Exact Solution of the Class-E Inverter for Wireless Power Transfer Systems

ABSTRACT

This paper introduces the exact solution of the Class-E WPT resonant inverter. Dominantly, the DC-AC inverters for WPT systems are analyzed by assuming a sinusoid waveform for the output variables. However, by allowing waveform light distortions, the magnetic element values can be reduced; which is provided by the proposed method. It is shown that the circuit waveforms are composed of sine, cosine and exponential terms. The obtained solutions are used to design a 500 kHz Class-E WPT resonant inverter which operates with ZVS. The design is experimentally verified and 74 % efficiency with 9.7 W output power is achieved.

KEYWORDS
dc-ac inverters; resonant converters; wireless power transfer

I. INTRODUCTION

By means of a variable magnetic field, energy can be wireless transferred, which avoids physical contact between transmitter and receiver devices. This concept is known as inductive wireless power transfer (WPT) and its applications are related to electric vehicle chargers [1]–[3], smart grids [4], portable electronics [5]–[7] and biomedical [8].

In order to drive the transmitter coil, a high-frequency inverter is commonly used, which provides the alternating current injection in the primary side. Several topologies have been studied as DC-AC inverter for the WPT system. Generally, resonant inverters are chosen to drive the transmitter. Class-D half-bridge or full-bridge topologies have low voltage and current stress on the switches, however, their gate-driver implementation is a big challenge due to the high-side switch referenced to a floating node [9]–[12]. On the other hand, the Class-E inverter has only one switch and it is referenced to the ground, making it easier to design the gate-driver circuit. Moreover, it has higher voltage and current stress than the Class-D counterpart [13]–[16]. In complement, the full potential of the resonant inverters, both Class-D and Class-E and their variants, is achieved by considering their design in optimal operating, i.e., by operating with soft-switching mechanisms as zero-voltage switching (ZVS) or zero-current switching (ZCS).

Considering the design complexity of the Class-E resonant inverter, originating from its soft-switching characteristic, the sinusoidal assumption for the output waveforms is chiefly considered. In [16], a ClassE ZVS inverter with load independence characteristic is analyzed by assuming the loaded quality factor high enough to ensure that the output voltage is represent by a sine function. Also, an inverse Class-E inverter applied to WPT system is design in [17] by using a sinusoid output current. Furthermore, Class-D, Class-E, ClassE⁻¹, Class-DE, Class-EF and constant output ClassE are mostly analyzed in the literature by using the aforementioned assumption [18]–[23].

In this paper, the Class-E resonant inverter is analyzed without assuming the sinusoid output waveform. Thus, the output obtained equations have sine, cosine and exponential terms. The proposed approach enhances the design range and allows to reduce the component values for the magnetic components. It is because the resonant inductor is directly proportional to the quality factor; by setting the output variables as a sinusoid waveform, the quality factor is restrained, limiting the resonant inductor component value. In addition, when considering WPT applications, the inductive coils are considered as an equivalent load for the power inverter. Therefore, by restraining the resonant inductor value, the inductive coils are also confined in an inductance range.

The proposed method consists in the exact solution of the Class-E inverter, which means that, the timedomain solutions are obtained for the, resonant inductor current, parallel capacitor voltage and resonant capacitor voltage, without assuming the output as a sine waveform. By equating the solutions for each operating stage, it is possible to design the hardware components for the WPT application.

In order to model the WPT system, a coupled inductor model based on lumped parameters is presented. In contrast to other approaches, like as the finite element method [24], the proposed WPT representation can be easily connected as an equivalent load to the 1D topology of the converter. Therefore, it is possible to analyze by parameter sweeping the influence of the coils distance into the electrical variables of the power stage.

This paper is organized as follows: In Section II, the Class-E WPT resonant inverter is presented. In Section III, the exact solution is obtained for the targeting electrical variables. The inductive coupling analysis is performed in Section IV. A design methodology is detailed in Section V. Finally, the experimental results and the conclusions are shown in Section VI and VII, respectively.

II. CLASS-E WPT RESONANT INVERTER

By cascading a Class-E resonant inverter to a wireless power transfer system and modeling the inductive link as an equivalent load, the Class-E WPT resonant inverter in Fig. 1 is obtained. The Class-E inverter is composed of: input voltage V_in, choke inductor L_c, switch S, inverter capacitor C_p, resonant inductor L_r and a resonant capacitor C_r. The WPT system models the load for the Class-E inverter, which has a transmitter coil L₁, a receiver coil L₂ and a mutual inductance M. Finally, the system supplies a load R_L. The circuit variables are described as: resonant current i_Lr, transmitter loop current I₁, receiver loop current I₂, switch voltage v_S, parallel capacitor voltage v_Cp, and resonant capacitor voltage v_Cr.

