boldsymbol{I_{L 1}} \overrightarrow{K_{1}}+\boldsymbol{I_{L 2}} \overrightarrow{K_{2}}+\frac{-j \omega_{0}\left(M_{13} \boldsymbol{I_{L 1}}+M_{23} \boldsymbol{I_{L 2}}\right)}{R_{L}} \overrightarrow{K_{3}}\right| \\ &=\left|\boldsymbol{I_{L 1}}\left(\overrightarrow{K_{1}}-j \frac{\omega_{0} M_{13}}{R_{L}} \overrightarrow{K_{3}}\right)+\boldsymbol{I_{L 2}}\left(\overrightarrow{K_{2}}-j \frac{\omega_{0} M_{23}}{R_{L}} \overline{K_{3}}\right)\right|. \end{aligned}\)       (9)

Here \(\vec{K_1}\), \(\vec{K_2}\) and \(\vec{K_3}\) are called magnetic flux density coefficients (MFDCs), which is defined as the magnetic flux density vector generated by 1 ampere of current in a coil when all of the other coil currents are zero. Clearly, the MFDC is a space vector and a function of the measurement position. B_sum is a function of three independent variables: |I_L1| ,|I_L2| and θ. After the output power constraint in Equ. (1) is imposed, there are two degrees of freedom left.

In the EVWCS prototype, due to the existence of a large metal plate that is used to simulate the car chassis, the magnetic flux lines generated by the three coils are almost coincident when the coil sizes are not too large and the coils are perfectly aligned (Fig. 7(a)). Thus, \(\vec{K_l}\) can be simplified as a scalar and the scalar form is adopted in the rest of this paper. Referring to Equ. (1), it is possible to rewrite Equ. (9) in two forms:

\(\begin{aligned} B_{sum} &=\left|\boldsymbol{I}_{\boldsymbol{{L} 1}} K_{1}+\boldsymbol{{I}_{L 2}} K_{2}+\boldsymbol{{I}_{L 3}} K_{3}\right| \\ &=\left|\boldsymbol{{I}_{L 1}} K_{1}+\frac{\boldsymbol{I_{L 3}} R_{L}+j \omega_{0} M_{13} \boldsymbol{I_{L 1}}}{-j \omega_{0} M_{23}} K_{2}+\boldsymbol{{I}_{L 3}} K_{3}\right| \\ &=\left|\boldsymbol{{I}_{L 1}}\left(K_{1}-\frac{M_{13}}{M_{23}} K_{2}\right)+\boldsymbol{{I}_{L 3}}\left(K_{3}+j \frac{R_{L}}{\omega_{0} M_{23}} K_{2}\right)\right| \end{aligned}\)       (10a)

\(\begin{aligned} B_{sum} &=\left|\boldsymbol{I}_{\boldsymbol{{L} 1}} K_{1}+\boldsymbol{{I}_{L 2}} K_{2}+\boldsymbol{{I}_{L 3}} K_{3}\right| \\ &=\left|\frac{\boldsymbol{I_{L 3}} R_{L}+j \omega_{0} M_{13} \boldsymbol{I_{L 2}}}{-j \omega_{0} M_{23}} K_{1}+\boldsymbol{{I}_{L 2}} K_{2}+\boldsymbol{{I}_{L 3}} K_{3}\right| \\ &=\left|\boldsymbol{{I}_{L 2}}\left(K_{2}-\frac{M_{23}}{M_{13}} K_{1}\right)+\boldsymbol{{I}_{L 3}}\left(K_{3}+j \frac{R_{L}}{\omega_{0} M_{23}} K_{1}\right)\right|. \end{aligned}\)       (10b)

When there is no restriction on the coil currents, any value of B_sum is attainable. Specifically, “B_sum = 0” is called the perfect LMF suppression (PLMFS) working point in this paper. Under this condition, I_L1 and I_L2 can be solved as:

\(\boldsymbol{I_{L 1}}=\frac{\boldsymbol{I_{L 3}}\left(\omega_{0} K_{3} M_{23}+j K_{2} R_{L}\right)}{\omega_{0}\left(K_{2} M_{13}-K_{1} M_{23}\right)}\)       (11a)

