A New ZVS Tuning Method for Double-Sided LCC Compensated Wireless Power Transfer System

Abstract

This paper presents a new zero voltage switching (ZVS) tuning method for the double-sided inductor/capacitor/capacitor (LCC) compensated wireless power transfer (WPT) system. An additional capacitor is added in the secondary side of the double-sided LCC compensation network in order to tune the network to realize ZVS operation for the primary-side switches. With the proposed tuning method, the turn off current of the primary-side switches at the low input voltage range can be reduced compared with the previous ZVS tuning method. Consequently, the efficiency of the WPT at the low input voltage range is improved. Moreover, the relationship between the input voltage and the output power is more linear than that of the previous ZVS tuning method. In addition, the proposed method has a lower start-up voltage. The analysis and validity of the proposed tuning method are verified by simulation and experimental results. A WPT system with up to 3.5 kW output power is built, and 95.9% overall peak efficiency is achieved.

1. Introduction

Wireless power transfer (WPT) technology can deliver electric power over certain distances without metal-to-metal contact by utilizing energy-contained fields [1,2]. WPT has attracted more and more research interest due to its convenience, safety and other advantages [3], particularly in electric vehicle (EV) charging [4,5].

Many academic papers have been published on WPT and many aspects of WPT have been studied in order to improve the performance of the WPT [4]. Among them, the compensation circuit is essential because it can be used to tune the resonant frequency of the system, reduce the reactive power in the power electronics converter, and improve the transferred power capability and efficiency [4,6]. According to the way in which the compensation capacitors are connected to the primary and secondary coils in the compensation networks, the basic compensation topologies can be classified into four types, that is, series–series (SS), series–parallel (SP), parallel–series (PS) and parallel–parallel (PP) [6,7]. A detailed analysis of these four basic compensation topologies has been conducted in [8,9]. It was concluded in [9] that SS and SP are superior to the other two compensation topologies. However, the SP compensation circuit is easily mistuned because the primary compensated capacitance relates to the coupling coefficient [10], which means the resonant frequency is dependent on the coupling coefficient. Therefore, SS compensation topology is superior to the other three compensation topologies because its resonant frequency is independent of the coupling coefficient and the load conditions [6]. The defect of the SS compensation topology is the poor output voltage regulation capability [11], so many other compensation topologies have been proposed in order to improve the efficiency or realize other purposes [12]. In [13], an inductor/capacitor (LC) compensation network is employed for the primary side and an inductor/capacitor/inductor (LCL) resonant network is formed for WPT. However, the resonant frequency of this compensation topology is dependent on the coupling coefficient and the load conditions [6]. In [14], LC compensation networks are employed for both the primary side and the secondary side and a bidirectional power transfer is achieved. In [15], an inductor/capacitor/inductor T-type (LCL-T) resonant converter is proposed which can be treated as a current source. Its primary side current is only determined by the input voltage, so the primary side current can be easily controlled. In [16], an LCC compensation network is adopted for the primary side and a zero current switching (ZCS) is achieved by tuning the parameters of the compensation network. In [17], the LCC compensation network is applied for the secondary side, which can minimize the reactive power at the secondary side and achieve a unit power factor pickup. A double-sided LCC compensation network is proposed for the WPT in [18]. Its resonant frequency is irrelevant to the coupling coefficient and the load conditions. Meanwhile, its tuning method for realizing zero voltage switching (ZVS) is also presented in [18]. A comparison study on SS and double-sided LCC compensation topology is carried out in [6]. It is concluded that double-sided LCC compensation topology is less sensitive to the self-inductance variations, lower voltage and current stress and higher efficiency compared with the SS compensation topology. Integrated LCC compensation topologies are proposed in [10,19] in order to reduce the coil size and make the system more compact with high efficiency. In [19], a numerical method to tune the compensation network parameters for ZVS is also presented. However, the turn off currents of the primary side switches are relatively large at the low input voltage condition which lowers the efficiency of the system [18,19]. In addition, the relationship between the input voltage and the output power is not so linear in the WPT system using the ZVS tuning method proposed by [18].