The WPT system can be represented by a coupled inductor model with elements that depend on the angular operating frequency ω. By equating I₁, the input impedance of the inductive link can be found. It is going to be presented that it has a resistance R_in part and an inductive reactance represented by an inductor L_in. Thus, due to the series connection, the resonant inductor L_r and the WPT system input inductor L_in can be replaced by an equivalent inductor L_eq = L_r + L_in.

FIGURE 1
Class-E WPT resonant inverter.

In this sense, the Class-E WPT resonant inverter to be analyzed has the same topology of the standard ClassE inverter as long as due care is taken. Therefore, the following premises are considered in the subsequently analysis: 1) All components are ideal; 2) V_in and L_c are replaced by a constant current source, I_in; 3) The exact solution is target on the resonant circuit variables, which means i_Lr, v_Cp and v_Cr; 4) The analyzed resonant inductance requires a value that is sufficient to incorporate the transmitter coil inductance value and also has enough inductance for the Class-E inverter implementation. It is because, the converter is analyzed by considering the combination of L_r and L_in, therefore, the physical inductor for the Class-E converter is the analyzed inductance in the theoretical approach minus the receiver coil inductance from the WPT system; 5) The converter has two operating stages related to switch S on/off states and defined by duty cycle D_c. In the end of stage II, switch S turns-on with zero-voltage switching (ZVS) and zero-derivative voltage switching (ZDVS); 6) Operating frequency is constant; 7) Steady-state operation. The theoretical waveforms are depicted in Fig. 2 and the equivalent circuits for each operating stage in Fig. 3.

FIGURE 2
Theoretical waveforms.

FIGURE 3
Class-E inverter topology. (a) Switch S on. (b) Switch S off.

In the next section, the Class-E resonant inverter is going to be analyzed targeting to find the exact solution for each circuit variable. By finding the time domain solution for the Class-E topology, it is possible to later on design the converter components by considering the equivalent load as a representation of the WPT system as conceptualized in Fig. 1.

EXACT SOLUTION OF THE CLASS-E INVERTER

The main concept is to use the reactive element governing equations in the complex frequency domain. The inductor voltage is given by:

$v_L\left(t\right)=L\frac{d{i}_L\left(t\right)}{dt}\stackrel{\mathcal{L}}{⟷}sL{I}_L\left(s\right)-L{i}_L\left(0\right)$ (1)

and a capacitator voltage:

$v_C\left(t\right)=\frac{1}{C}\int i_C\left(t\right)dt\stackrel{\mathcal{L}}{⟷}\frac{1}{sC}{I}_C\left(t\right)+\frac{1}{s}{v}_C\left(0\right).$ (2)

The loop equation for stage 1 is written as

$v_{L_r}+v_{C_r}+v_R=0$ (3)

which can be expressed by

$s{L}_r{I}_2\left(s\right)-L_r{i}_{L_r}\left(0\right)+\frac{I_2\left(s\right)}{s{C}_r}+\frac{v_{C_r}\left(0\right)}{s}+I_2\left(s\right)R=0.$ (4)

I₂(s) represents the current in the resonant loop, composed of C_p − L_r − C_r − R, and it can be written in terms of the initial conditions, i_Lr(0) e v_Cr(0), the complex frequency s and the circuit components,

$I_2\left(s\right)=\frac{s{i}_{L_r}\left(0\right)-\frac{v_{C_r}\left(0\right)}{L_r}}{s^2+s\frac{R}{L_r}+\frac{1}{L_r{C}_r}}$ (5)

Due to the second order polynomial, the completing the square method can be used to rewrite the denominator aiming to find well-known Laplace transformations. Thus, I₂(s) is represented by:

$I_2\left(s\right)=\frac{s{i}_{L_r}\left(0\right)-\frac{v_{C_r}\left(0\right)}{L_r}}{a_1\left[{\left(s+{\alpha }_1\right)}^2+{\omega }_1^2\right]}$ (6)

where:${\alpha }_1=\frac{b_1}{2a_1},{\omega }_1=\sqrt{{\beta }_1} \mathrm{and} {\beta }_1=\frac{c_1}{a_1}-{\alpha }_1^2,$ with the coefficients: a₁ = 1, $b_1=\frac{R}{L_r} \mathrm{e} c_1=\frac{1}{L_r{C}_r}.$