\(\boldsymbol{I_{L 2}}=\frac{\boldsymbol{I_{L 3}}\left(\omega_{0} K_{3} M_{13}+j K_{1} R_{L}\right)}{\omega_{0}\left(K_{1} M_{23}-K_{2} M_{13}\right)}.\)       (11b)

For a given B_limit > 0 , there are many possible combinations of I_L1 and I_L2 that make B_sum < B_limit. To maximize efficiency, a proper control strategy is needed, which will be developed in Section Ⅲ. Fig. 4 shows the LMF suppression principle using phasor diagrams. Multiplying I_Li and K_i yields the magnetic flux density phasors in Fig. 4(a). In addition, the two induced voltage components in the RX coil and the two magnetic flux density phasor components are also shown. In Fig. 4(b), B_sum is the sum of three phasors. When B_limit becomes stricter, the highest achievable η_trans becomes lower. The reason for this is simple. If there is no B_limit restriction, the maximum η_trans is obtained at θ=0 (Equ. (6)). However, θ should be greater than \(\frac{\pi}{2}\) to suppress B_sum. Thus, there exists a trade-off between η_trans and B_sum.

Fig. 4. Phasor diagram of the LMF suppression principle. (a) Phasor components. (b) Calculation of B_sum.

In comparison, the ST-EVWCS has only two coil current phasors. There is only one control variable (the inverter output voltage). Therefore, after the output power constraint is imposed there are no degrees of freedom left. To realize PLMFS, the two coil current phasors need to be opposite in direction and identical in magnitude. Otherwise, they should be null phasors themselves. Both of these conditions are hard to satisfy because the coil current phasors are orthogonal (\(\boldsymbol{{I}_{L 3}}=\frac{-j \omega_{0} M_{13} \boldsymbol{I_{L 1}}}{R_{L}}\)) when L₃ and C₃ resonate. Hence, there should be an upper limit in the LMF suppression capability for the ST-EVWCS.

III. LMF SUPPRESSION PRINCIPLE

A. LMF Distribution Patterns

Finite element analysis (FEA) is used to simulate the LMF distribution patterns, i.e., the direction and magnitude of the magnetic flux density vectors in the ISR generated by the unit coil currents. The geometric parameters of the EVWCS prototype are listed in Table I. The geometry of the single-layer circular coils, which are made of Litz wire, is shown in Fig. 8(a). A large steel plate on the RX side is used to simulate the car chassis. The TX coil and RX coil are separated by roughly 150mm in the vertical direction, and the ferrite slabs on both sides are arranged as illustrated in Fig. 5.

TABLE I GEOMETRIC PARAMETERS OF THE EVWCS PROTOTYPE

Fig. 5. FEA model. (a) Front view. (b) Bottom view.

Some simplifications are made in the FEA model. The thicknesses of the steel plate and aluminum plate, which are 1mm and 2mm, are changed to 10mm and 5mm, respectively. Otherwise, a very fine mesh is required due to the large length-to-thickness ratio of the steel plate (1800:1). L₁, L₂ and L₃ have only one turn in the FEA model, with diameters of 210mm, 310mm and 210mm, respectively. This is also for reducing the model complexity. Due to the employed simplifications, the FEA results are only used to show the overall trend. In Fig. 5, there is also a side-mounted rectangular coil, which is only used for the LMF calculation and is absent in the prototype. The FEA is conducted on COMSOL Multiphysics. The ISR and the measurement points are shown in Fig. 6. The ISR is part of the space outside the car where users are likely to enter under normal circumstances. This should include the regions where the highest LMF strength appears. It is verified in Fig. 7(d) that the ISR chosen in Fig. 6 can satisfy this requirement.

Fig. 6. Intended shielding region and measurement point locations.(a) Definition of ISR. (b) Measurement points.

Fig. 7. Simulated LMF distribution patterns. (a) Magnetic flux lines of coils 1-3. (b) Magnetic flux lines of coil 1 and the side-mounted coil. (c) MFDC ratios. (d) Magnetic flux density distribution of coil 1.