For high voltage high frequency metallic oxide semiconductor field effect transistor (MOSFET) switching applications, the zero voltage turn on soft switching (ZVS) condition is very important due to the poor reverse recovery characteristics of the MOSFET’s body diode [20]. At the ZVS condition, the turn on switching loss is almost zero and the turn off loss is proportional to the turn off current [21]. In most of the WPT studies, the system is simply tuned to the inductive region to realize ZVS [22,23]. The turn off currents are not accurately designed. It is important to optimize the turn off current while maintaining the ZVS condition to minimize the switching loss.

In this paper, a new ZVS tuning method for double-sided LCC-compensated WPT system is proposed. For the secondary side, an additional capacitor is added to the compensation topology in order to realize ZVS operation of the primary side switches. With the proposed ZVS tuning method, the resonant frequency of the resonant network is still irrelevant with the coupling coefficient and the load conditions. Meanwhile, the turn off current of the primary-side switches at the low input voltage range can be reduced. Consequently, the efficiency of the WPT at the low input voltage range is improved. Moreover, the relationship between the input voltage and the output power is more linear than that of the previous ZVS tuning method. The proposed new ZVS tuning method and the parameter design method are also analyzed and presented in this paper, respectively. Simulation and experimental results verify the analysis and validity of the proposed circuit and the tuning method. A WPT system with up to 3.5 kW output power and 95.9% peak efficiency is built to verify the proposed ZVS tuning method.

The remainder of this paper is organized as follows. The basic circuit structure of the double-sided LCC compensation network and the new tuning method for ZVS operation are presented in Section 2. The parameter design of the proposed new ZVS tuning method is presented in Section 3. Simulated and experimental results are presented in Section 4. The conclusions are drawn in Section 5.

2. Double-Sided LCC Compensation Network and Parameter Tuning for ZVS

2.1. Double-Sided LCC Compensation Network

Figure 1 shows the double-sided LCC compensation network and corresponding power electronics circuit components. S₁–S₄ are the primary side MOSFEs. L_ps, C_pp and C_p form the primary side compensation network. L_p is the self-inductance of the transmitting coils. M is the mutual inductance of the transmitting coil and the receiving coil. L_s represent the self-inductance of the transmitting coils. L_ss, C_sp and C_s form the secondary side compensation network. C_d is a capacitor added in order to achieve ZVS for the primary side metallic oxide semiconductor field effect transistors (MOSFEs). D₁–D₄ represent the secondary side rectifier diodes. u_AB is the input voltage of the primary side compensation network, which is the voltage generated by V_in and the switching MOSFET S₁–S₄. u_ab represents the output voltage of the secondary side compensation network, which is related to the output voltage V_o. The currents on L_p, L_s, L_ps and L_ss are represented by i_p, i_s, i_Lps, i_Lss as shown in the Figure 1.

According to [18], the equivalent circuit to the circuit in Figure 1 referring to the primary side can be derived as shown in Figure 2; the circuit status at resonant frequency can be derived as shown in Figure 3. In Figure 2 and Figure 3, U_AB, U_ab, I_P, I_S, I_Lps, I_Lss represent the phasor form of the corresponding variables.

The variables in Figure 2 can be expressed by the following equations

$$\{\begin{array}{c}n=\sqrt{L_s/L_p}\\ L_m=k\cdot L_p,L_{p1}=\left(1-k\right)\cdot L_p\\ L_{p2}^{\prime }=\left(1-k\right)\cdot L_s/n^2,L_{ss}^{\prime }=L_{ss}/n^2\\ C_s^{\prime }=n^2\cdot C_s,C_{sp}^{\prime }=n^2\cdot C_{sp}\\ \begin{array}{l}{U}_{ab}^{\prime }=U_{ab}/n\\ U_{AB}=U_{AB}\end{array}\end{array}$$

(1)

where k is the coupling coefficient between the transmitting and receiving coils, U_AB represent the root mean square value of U_AB.