Equation (6) is algebraically adjusted to find s+α₁ in the numerator:

$\begin{array}{r}{I}_2\left(s\right)=\frac{s-\frac{v_{C_r}\left(0\right)}{L_r{i}_{L_r}\left(0\right)}}{\frac{a_1}{i_{L_r}\left(0\right)}\left[{\left(s+{\alpha }_1\right)}^2+{\omega }_1^2\right]}\\ =\frac{s+{\alpha }_1-{\alpha }_1}{\frac{a_1}{i_{L_r}\left(0\right)}\left[{\left(s+{\alpha }_1\right)}^2+{\omega }_1^2\right]}-\frac{\frac{v_{C_r}\left(0\right)}{L_r{i}_{L_r}\left(0\right)}}{\frac{a_1}{i_{L_r}\left(0\right)}\left[{\left(s+{\alpha }_1\right)}^2+{\omega }_1^2\right]}.\end{array}$ (7)

The first term in (7) is separated into two new terms that are known as cosine and sine portions in the Laplace transformations:

$\begin{array}{r}{I}_2\left(s\right)=\frac{i_{L_r}\left(0\right)}{a_1}\frac{s+{\alpha }_1}{\left[{\left(s+{\alpha }_1\right)}^2+{\omega }_1^2\right]}-\frac{i_{L_r}\left(0\right){\alpha }_1}{a_1{\omega }_1}\frac{{\omega }_1}{\left[{\left(s+{\alpha }_1\right)}^2+{\omega }_1^2\right]}\\ -\frac{v_{C_r}\left(0\right)}{a_1{L}_r{\omega }_1}\frac{{\omega }_1}{\left[{\left(s+{\alpha }_1\right)}^2+{\omega }_1^2\right]}.\end{array}$ (8)

By applying the Inverse Laplace Transform by means of the direct replacement of the well-known transformations, the time domain solution can be achieved:

$\begin{array}{r}{i}_{L_r}^{\mathrm{I}}\left(t\right)=\frac{i_{L_r}\left(0\right)}{a_1}{\mathrm{e}}^{-{\alpha }_{1}t}\cos \left({\omega }_{1}t\right)-\frac{i_{L_r}\left(0\right){\alpha }_1}{a_1\omega }{\mathrm{e}}^{-{\alpha }_{1}t}\sin \left({\omega }_{1}t\right)\\ -\frac{v_{C_r}\left(0\right)}{a_1{L}_r\omega }{\mathrm{e}}^{-{\alpha }_{1}t}\sin \left({\omega }_{1}t\right)\mathrm{A}.\end{array}$ (9)

Equation (9) describes the exact solution of the loop current I₂ in time domain, which also describes the current i_Lr(t) in the resonant inductor L_r. The upper-script I denotes the first operating stage.

Since the switch S is turned-on in the first stage, the parallel capacitor voltage is zero. Therefore:

$v_{C_p}^{\mathrm{I}}\left(t\right)=0\mathrm{V}$ (10)

The resonant capacitor voltage v_Cr in the complex frequency domain can be obtained by considering I₂(s), described by (5), times 1/(sC_r):

${\mathrm{V}}_{{\mathrm{C}}_{\mathrm{r}}}\left(\mathrm{s}\right)=\frac{1}{{\mathrm{sC}}_{\mathrm{r}}}{\mathrm{I}}_2\left(\mathrm{s}\right)+\frac{1}{\mathrm{s}}{\mathrm{v}}_{{\mathrm{C}}_{\mathrm{r}}}\left(0\right)$ (11)

ergo,

${\mathrm{V}}_{{\mathrm{C}}_{\mathrm{r}}}\left(\mathrm{s}\right)=\left(\frac{1}{{\mathrm{sC}}_{\mathrm{r}}}\right)\left(\frac{{\mathrm{si}}_{{\mathrm{L}}_{\mathrm{r}}}\left(0\right)-\frac{{\mathrm{v}}_{{\mathrm{C}}_{\mathrm{r}}}\left(0\right)}{{\mathrm{L}}_{\mathrm{r}}}}{{\mathrm{s}}^2+\mathrm{s}\frac{\mathrm{R}}{{\mathrm{L}}_{\mathrm{r}}}+\frac{1}{{\mathrm{L}}_{\mathrm{r}}{\mathrm{C}}_{\mathrm{r}}}}\right)+\frac{{\mathrm{v}}_{{\mathrm{C}}_{\mathrm{r}}}\left(0\right)}{\mathrm{s}}.$ (12)