Simulated LMF distribution patterns are shown in Fig. 7. The MFDC of coil i is obtained when the current in coil i is nonzero and all of the other coil currents are zero. Because coils 1 and 3 have 17 turns in the prototype but only 1 turn in the FEA model, a 17A current is injected into each of them. For coil 2, 7A is injected, where “7” is the originally-chosen turn number of coil 2 (later on “7” is changed to “8” in the prototype). When there is no coil misalignment, the magnetic field vectors generated by the three coils have almost the same directions at all of the points in the ISR. It is worth noting that without the steel plate, the flux lines no longer coincide. When the diameter of coil 2 increases, its flux lines also tend to diverge from those of coil 1. If coil 2 is mounted in other ways, such as the side-mounted type proposed in Fig. 9 in [14], the LMF distribution pattern of coil 2 is different from that of the other coils. Apparently, the flux lines intersect each other. This shows that perfect LMF suppression is impossible because the sum of two non-parallel space vectors cannot be a null vector unless both of them are null vectors. Theoretically, there should be an upper limit to the LMF suppression capability. The geometric parameters of the single-turn side-mounted coil are shown in Fig. 5. Therefore, the “coincident flux lines” are a precondition for a highly effective LMF suppression. In Fig. 7(c), the MFDC ratios are nearly constant, which is another precondition for high LMF suppression effectiveness. Here K₂ = 1.2041K₁ and K₃ = 0.5722K₁. Constant MFDC ratios, combined with Equ. (10-a), leads to the fact that if K₁ on point A is x times that on point B, then B_sum on point A is also x times that on point B (here A and B are two arbitrary points in the ISR). Theoretically, only the B_sum on one measurement point is required to evaluate the LMF suppression effectiveness in the whole ISR. If the MFDC ratios are not constant, there exist at least two points in the ISR whose B_sum cannot simultaneously reach a sufficiently low value. Due to symmetry, only the measurement planes 1-3 need to be analyzed. In Fig. 7(c), among all of the measurement points, mp₃₂ has the highest MFDCs (K₁, K₂ and K₃ are 0.2275, 0.2745 and 0.1316 μT/A, respectively). As a result, this point is chosen as a representative. If B_sum is below B_limit on mp₃₂, then theoretically B_sum should be still lower on other points.

In Fig. 8(b), the MFDCs are obtained by measuring the magnetic flux density using a Narda EHP-50 TS when a 1A-amplitude sinusoidal current (85kHz) is injected into the coil under test and the other coils are open-circuited.

According to the linear fitting result, \(\frac{\mathrm{K}_{2}}{\mathrm{K}_{1}}\) and \(\frac{\mathrm{K}_{3}}{\mathrm{K}_{1}}\) are assumed to be 1.0094 and 0.4778. The strongest LMF strength among all of the measurement points appears on mp₃₂, whose MFDCs are 0.2715, 0.2711 and 0.1309 μT/A, respectively.

Fig. 8. Coil geometry and measured MFDCs. (a) Coil structure.(b) Measured MFDCs.

Fig. 9. Summation of induced voltage phasors. (a) 0< θ < \(\frac{{\pi}}{2}\). (b) \(\frac{{\pi}}{2}\) < θ < π.

B. Control Strategy

When optimizing the three control variables, it is necessary to pay attention to the following two constraints: output power requirement and B_sum < B_limit. The number of control variables is three, so there is still one degree of freedom left. This offers the possibility of efficiency optimization.

The optimum combination of I_L1 and I_L2 is hard to solve analytically. However, the following calculation procedure is applicable. Table II gives the electrical parameters, some of which will be used in the calculation. The coil ESRs are measured with an Agilent U1733C at 100kHz (because it does not work at 85kHz).