The variables $L_{e1}$ and $L_{e2}^{\prime }$ in Figure 3 can be expressed by the following equations

$$\{\begin{array}{c}{L}_{e1}=L_{ss}-k\cdot L_p\\ L_{e2}^{\prime }=L_{ps}^{\prime }-k\cdot L_p\end{array}$$

(2)

For Equations (1) and (2), the influence of C_d is implied in ${\mathbf{U}}_{ab}^{\prime }$. The relation between C_d and ${\mathbf{U}}_{ab}^{\prime }$ will be analyzed in the next section. It can be seen from Figure 2 and Figure 3 that the added C_d has no impact on the equivalent circuit of the double-sided LCC compensation network which is presented in [18]. Therefore, the similar analysis process presented in [18] can be used in this paper. However, the added C_d has impacts on ${\mathbf{U}}_{ab}^{\prime }$, so the waveform of ${\mathbf{U}}_{ab}^{\prime }$ in Figure 3 is different from the waveform of ${\mathbf{U}}_{ab}^{\prime }$ in [18]. Moreover, the parameter tuning method for ZVS operation is also different.

2.2. Parameter Tuning for ZVS

For the MOSFETs, the turn off switching loss is much smaller than its turn on loss at hard switching [20]. Therefore, it is better that all the MOSFETs are turned on at ZVS conditions in order to reduce the overall switching loss. To realize ZVS for a MOSFET, the MOSFET should be turned on when the body diode is conducting. This means the resonant current of the primary side should lag the resonant voltage in order to form the ZVS operation condition for all MOSFETs. The added C_d in the secondary side of the compensation network should be properly designed in order to achieve the ZVS operation.

According to the Figure 3a,b, the following equations can be derived by superposition principle

$${\mathbf{I}}_{Lss\_1\mathrm{st}}^{\prime }={\mathbf{I}}_{LssAB}^{\prime }+{\mathbf{I}}_{Lssab}^{\prime }=\frac{k{U}_{AB}{L}_p}{j{\omega }_0{L}_{ps}{L}_{ss}^{\prime }}$$

(3)

$$I_{Lss\_1\mathrm{st}}=\frac{{\mathbf{I}}_{Lss\_1\mathrm{st}}^{\prime }}{n}=\frac{k{U}_{AB}\sqrt{L_p{L}_s}}{j{\omega }_0{L}_{ps}{L}_{ss}}$$

(4)

where the subscript ‘1st’ represents the first harmonic component of the corresponding item.

According to Equation (3), I_{Lss_1st} leads U_AB by a phase ϕ₁, which can be expressed as

$${\phi }_1=-\frac{\pi }{2}$$

(5)

To achieve ZVS conditions at a small turn off current, the analysis accuracy of the turn off current is very important. Therefore, the high order harmonics of the square voltage cannot be neglected. The effect of the high order harmonics current is shown in Figure 4. The phase between u_ab zero crossing time t_b and its maximum value time t_c is represented by a. The phase between the zero crossing of i_Lss and u_ab is represented by ϕ_ab. Meanwhile, i_Lss contains not only the first order current but also the high order harmonic currents. The interaction of the high order harmonics between the primary and secondary side can be neglected because the compensation networks form a high order filter between the primary side and the secondary side. Moreover, a is close to zero as shown in Figure 4, so the waveform of u_ab is firstly approximated by a square wave. Therefore, the high order harmonic currents of the L_ss can be roughly expressed as:

$$\begin{array}{cc}\hfill {\mathbf{I}}_{Lss\_3\mathrm{rd}}& \approx -\frac{U_{ab\_3\mathrm{rd}}}{j\cdot 3{\omega }_0{L}_{ss}+\frac{1}{j\cdot 3{\omega }_0{C}_{sp}}}=j\frac{3U_{ab\_3\mathrm{rd}}}{8{\omega }_0{L}_{ss}}\hfill \\ \hfill {\mathbf{I}}_{Lss\_5\mathrm{th}}& \approx -\frac{U_{ab\_5\mathrm{th}}}{j\cdot 5{\omega }_0{L}_{ss}+\frac{1}{j\cdot 5{\omega }_0{C}_{sp}}}=j\frac{5U_{ab\_5\mathrm{th}}}{24{\omega }_0{L}_{ss}}\hfill \\ & \cdots \hfill \\ \hfill {\mathbf{I}}_{Lss\_(2k+1)\mathrm{th}}& \approx -\frac{U_{ab\_(2k+1)\mathrm{th}}}{j\cdot (2k+1){\omega }_0{L}_{ss}+\frac{1}{j\cdot (2k+1){\omega }_0{C}_{sp}}}\hfill \\ & =j\frac{(2k+1)U_{ab\_(2k+1)\mathrm{th}}}{({(2k+1)}^2-1){\omega }_0{L}_{ss}}\hfill \\ & \cdots \hfill \end{array}$$