Due to the polynomial multiplication in the first term of (12), it is necessary to apply partial fractions by considering the finding constants A₁, A₂ and B₂,

${\mathrm{V}}_{{\mathrm{C}}_{\mathrm{r}}}\left(\mathrm{s}\right)=\frac{1}{{\mathrm{C}}_{\mathrm{r}}}\left(\frac{{\mathrm{A}}_1}{\mathrm{s}+0}+\frac{{\mathrm{A}}_2\mathrm{s}+{\mathrm{B}}_2}{{\mathrm{s}}^2+\mathrm{s}\frac{\mathrm{R}}{{\mathrm{L}}_{\mathrm{r}}}+\frac{1}{{\mathrm{L}}_{\mathrm{r}}{\mathrm{C}}_{\mathrm{r}}}}\right)+\frac{{\mathrm{v}}_{{\mathrm{C}}_{\mathrm{r}}}\left(0\right)}{\mathrm{s}},$ (13)

hence,

$\begin{array}{r}{\mathrm{V}}_{{\mathrm{C}}_{\mathrm{r}}}\left(\mathrm{s}\right)=\frac{1}{{\mathrm{C}}_{\mathrm{r}}}\left[\frac{{\mathrm{A}}_1{\mathrm{s}}^2+{\mathrm{A}}_1\mathrm{s}\frac{\mathrm{R}}{{\mathrm{L}}_{\mathrm{r}}}+{\mathrm{A}}_1\frac{1}{{\mathrm{L}}_{\mathrm{r}}{\mathrm{C}}_{\mathrm{r}}}+{\mathrm{A}}_2{\mathrm{s}}^2+{\mathrm{B}}_2\mathrm{s}}{(\mathrm{s}+0)\left({\mathrm{s}}^2+\mathrm{s}\frac{\mathrm{R}}{{\mathrm{L}}_{\mathrm{r}}}+\frac{1}{{\mathrm{L}}_{\mathrm{r}}{\mathrm{C}}_{\mathrm{r}}}\right)}\right]\\ +\frac{{\mathrm{v}}_{{\mathrm{C}}_{\mathrm{r}}}\left(0\right)}{\mathrm{s}}.\end{array}$ (14)

Therefore, the following linear system is considered:

$\left\{\begin{array}{l}{A}_1+A_2=0\\ A_1\frac{R}{L_r}+B_2=i_{L_r}\left(0\right)\\ A_1\frac{1}{L_r{C}_r}=-\frac{v_{C_r}\left(0\right)}{L_r}\end{array}\right.$ (15)

which returns A₁ = −C_rv_Cr(0), A₂ = C_rv_Cr(0) and $B_2=\frac{L_r{i}_{L_r}\left(0\right)+R{C}_r{v}_{C_r}\left(0\right)}{L_r}$ A₁, A₂ and B₂ are replaced in (14):

$\begin{array}{r}{V}_{C_r}\left(s\right)=-v_{C_r}\left(0\right)\left(\frac{1}{s+0}\right)+v_{C_r}\left(0\right)\left(\frac{s}{s^2+s\frac{R}{L_r}+\frac{1}{L_r{C}_r}}\right)\\ +\frac{L_r{i}_{L_r}\left(0\right)+R{C}_r{v}_{C_r}\left(0\right)}{L_r{C}_r}\left(\frac{s}{s^2+s\frac{R}{L_r}+\frac{1}{L_r{C}_r}}\right)\\ +\frac{v_{C_r}\left(0\right)}{s}\end{array}$ (16)

The same completing the square that was applied in the resonant current i_Lr analysis is used in the resonant capacitor voltage v_Cr. In addition, the terms should be arranged aiming to find the Laplace transformations. Therefore:

$\begin{array}{r}{V}_{C_r}\left(s\right)=-v_{C_r}\left(0\right)\left(\frac{1}{s}\right)+\frac{v_{C_r}\left(0\right)}{a_1}\left(\frac{s+{\alpha }_1}{\left[{\left(s+{\alpha }_1\right)}^2+{\omega }_1^2\right]}\right)\\ -\frac{v_{C_r}\left(0\right){\alpha }_1}{a_1{\omega }_1}\left(\frac{{\omega }_1}{\left[{\left(s+{\alpha }_1\right)}^2+{\omega }_1^2\right]}\right)\\ +\frac{L_r{i}_{L_r}\left(0\right)+R{C}_r{v}_{C_r}\left(0\right)}{L_r{C}_r{a}_1{\omega }_1}\left(\frac{{\omega }_1}{\left[{\left(s+{\alpha }_1\right)}^2+{\omega }_1^2\right]}\right)\\ +\frac{v_{C_r}\left(0\right)}{s}.\end{array}$ (17)

Finally, the equation can be converted to the time domain as:

$\begin{array}{r}{}_{v_C}^{\mathrm{I}}\left(t\right)=\frac{v_{C_r}\left(0\right)}{a_1}{\mathrm{e}}^{-{\alpha }_{1}t}\cos \left({\omega }_{1}t\right)-\frac{v_{C_r}\left(0\right){\alpha }_1}{a_1{\omega }_1}{\mathrm{e}}^{-{\alpha }_{1}t}\sin \left({\omega }_{1}t\right)\\ +\frac{L_r{i}_{L_r}\left(0\right)+R{C}_r{v}_{C_r}\left(0\right)}{L_r{C}_r{a}_1{\omega }_1}{\mathrm{e}}^{-{\alpha }_{1}t}\sin \left({\omega }_{1}t\right)\mathrm{V}.\end{array}$ (18)

In the stage II, switch S is off and the circuit shown in Fig. 3(b) should be considered. In order to find the resonant current, the loop equation is used as follows

$-v_{C_p}+v_{L_r}+v_{C_r}+v_R=0$ (19)

so

$\begin{array}{r}-\frac{1}{s{C}_p}\left[I_1\left(s\right)-I_2\left(s\right)\right]-\frac{1}{s}{v}_{C_p}\left(D_{c}T\right)+s{L}_r{I}_2\left(s\right)\\ -L_r{i}_{L_r}\left(D_{c}T\right)+\frac{1}{s{C}_r}{I}_2\left(s\right)+\frac{1}{s}{v}_{C_r}\left(D_{c}T\right)+I_2\left(s\right)R=0\end{array}$ (20)

One can see that the initial conditions for stage II are related to the instant D_cT, where T is the period. In addition, $I_1\left(s\right)=\frac{I_{in }}{s}$. Equating I₂(s):

$I_2\left(s\right)=\frac{\frac{I_{\mathrm{in}}}{s}\frac{1}{s{C}_p}+L_r{i}_{L_r}\left(D_{c}T\right)-\frac{v_{C_r}\left(D_{c}T\right)}{s}}{s{L}_r+\frac{1}{s{C}_r}+R+\frac{1}{s{C}_p}}$ (21)

which should be adjusted as

$I_2\left(s\right)=\left[\frac{1}{s}\right]\left[\frac{s^2{L}_r{C}_p{C}_r{i}_{L_r}\left(D_{c}T\right)-s{C}_p{C}_r{v}_{C_r}\left(D_{c}T\right)+I_{\mathrm{in}}{C}_r}{s^2{L}_r{C}_p{C}_r+sR{C}_p{C}_r+\left(C_r+C_p\right)}\right]$ (22)

By applying partial fraction decomposition considering one linear term and one quadratic term, the following is obtained:

$I_2\left(s\right)=\frac{A_1}{s+0}+\frac{A_{2}s+B_2}{s^2{L}_r{C}_p{C}_r+sR{C}_p{C}_r+\left(C_r+C_p\right)}$ (23)

then,

$I_2\left(s\right)=\frac{s^2{L}_r{C}_p{C}_r{A}_1+sR{C}_p{C}_r{A}_1+\left(C_r+C_p\right)A_1+s^2{A}_2+s{B}_2}{(s+0)\left(s^2{L}_r{C}_p{C}_r+sR{C}_p{C}_r+C_r+C_p\right)}.$ (24)

The following system of equations should be solved in order to find the constants:

$\left\{\begin{array}{l}{L}_r{C}_p{C}_r{A}_1+A_2=L_r{C}_p{C}_r{i}_{L_r}\left(D_{c}T\right)\\ R{C}_p{C}_r{A}_1+B_2=-C_p{C}_r{v}_{C_r}\left(D_{c}T\right)\\ \left(C_r+C_p\right)A_1=I_{\mathrm{in}}{C}_r\end{array},\right.$ (25)

which leads to

$A_1=\frac{I_{\mathrm{in}}{C}_r}{C_p+C_r}$ (26)