TABLE II ELECTRICAL PARAMETERS OF THE EVWCS PROTOTYPE

Step 1. Set the initial value of θ and calculate the highest possible current amplitude |I_L1|_max under this θ.For 0 < θ < \(\frac{\pi}{2}\), |I_L1|_max = \(\frac{\left|\boldsymbol{I_{L 3}}\right| R_{L}}{\omega_{0} M_{13}}\). For \(\frac{\pi}{2}\)  < θ < π, |I_L1|_max = \(\frac{\left|\boldsymbol{I_{L 3}}\right| R_{L}}{\sin(\theta) \omega_{0} M_{13}}\). This principle is illustrated in Fig. 9. In the first case, the gap between |I_L3| and |I_L1| widens as |I_L2| increases. Thus, the maximum |I_L1| under a given |I_L3| is achieved when |I_L2| decreases to zero. In the second case, according to the sine theorem, \(\frac{\omega_{0} M_{13}\left|\boldsymbol{I_{L 1}}\right|}{\sin (\alpha)}=\frac{\left|\boldsymbol{I_{L 3}}\right| R_{L}}{\sin (\pi-\theta)}\)(\(\frac{\pi}{2}\)  < θ < π).Therefore, |I_L1| is maximized at α  = \(\frac{\pi}{2}\).

Step 2. Decrease |I_L1| by an iteration step and calculate |I_L2| so that |I_L3| = \(\frac{-j \omega_{0}\left(M_{13} \boldsymbol{I_{L 1}}+M_{23} \boldsymbol{I_{L 2}}\right)}{R_{L}}\) can meet the output power requirement. As mentioned above, B_sum is a function of the three control variables |I_L1|, |I_L2| and θ. Since θ is given in Step 1 and |I_L1| is given here, and because of the output power constraint, all three of the degrees of freedom are consumed. Thus, B_sum can be uniquely determined.

Step 3. Calculate B_sum using Equ. (10a), and check whether B_sum ≤ B_limit. If so, calculate η_trans with the help of Equ. (6), and calculate η_inv with the help of Equ. (8), otherwise discard this combination. Then return to Step 2.

Step 4. When |I_L1| reaches 0, this increase or decrease θ by an iteration step. Then return to Step 1. After all of the reasonable θ values have been tested, pick the combination of θ, |I_L1| and |I_L2| that results in the highest η_trans ∙ η_inv.

Based on these four steps, an open-loop and a close-loop control scheme are proposed and shown in Fig. 10. The former is suitable for applications where the MFDCs, mutual inductances and ESRs are known or can be accurately estimated. B_limit is a given constant, B_OUT is determined by the charging status, and ܴR_L can be calculated from the RX-side information. Based on these quantities, the control strategy in Step 1 – Step 4 can be used to obtain the optimum |I_L1| − |I_L2| − θ combination. Phase shift control and DC-link voltage control are used together to generate the desired |U₁| − |U₂| − θ parameters. The close-loop control in Fig. 10(b), uses a feedback mechanism. As a result, the “constants” in Fig. 10(a) are not required. There are two constraints: P_OUT = \(P_{OUT}^{*}\) and B_sum < B_limit. The efficiency maximization algorithm decreases θ when B_sum is low enough and it searches for the optimum |U₁| − |U₂| combination to maximize efficiency while maintaining the required output power. When B_sum > B_limit, θ should be increased. In practice, the θ-B_limit curve in Fig. 11(a), which is the basis of the control algorithm, should be smoothed to make θ a monotonic function of B_limit.

Fig. 10. Control block diagrams. (a) Open-loop control. (b)Close-loop control.

Fig. 11. Calculated control variables. (a) TX input voltage. (b) TX input current and impedance angle.