(6)

In the time domain, the high order harmonic currents of the L_ss can be expressed as

$${\displaystyle \sum i_{Lss\_m\mathrm{th}}(t)}\approx \frac{\sqrt{2}{U}_{ab}}{{\omega }_0{L}_{ss}}\left[\frac{\cos (3{\omega }_{0}t)}{8}+\frac{\cos (5{\omega }_{0}t)}{24}+\dots +\frac{\cos ((2k+1){\omega }_{0}t)}{({(2k+1)}^2-1)}+\dots \right] (k=1,2,\dots ,+\infty )$$

(7)

According to Figure 4, the summation of all the high order harmonics is zero at t₀ = $-{\phi }_0/{\omega }_0$,

$${{\displaystyle \sum i_{Lss\_m\mathrm{th}}(t)}|}_{t=\frac{-{\phi }_0}{{\omega }_0}}={{\displaystyle \sum _{k=1}^{\infty }{i}_{Lss\_m\mathrm{th}}(t)}|}_{t=\frac{-{\phi }_0}{{\omega }_0}}\approx {{\displaystyle \sum _{k=1}^9{i}_{Lss\_m\mathrm{th}}(t)}|}_{t=\frac{-{\phi }_0}{{\omega }_0}}=0$$

(8)

Solve the Equation (8), then the following equation can be derived

$${\phi }_0\approx 0.412$$

(9)

According to Equation (6), $I_{L\mathrm{ss}\_m\mathrm{th}}$ leads $U_{ab\_m\mathrm{th}}$ by a phase which equals to 90°. So $i_{L\mathrm{ss}\_m\mathrm{th}}$ reaches the peak when $U_{ab}$ jumps at the time t_b. The peak value of the $i_{Lss\_m\mathrm{th}}$ can be calculated by

$$\begin{array}{ll}\max \left\{{\displaystyle \sum i_{Lss\_m\mathrm{th}}}\right\}& =\sqrt{2}\cdot {\displaystyle \sum _{k=1}^{\infty }{I}_{Lss\_(2k+1)\mathrm{th}}}\\ & =\sqrt{2}\cdot {\displaystyle \sum _{k=1}^{\infty }\frac{1}{({(2k+1)}^2-1)}}\frac{U_{ab}}{{\omega }_0{L}_{ss}}\\ & =\frac{\sqrt{2}}{4}\cdot \frac{U_{ab}}{{\omega }_0{L}_{ss}}\end{array}$$

(10)

According to Equations (8)–(10), the high order harmonics can be approximated by

$$\begin{array}{ll}{\displaystyle \sum i_{Lss\_m\mathrm{th}}(t)}& \approx h_p+\frac{h_p{\omega }_0}{{\phi }_0}t\\ & =\frac{\sqrt{2}}{4}\cdot \frac{U_{ab}}{{\omega }_0{L}_{ss}}\left(1+\frac{{\omega }_0}{0.412}t\right)\end{array}$$

(11)

where $h_p$ is the peak value of the total high order current.

Assuming t_b = 0, t_a = $-{\phi }_{ab}/{\omega }_0$, then the following equations can be derived:

$$c={\displaystyle \sum i_{Lss\_m\mathrm{th}}(t_a)}\approx \frac{\sqrt{2}}{4}\cdot \frac{U_{ab}}{{\omega }_0{L}_{ss}}\left(1+\frac{{\omega }_0}{0.412}{t}_a\right)=\frac{\sqrt{2}}{4}\cdot \frac{U_{ab}}{{\omega }_0{L}_{ss}}\left(1-\frac{{\phi }_{ab}}{0.412}\right)$$

(12)

$${\displaystyle \underset{t_a}{\overset{t_b}{\int }}{i}_{Lss}(t)dt}={\displaystyle \underset{\frac{-{\phi }_{ab}}{{\omega }_0}}{\overset{0}{\int }}{i}_{Lss\_1\mathrm{st}}(t)+{\displaystyle \sum i_{Lss\_m\mathrm{th}}(t)}dt}=C_d{V}_o$$