$A_2=\frac{-I_{\mathrm{in}}{C}_p{C}_r^2{L}_r+C_p^2{C}_r{L}_r{i}_{L_r}\left(D_{c}T\right)+C_p{C}_r^2{L}_r{i}_{L_r}\left(D_{c}T\right)}{C_p+C_r}$ (27)

and

$B_2=\frac{-I_{\mathrm{in}}{C}_p{C}_r^{2}R-C_p^2{C}_r{v}_{C_r}\left(D_{c}T\right)-C_p{C}_r^2{v}_{C_r}\left(D_{c}T\right)}{C_p+C_r}.$ (28)

By replacing A₁, A₂ e B₂:

$\begin{array}{r}{I}_2\left(s\right)=\frac{I_{\mathrm{in}}{C}_r}{C_p+C_r}\left(\frac{1}{s+0}\right)\\ +\frac{-I_{\mathrm{in}}{C}_p{C}_r^2{L}_r+C_p^2{C}_r{L}_r{i}_{L_r}\left(D_{c}T\right)+C_p{C}_r^2{L}_r{i}_{L_r}\left(D_{c}T\right)}{C_p+C_r}\\ \times \left(\frac{s}{s^2{L}_r{C}_p{C}_r+sR{C}_p{C}_r+\left(C_r+C_p\right)}\right)\\ +\frac{-I_{\mathrm{in}}{C}_p{C}_r^{2}R-C_p^2{C}_r{v}_{C_r}\left(D_{c}T\right)-C_p{C}_r^2{v}_{C_r}\left(D_{c}T\right)}{C_p+C_r}\\ \times \left(\frac{1}{s^2{L}_r{C}_p{C}_r+sR{C}_p{C}_r+\left(C_r+C_p\right)}\right),\end{array}$ (29)

which is rewritten by the completing the square method as

$\begin{array}{r}{I}_2\left(s\right)=\frac{I_{\mathrm{in}}{C}_r}{C_p+C_r}\left(\frac{1}{s+0}\right)\\ +\frac{-I_{\mathrm{in}}{C}_p{C}_r^2{L}_r+C_p^2{C}_r{L}_r{i}_{L_r}\left(D_{c}T\right)+C_p{C}_r^2{L}_r{i}_{L_r}\left(D_{c}T\right)}{C_p+C_r}\\ \times \left(\frac{s}{a_2\left[{\left(s+{\alpha }_2\right)}^2-{\omega }_2^2\right]}\right)\\ +\frac{-I_{\mathrm{in}}{C}_p{C}_r^{2}R-C_p^2{C}_r{v}_{C_r}\left(D_{c}T\right)-C_p{C}_r^2{v}_{C_r}\left(D_{c}T\right)}{C_p+C_r}\\ \times \left(\frac{1}{a_2\left[{\left(s+{\alpha }_2\right)}^2-{\omega }_2^2\right]}\right),\end{array}$ (30)

where ${\alpha }_2=\frac{b_2}{2a_2},{\omega }_2=\sqrt{-{\beta }_2} \mathrm{and} {\beta }_2=\frac{c_2}{a_2}-{\alpha }_2^2$ with coefficients: a₂ = L_rC_pC_r, b₂ = RC_pC_r e c₂ = C_r + C_p.

By adjusting the terms in order to find the Laplace transformations, I₂(s) becomes

$\begin{array}{r}{I}_2\left(s\right)=\frac{I_{\mathrm{in}}{C}_r}{C_p+C_r}\left(\frac{1}{s}\right)\\ +\frac{-I_{\mathrm{in}}{C}_p{C}_r^2{L}_r+C_p^2{C}_r{L}_r{i}_{L_r}\left(D_{c}T\right)+C_p{C}_r^2{L}_r{i}_{L_r}\left(D_{c}T\right)}{C_p+C_r}\\ \times \frac{1}{a_2}\left(\frac{s+{\alpha }_2}{\left[{\left(s+{\alpha }_2\right)}^2-{\omega }_2^2\right]}\right)\\ -\frac{-I_{\mathrm{in}}{C}_p{C}_r^2{L}_r+C_p^2{C}_r{L}_r{i}_{L_r}\left(D_{c}T\right)+C_p{C}_r^2{L}_r{i}_{L_r}\left(D_{c}T\right)}{C_p+C_r}\\ \times \frac{{\alpha }_2}{a_2{\omega }_2}\left(\frac{{\omega }_2}{\left[{\left(s+{\alpha }_2\right)}^2-{\omega }_2^2\right]}\right)\\ +\frac{-I_{\mathrm{in}}{C}_p{C}_r^{2}R-C_p^2{C}_r{v}_{C_r}\left(D_{c}T\right)-C_p{C}_r^2{v}_{C_r}\left(D_{c}T\right)}{C_p+C_r}\\ \times \frac{1}{a_2{\omega }_2}\left(\frac{{\omega }_2}{a_2\left[{\left(s+{\alpha }_2\right)}^2-{\omega }_2^2\right]}\right).\end{array}$ (31)