Open-loop control is used in this paper. For the DT-EVWCS, the optimum control variables are shown in Fig. 11. When B_limit gets less strict, the overall trend is that the optimum θ decreases and the DC to DC efficiency increases. However, after B_limit increases beyond 1.23μT, θ no longer decreases. This is due to the four compensation parameters L_f1, L_f2, C₁ and C₂, whose impact on the TX input power factors were explained in part A of section II, are designed in such a way that the unity power factor of both TXs are reached at B_limit ≈ 1.23μT. If θ is further decreased, a lower η_inv is obtained due to the lower TX input power factors. However, η_trans might increase. In real applications, another reason for not further decreasing θ is the ZVS condition. The TX input impedance angle becomes negative when θ decreases further, which can also be inferred from Fig. 11(b). The highest theoretical DC to DC efficiency is achieved at the θ = 2.225 point, where both of the TXs have a nearly unity input power factor. Here the receiver efficiency (\(\eta_{R X}=\frac{R_{L}}{R_{L, e q}+R_{L 3}}\)) is also taken into consideration in the efficiency calculation. However, the rectifier loss is not considered. Fig. 14(b) shows that the LMF on mp32 in the DT-EVWCS can be as low as 0.02μT, which is less than 2% of that in the ST-EVWCS. This means that the LMF can meet strict B_limit constraints even at extremely high output power levels. In theory, the LMF strength can reach 0 when the two conditions in Eqns. (11a and b) are satisfied. However, this is not realized here due to discontinuity in the iteration process. In practical applications, the lowest achievable B_sum is limited by the voltage regulation accuracy of the inverters. Based on the calculated efficiency-B_limit curve (at P_OUT = 1ܹ݇) in Fig. 14(a), the theoretical efficiency-P_OUT curve (at B_limit = 0.85ߤ ( ܶis shown in Fig. 12. It is assumed that P_OUT is a quadratic function of the coil current amplitudes, while P_sum is a linear function of them. Of course, a real WPT system might exhibit very different behavior under very high-power levels, and the DC link voltage and phase shift angle might deviate from the trajectories in Fig. 11. Thus, Fig. 12 shows only the possible trend. Due to electrical stress and thermal stress considerations, P_OUT is fixed at 1kW in the experiments.

Fig. 12. Theoretical efficiency versus output power at B_limit=0.85μT.

IV. MEASUREMENT RESULTS

A. Experimental Setup

The geometric parameters of the EVWCS prototype are shown in Fig. 5 and Table I. The electrical parameters are given in Table II. The ST-EVWCS is identical to the DT-EVWCS, except that it has no TX2 or inverter 2, and it uses the fully resonant LCC compensation topology in TX1 (C₁ = 12.32݊nF). Meanwhile in the DT-EVWCS the compensation capacitor C₁ is redesigned (C₁ = 11nF). SiC MOSFETs with R_ds,on=0.12Ω (Cree C3M0120090J) are used in both of the inverters (2×2 MOSFETs in each leg), which are connected to the same DC source (maximum U_dc=120V). The rectifier diodes on the RX side are IXYS DPG15I400PM. Phase shift control is used to regulate the inverter output voltage and the output power is fixed at 1kW. Meanwhile, the DC link voltage is manually regulated. A Teledyne LeCroy Waverunner 610Zi is used for signal measurement. The LMF strength is measured using a NARDA EHP-50 TS. The two inverters in the DT-EVWCS are controlled by the same DSP (TMS320F28335) controller. As a result, driving signal synchronization is easily achieved. The experimental setup is shown in Fig. 13.

Fig. 13. Experimental setup. (a) Whole system. (b) Measurement position.

B. Measurement Results

In the ST-EVWCS, the DC link voltage is increased gradually until the output power reaches 1kW while the inverter conduction angle is fixed at 180 degrees. Inverter soft switching is realized in the experiment. In the DT-EVWCS, the optimum combinations of U₁ and U₂ in Fig. 11(a) are used as a reference. Due to unavoidable errors, the actual inverter output voltages are tuned around their theoretical optimum values. The DC link voltage control and the phase shift control are used together. Thus, only one of the inverters requires a phase shift. The 120V limit was not exceeded in all of the tests. The results are shown in Figs. 14(a and b). The measured DC to DC efficiency is lower than the theoretical values. However, the trend is the same. The lowest B_sum achieved is 0.06μT, and the DC to DC efficiency at this working point is 92.1%. When compared to 1.97μT in the ST-EVWCS, the LMF strength can be reduced by 97%. The highest DC to DC efficiency is 93.6%, which is 2% higher when compared to that of the ST-EVWCS. The trade-off between the LMF strength and efficiency is experimentally validated, and the effectiveness of the openloop control strategy is also shown, i.e., a higher efficiency is achievable when B_limit increases. Although η_trans and η_inv are not measured separately due to practical difficulties, it can be seen from the theoretical analysis that η_trans is lower in the DT-EVWCS, whereas η_inv is higher, as explained in part C of section II. In the LTspice simulation, it was observed that at the B_sum = 0.06μT ܶworking point, η_inv = 97.30% and η_trans ∙η_RX = 94.96% . In addition, in the ST-EVWCS the two numbers are 95.36% and 96.34%, respectively. At the B_sum > 1.23μT ܶworking points, the three control variables are constant. The two numbers are 98.53% and 95.26% respectively. The price paid for a high LMF suppression capability is an extra TX and lower η_trans when compared to the ST-EVWCS. In addition, the price paid for a higher η_inv is twice the inverter cost and a higher demand on the DC link voltage (when compared to the data in Fig. 11(a), the fundamental inverter output voltage in the ST-EVWCS is 95.6V in amplitude). ZVS is realized in most of the tests due to the variable DC link voltage control, which reduces the required phase shift angle. In some tests one of the four half-bridges is hard-switched. However, the effect on efficiency is small and the overall efficiency performance is satisfactory.