(13)

In Figure 4, ϕ_ab is close to zero, so $i_{Lss\_1\mathrm{st}}(t)$ in Equation (13) can be approximately expressed as

$$i_{Lss\_1\mathrm{st}}(t)=A_{m}t+A_m\frac{{\phi }_{ab}}{{\omega }_0}$$

(14)

where

$$A_m=\frac{\sqrt{2}k{U}_{AB}\sqrt{L_p{L}_s}}{{\omega }_0{L}_{ps}{L}_{ss}}$$

(15)

Integrating the left side of Equation (13), then the following equation can be derived:

$$\frac{1}{2}\frac{{\phi }_{ab}{}^2\left(0.412A_m+h_p\right)}{0.412{\omega }_0}=C_d{V}_o$$

(16)

where V_o is the output voltage.

Solving the Equation (16), then the phase shift ϕ_ab can be expressed as

$${\phi }_{ab}=\sqrt{\frac{0.824C_d{\omega }_0{V}_o}{\left(0.412A_m+h_p\right)}}$$

(17)

According to Figure 4, the following equation can be derived:

$${\phi }_2\approx {\phi }_{ab}-\frac{h_p(1-2.43{\phi }_{ab})}{A_m}$$

(18)

However, the effect of the added C_d on the magnitude of h_p should be taken into consideration in order to calculate ϕ₂ accurately. According to Figure 4, the charge time from t_b to t_c is around 0.4 times of the charge time from t_a to t_b, so the following equations can be derived:

$$a\approx 0.4{\phi }_{ab}$$

(19)

$$\begin{array}{ll}{h}_p^{\prime }& \approx \frac{4V_o}{a\pi {\omega }_0{L}_{ss}}\cdot \left(\frac{3}{8}\cdot \frac{1}{9}\sin (3a)+\frac{5}{24}\cdot \frac{1}{25}\sin (5a)+\dots \frac{19}{360}\cdot \frac{1}{361}\sin (19a)\right)\\ & =\frac{8V_o}{{\phi }_{ab}\pi {\omega }_0{L}_{ss}}\cdot \left(\frac{1}{24}\sin (\frac{3{\phi }_{ab}}{2})+\frac{1}{120}\sin (\frac{5{\phi }_{ab}}{2})+\dots \frac{1}{6840}\sin (\frac{19{\phi }_{ab}}{2})\right)\end{array}$$

(20)

where $h_p^{\prime }$ is the modified peak value of the total high order current.

Substituting Equation (20) into (17), then the following equations can be derived:

$${\phi }_{ab}^{\prime }=\sqrt{\frac{0.824C_d{\omega }_0{V}_{\mathrm{o}}}{\left(0.412A_m+h_p^{\prime }\right)}}$$

(21)

$$a^{\prime }\approx 0.4{\phi }_{ab}^{\prime }$$

(22)

$$h_p^{″}\approx \frac{8V_o}{{\phi }_{ab}^{\prime }\pi {\omega }_0{L}_{ss}}\cdot \left(\frac{1}{24}\sin (\frac{3{\phi }_{ab}^{\prime }}{2})+\frac{1}{120}\sin (\frac{5{\phi }_{ab}^{\prime }}{2})+\dots \frac{1}{6840}\sin (\frac{19{\phi }_{ab}^{\prime }}{2})\right)$$

(23)

Finally, the phase by which $i_{L\mathrm{ss}\_1st}$ leads $u_{ab}$ can be obtained:

$${\phi }_2\approx {\phi }_{ab}^{\prime }-\frac{h_p^{″}(1-2.43{\phi }_{ab}^{\prime })}{A_m}$$

(24)

In Figure 4, I_{Lss_1st} leads U_ab by a phase ϕ₂. According to Equation (5), the following equations can be derived:

$$\cos \phi =\cos \left({\phi }_1-{\phi }_2\right)=\cos \left(-\frac{\pi }{2}-{\phi }_2\right)=-\sin {\phi }_2$$