In this configuration, the Inverse Laplace Transform can be applied. The time-domain exact equation for the resonant current in stage II is obtained as:

$\begin{array}{r}{i}_{L_r}\left(t\right)=\frac{I_{\mathrm{in}}{C}_r}{C_p+C_r}\\ +\frac{-I_{\mathrm{in}}{C}_p{C}_r^2{L}_r+C_p^2{C}_r{L}_r{i}_{L_r}\left(D_{c}T\right)+C_p{C}_r^2{L}_r{i}_{L_r}\left(D_{c}T\right)}{C_p+C_r}\\ \times \frac{1}{2a_2}\left({\mathrm{e}}^{-\left({\alpha }_2-{\omega }_2\right)t}+{\mathrm{e}}^{-\left({\alpha }_2+{\omega }_2\right)t}\right)\\ -\frac{-I_{\mathrm{in}}{C}_p{C}_r^2{L}_r+C_p^2{C}_r{L}_r{i}_{L_r}\left(D_{c}T\right)+C_p{C}_r^2{L}_r{i}_{L_r}\left(D_{c}T\right)}{C_p+C_r}\\ \times \frac{{\alpha }_2}{2a_2{\omega }_2}\left({\mathrm{e}}^{-\left({\alpha }_2-{\omega }_2\right)t}-{\mathrm{e}}^{-\left({\alpha }_2+{\omega }_2\right)t}\right)\\ +\frac{-I_{\mathrm{in}}{C}_p{C}_r^{2}R-C_p^2{C}_r{v}_{C_r}\left(D_{c}T\right)-C_p{C}_r^2{v}_{C_r}\left(D_{c}T\right)}{C_p+C_r}\\ \times \frac{1}{2a_2{\omega }_2}\left({\mathrm{e}}^{-\left({\alpha }_2-{\omega }_2\right)t}-{\mathrm{e}}^{-\left({\alpha }_2+{\omega }_2\right)t}\right)\mathrm{A}.\end{array}$ (32)

In stage II, switch S is off, so, there is voltage in capacitor C_p. The frequency domain equation for its voltage is

$v_{C_p}\left(s\right)=\frac{I_1\left(s\right)-I_2\left(s\right)}{s{C}_p}+\frac{v_{C_p}\left(D_{c}T\right)}{s},$ (33)

then

$\begin{array}{r}{v}_{C_p}\left(s\right)=\frac{I_{\mathrm{in}}}{s^2{C}_p}-\left(\frac{1}{s{C}_p}\right)\left(\frac{1}{s}\right)\\ \times \left(\frac{s^2{L}_r{C}_p{C}_r{i}_{L_r}\left(D_{c}T\right)-s{C}_p{C}_r{v}_{C_r}\left(D_{c}T\right)+I_{\mathrm{in}}{C}_r}{s^2{L}_r{C}_p{C}_r+sR{C}_p{C}_r+C_r+C_p}\right).\end{array}$ (34)

The same procedure shown for the resonant inductor analysis should be applied in (34), which consists of: partial fraction decomposition, completing the square method and Inverse Laplace Transform. The terms α₂, ω₂ e a₂ are the same terms obtained in the inductor L_r equating for stage II.

The analysis for capacitor C_r is the same as shown for capacitor C_p, except for the influence of current I₁(s). Note that, the loop current I₁(s) is the same as the DC input current I_in. Therefore, the parallel capacitor voltage in stage II and the time-domain exact equation for v_Cr(t) in stage II are shown in (35) and (36), respectively.