Fig. 14. Calculated and measured efficiency and LMF strength on mp₃₂ in the DT-EVWCS and ST-EVWCS. (a) DC-to-DC efficiency. (b) LMF strength.

C. Discussion

Since the ISR is in 3-D space, which increases the difficulty in accurately measuring and clearly presenting the LMF strength on all of the measurement points, the LMF data on only one point (mp₃₂) are experimentally measured. To complement the experimental results, the simulated LMF distribution at the PLMFS working point is presented in Fig. 15. The coil currents used in the simulation are calculated based on simulated MFDCs. The LMF on mp32 is 0.016μT, while the highest strength in the ISR is 0.042μT. The highest LMF strength does not appear on mp₃₂. This is due to the fact that the flux lines generated by the three coils are not strictly coincident and the MFDC ratios are not perfectly constant. At the PLMFS point, a slight error in the MFDC can result in a huge difference in B_sum.

Fig. 15. Simulated LMF distribution at the PLMFS point.

At the same time, a manual evaluation process was conducted by moving the field probe in the ISR. It was observed that the magnetic flux density on most of the points in the ISR is lower than that on mp₃₂. This is especially true on points further away from the coil center in the y-coordinate direction. There are some points located near mp₃₂ whose B_sum is slightly higher than mp₃₂. This is due to the fact that the point with the highest MFDC appears on a point other than mp₃₂, as is shown in Fig. 7(d). The point mp₃₂ has the highest MFDCs among all of the measurement points instead of among all the points. Overall, the experimental results coincide well with the theoretical analysis.

However, there are still some drawbacks. (1) Only the perfect alignment case is considered here. However, in practical applications, LMF suppression effectiveness is seriously impaired by coil misalignment, as has been observed in the experiment. Due to page limits, this topic is not discussed in this paper. (2) A steel plate is used to simulate the car chassis. However, a real car chassis is a lot more complex in terms of geometry. How this would affect the LMF distribution patterns deserves more in-depth research. (3) The inverter efficiency analysis should be improved to include both harmonic current loss and switching loss.

Despite these drawbacks, the most important finding is that LMF strength can be effectively suppressed by adding an extra TX coil driven by a separate inverter, and that the trade-off between efficiency and LMF suppression can be balanced by a simple control strategy. The DT-EVWCS is superior to the ST-EVWCS in that it can theoretically realize perfect LMF suppression without seriously lowering efficiency due to more control degrees of freedom.

V. CONCLUSION

This paper proposes a novel LMF suppression scheme using the dual-transmitter topology, which is aimed at high-power EV wireless charging applications. The operating principles of the DT-EVWCS are explained. The LMF distribution patterns are simulated via finite element analysis. When there is a large steel plate and the coils are perfectly aligned, the magnetic flux lines generated by different coils are almost coincident and the MFDC ratios are stable, which are the two preconditions for high LMF suppression capability. A 97% reduction in leakage magnetic flux density is experimentally achieved. Meanwhile, a simple control strategy is proposed to balance the LMF suppression and efficiency.

Due to practical restrictions, only open-loop control was implemented in this paper, and the magnetic couplers (including the coils, ferrite slabs and metal plates) were constructed based on experience. Further research is required in terms of close-loop feedback control, performance under coil misalignment, variable-load/variable-power conditions, circuit parameter optimization and magnetic coupler optimization.