(25)

$$\begin{array}{cc}{\mathbf{I}}_{Lps\_1\mathrm{st}}& =-\frac{k{\mathbf{U}}_{ab}^{\prime }{L}_P}{j{\omega }_0{L}_{ps}{L}_{ss}^{\prime }}=-\frac{k{U}_{ab}^{\prime }(\cos \phi +j\cdot \sin \phi )L_p}{j{\omega }_0{L}_{ps}{L}_{ss}^{\prime }}\\ & =-\frac{k{U}_{ab}^{\prime }\sin \phi L_p}{{\omega }_0{L}_{ps}{L}_{ss}^{\prime }}-\frac{k{U}_{ab}^{\prime }\cos \phi L_p}{j\cdot {\omega }_0{L}_{ps}{L}_{ss}^{\prime }}\\ & \approx \frac{k{U}_{ab}\sqrt{L_p{L}_s}}{{\omega }_0{L}_{ps}{L}_{ss}}+\frac{k{U}_{ab}\sin {\phi }_2\sqrt{L_p{L}_s}}{j\cdot {\omega }_0{L}_{ps}{L}_{ss}}\\ & \approx \frac{k{U}_{ab}\sqrt{L_p{L}_s}}{{\omega }_0{L}_{ps}{L}_{ss}}+\frac{k{U}_{ab}{\phi }_2\sqrt{L_p{L}_s}}{j\cdot {\omega }_0{L}_{ps}{L}_{ss}}\end{array}$$

(26)

According to [18], the total high order current peak value of the primary side can be calculated by:

$$\max \left\{{\displaystyle \sum i_{Lss\_m\mathrm{th}}}\right\}=\frac{\sqrt{2}}{4}\cdot \frac{U_{AB}}{{\omega }_0{L}_{ss}}$$

(27)

The turn off current of the MOSFET is determined by both the first order and the high order harmonic currents. According to Equations (26) and (27), the turn off current of the MOSFET can be calculated as:

$$i_{OFF}=\sqrt{2}\left(\frac{k{U}_{ab}{\phi }_2\sqrt{L_p{L}_s}}{{\omega }_0{L}_{ps}{L}_{ss}}+\frac{U_{AB}}{4{\omega }_0{L}_{ps}}\right)$$

(28)

According to the MOSFET parameters, a minimum turn off current I_{OFF_min} which can achieve ZVS can be determined [21]. Then,

$$i_{OFF\_\min }\le \sqrt{2}\left(\frac{k{U}_{ab}{\phi }_2\sqrt{L_p{L}_s}}{{\omega }_0{L}_{ps}{L}_{ss}}+\frac{U_{AB}}{4{\omega }_0{L}_{ps}}\right)$$

(29)

According to Equations (21), (24) and (28), it can be seen that a turn off current can be calculated for a given C_d. When the value of C_d is increased, the phase shift ϕ_ab and ϕ₂ will also be increased. Consequently, the turn off current will also be increased.

3. Parameter Design

In this section, a 3.6 kW WPT system is designed according to the above analysis. The parameters of the WPT system are shown in

Table 1. Parameters of wireless power transfer (WPT) system.

Spec/Parameter	Values
Input DC voltage	<500 V
Output voltage	300–450 V
Nominal gap	120 mm
Coupling coefficient	0.2–0.3
Transmitting coil inductance	300 μH
Transmitting coil AC resistance	~500 mΩ
Receiving coil inductance	202.7 μH
Receiving coil AC resistance	~500 mΩ
Switching frequency	85 kHz
Maximum power	~3.6 kW

.

If U_AB is not too low, Figure 4 and Equations (7)–(9) satisfies. Thus ϕ₂ < ϕ₀/2 ≈ 0.2, and cosϕ₂ ≈ 1. From Table 1 and according to the design principle for minimizing the total reactive power in L_p and L_s [24], the L_ps and L_ss can be designed as

$$\begin{array}{ll}{L}_{ps}& =U_{AB}\sqrt{\frac{k_{\max }}{{\omega }_0{P}_{\max }}\cdot L_p}\\ & =\frac{2\sqrt{2}}{\pi }\times 500\times \sqrt{\frac{0.3}{2\pi \times 85\times {10}^3\times 3.6\times {10}^3}\cdot 300\times {10}^{-6}}H\\ & \approx 97.40 \mu H\end{array}$$

(30)

$$\begin{array}{ll}{L}_{ss}& =U_{ab}\sqrt{\frac{k_{\max }}{{\omega }_0{P}_{\max }}\cdot L_s}\\ & =\frac{2\sqrt{2}}{\pi }\times 450\times \sqrt{\frac{0.3}{2\pi \times 85\times {10}^3\times 3.6\times {10}^3}\cdot 202.7\times {10}^{-6}}H\\ & \approx 72.05 \mu H\end{array}$$

(31)

The values of C_pp and C_sp can be derived from the following equations,

$$\{\begin{array}{c}{C}_{pp}=\frac{1}{{\omega }_0^2{L}_{ps}}\approx 36 nF\\ C_{sp}=\frac{1}{{\omega }_0^2{L}_{ss}}\approx 48.66 nF\end{array}$$

(32)

The values of C_ps and C_ss can also be calculated from

$$\{\begin{array}{c}{C}_{ps}=\frac{1}{{\omega }_0^2(L_p-L_{ps})}\approx 17.30 nF\\ C_{ss}=\frac{1}{{\omega }_0^2(L_p-L_{ss})}\approx 26.83 nF\end{array}$$

(33)

The minimum turn off current is different for different MOSFET and dead-time settings. The turn off current should be large enough to discharge the junction capacitors within the dead-time in order to achieve ZVS, which can be expressed as [21]:

$$i_{OFF}\ge \frac{4{\displaystyle {\int }_0^{U_{AB,\max }}{C}_{oss}(u_{ds})d{u}_{ds}}}{t_d}$$

(34)

where $U_{AB,\max }$ is the maximum input voltage, u_ds is the drain-source voltage of the MOSFET, $C_{oss}$ is the junction capacitance which is a function of u_ds, and $t_d$ is the dead-time. In this paper, Cree C3M0065090D SiC MOSFET (Cree, Research Triangle Park, NC, USA) is chosen as the primary side switches. According to the MOSFET parameters from the datasheet and Equation (34), the turn off current should be larger than 1A in order to realize ZVS. Therefore, the minimum turn off current I_{OFF_min} is designed to be 2A.

Theoretical results of the MOSFETs’ turn off current I_OFF with different values of C_d are shown in Figure 5. For a given C_d, the turn off current varies as the input voltage increasing. Therefore, the selection of C_d should satisfy the minimal turn off current requirement for all the input voltage. It can be seen from the Figure 5a that the value of C_d can be designed to be 2.8 nF in order to ensure that the turn off current is larger than 2A.

According to the above calculations, the designed parameters of the compensation network is shown in

Table 2. Parameters of compensation network.

Parameter	Values
Lps	97.4 μH
Lss	72.05 μH
Cpp	36 nF
Csp	48.66 nF
Cps	17.3 nF
Css	26.83 nF
Cd	2.8 nF

.

4. Simulation and Experimental Results

4.1. Simulated Results

To evaluate the performance of the proposed ZVS tuning method, a simulation model is built in LTspice. The parameters of the simulation model are shown in Table 1 and Table 2. The various misalignments between transmitting coil and the receiving coil can be reflected by the different coupling coefficients. Therefore, three coupling coefficients, namely, k = 0.2, 0.25, 0.3, and three output voltages, V_o = 300, 400, and 450 V, are chosen as case studies in the simulations. Figure 6 shows the comparison of simulation and theoretical calculation results for the output power as a function of input voltage for three coupling coefficients and three output voltage. It can be seen from the Figure 6 that the simulation results agree well with the theoretical analysis and the output power varies linearly with the input voltage from 100 V to 500 V.

Figure 7 shows the comparison of simulation and theoretical calculation results for the turn off currents as a function of input voltage for three coupling coefficients and three output voltages. It can be seen from the Figure 7 that the simulation results agree well with the theoretical analysis, which verifies the validity of the theoretical analysis.

4.2. Experimental Results

A WPT experimental setup was built as shown in Figure 8a in order to verify the theoretical analysis and simulation results. The parameters of the prototype are tabulated in Table 1 and Table 2. The widths of the transmitter coil and the receiver coil are 500 mm and 450 mm, respectively. The air gap between two coils is 120 mm. The coupling coefficient k = 0.3, and three output voltages, V_o = 300, 400, and 450 V, are chosen as the case studies in the experiment. An electronic load at constant voltage mode is used to represent the battery pack for flexible voltage adjustment. The efficiency screen shot from power meter WT1800 (Yokogawa, Tokyo, Japan) at output power of 3.5 kW is shown in Figure 8b. Urms1, Irms1 and Urms5, Irms5 represent the voltages and currents of the input side and the output side, respectively. P1 and P5 represent the input and output power, respectively. ${\eta }_2$ is the efficiency of the WPT system. Figure 9 shows comparison of theoretical and experimental output power as a function of input voltage for three output voltages. The experimental output power matches well with the theoretical calculation. They vary linearly with the input voltage very well even at the low input voltage. The comparison of the experimental output power of the proposed new ZVS tuning method and the previous method proposed in [18] are shown in Figure 10. The relationship between the input voltage and the output power of the proposed new ZVS tuning method is more linear than that of the previous ZVS tuning method, especially at the low input voltage conditions. Moreover, the WPT system using the previous ZVS tuning method cannot start up at V_in = 100 V when V_o = 400 and V_o = 450 V. This is why the output power data of the WPT system using the previous ZVS tuning method are not shown at V_in = 100 V in Figure 10b,c. It is obvious that the proposed new ZVS tuning method has a lower start-up voltage.

Figure 11 shows the theoretical calculation and experimental results of MOSFETs’ turn off currents for different output voltages. The experimental results agree well with the theoretical analysis. The primary-side waveforms are shown in Figure 12 when the WPT system steadily delivers 3.5 kW to the load. At this operating point, the input voltage equals to 500 V, while the output voltage equals to 450 V. The turn off current of the MOSFET I_OFF equals 4.5 A and a good ZVS condition is achieved. Although the turn off current is higher than required, it is quite small compared with the peak current. The comparison of MOSFETs turn off currents for the proposed new ZVS tuning method and the previous ZVS tuning method is shown in Figure 13. It can be seen that the MOSFETs turn off currents of the proposed new ZVS tuning method are smaller those that of the previous ZVS tuning method, especially at the low input voltage conditions.

Figure 14 shows the comparison of the experimental efficiencies from the direct current (DC) power source to the battery load for the proposed new ZVS tuning method and the previous ZVS tuning method. In the experiment, an electronic load at constant voltage mode is employed to take the position of a real battery pack in order to facilitate voltage adjustment. A power meter WT1800 from Yokogawa is used to measure the efficiency of the WPT system. The maximum measured efficiency of the proposed new ZVS tuning method is 95.9% when V_in = 450 V, V_o = 400 V, as shown in Figure 14b. At this point, an efficiency improvement of about 0.15% is achieved compared with the previous ZVS tuning method. For the low output power range, the efficiency of these two ZVS tuning methods are almost the same when V_o = 400 V, while a significant efficiency improvement can be achieved by using the proposed new ZVS method when V_o = 300 V and V_o = 450 V. For the high output power range (>1.5 kW), the proposed new ZVS tuning method has a little higher efficiency for all the output voltages.

Besides the efficiency and better operations at low input voltage, the proposed method by employing C_d provides an easier, more flexible way to optimize the turn-off current. In the previous method, the resonant capacitor C₂ needs to be tuned. When the resonant capacitor and compensation inductors are integrated together with the main coils, it is very difficult to tune C₂ as it is already packed inside the coil pad [19]. Also, C₂ is in series with the main coil, which usually suffers from a much higher voltage than the output DC voltage. Sometimes C₂ is even formed by a series connected low voltage capacitors [10].

5. Conclusions

A new ZVS tuning method and its parameter design procedure for the double-sided LCC compensated WPT system is proposed in this paper. The new method ensures that the resonant frequency is irrelevant to the coupling coefficient and the load conditions, and the ZVS operation for the primary side MOSFETs is achieved. Meanwhile, the efficiency of the WPT at the low input voltage range is improved because the turn off current of the primary side switches at the low input voltage range is reduced. Moreover, a linear relationship between the input voltage and the output power is exhibited. In addition, the WPT system can start up at a lower voltage. A WPT system with output power about 3.5 kW is designed and built. Simulation and experimental results verify the analysis and validity of the proposed ZVS tuning method.