The provided exact solution of the Class-E WPT converter is related to its resonant electrical variables: resonant current i_Lr(t), parallel capacitor voltage v_Cp(t) and resonant capacitor voltage v_Cr(t). The proposed methodology has the following features:

1. The exact solution of i_Lr(t) is not limited to the high load quality factor, i.e., the solution is not limited to the sinusoidal approximation;

2. The resonant inductor L_r and load R can be evaluated considering the equivalent input impedance of the WPT system. Therefore, allowing to investigate the converter behavior considering the inductive coupling;

3. By performing a parameter sweeping and considering the steady-state conditions, the power stage can be designed based on the exact time-domain equations.

$\begin{array}{r}{v}_{C_p}^{II }\left(t\right)=\frac{I_{in }}{C_p}t+\frac{I_{in }{C}_r^{2}R+C_p{C}_r{v}_{C_r}\left(D_{c}T\right)+C_r^2{v}_{C_r}\left(D_{c}T\right)}{{\left(C_p+C_r\right)}^2}\\ -\frac{1}{C_p}\frac{I_{in }{C}_r}{C_p+C_r}t\\ -\frac{C_p{L}_r\left[I_{in }{C}_r^{3}R+C_p{C}_r^2{v}_{C_r}\left(D_{c}T\right)+C_r^3{v}_{C_r}\left(D_{c}T\right)\right]}{{\left(C_p+C_r\right)}^2}\\ \times \frac{1}{2a_2}\left({\mathrm{e}}^{-\left({\alpha }_2-{\omega }_2\right)t}+{\mathrm{e}}^{-\left({\alpha }_2+{\omega }_2\right)t}\right)\\ +\frac{C_p{L}_r\left[I_{in }{C}_r^{3}R+C_p{C}_r^2{v}_{C_r}\left(D_{c}T\right)+C_r^3{v}_{C_r}\left(D_{c}T\right)\right]}{{\left(C_p+C_r\right)}^2}\\ \times \frac{{\alpha }_2}{2a_2{\omega }_2}\left({\mathrm{e}}^{-\left({\alpha }_2-{\omega }_2\right)t}-{\mathrm{e}}^{-\left({\alpha }_2+{\omega }_2\right)t}\right)+\frac{1}{C_p}\\ \times \left[\frac{I_{in }{C}_p^2{C}_r^2{L}_r+I_{in }{C}_p{C}_r^3{L}_r-C_p^3{C}_r{L}_r{i}_{L_r}\left(D_{c}T\right)}{{\left(C_p+C_r\right)}^2}\right.\\ -\frac{2C_p^2{C}_r^2{L}_r{i}_{L_r}\left(D_{c}T\right)+C_p{C}_r^3{L}_r{i}_{L_r}\left(D_{c}T\right)+I_{in }{C}_p^2{C}_r^3{R}^2}{{\left(C_p+C_r\right)}^2}\\ \left.+\frac{-C_p^3{C}_r^{2}R{v}_{C_r}\left(D_{c}T\right)-C_p^2{C}_r^{3}R{v}_{C_r}\left(D_{c}T\right)}{{\left(C_p+C_r\right)}^2}\right]\mathrm{V}.\\ \times \frac{1}{a_2{\omega }_2}\left({\mathrm{e}}^{-\left({\alpha }_2-{\omega }_2\right)t}-{\mathrm{e}}^{-\left({\alpha }_2+{\omega }_2\right)t}\right)\mathrm{V}\end{array}$ (35)

FIGURE 4
Coupled inductor model.

$\begin{array}{r}{v}_{C_r}^{\mathrm{II}}\left(t\right)=-\frac{C_p}{C_r}\frac{I_{\mathrm{in}}{C}_r^{2}R+C_p{C}_r{v}_{C_r}\left(D_{c}T\right)+C_r^2{v}_{C_r}\left(D_{c}T\right)}{{\left(C_p+C_r\right)}^2}\\ +\frac{I_{\mathrm{in}}{C}_r}{C_p+C_r}t\\ +\frac{C_p^2{L}_r\left[I_{\mathrm{in}}{C}_r^{3}R+C_p{C}_r^2{v}_{C_r}\left(D_{c}T\right)+C_r^3{v}_{C_r}\left(D_{c}T\right)\right]}{{\left(C_p+C_r\right)}^2}\\ \times \frac{1}{2a_2}\left({\mathrm{e}}^{-\left({\alpha }_2-{\omega }_2\right)t}+{\mathrm{e}}^{-\left({\alpha }_2+{\omega }_2\right)t}\right)\\ -\frac{C_p^2{L}_r\left[I_{\mathrm{in}}{C}_r^{3}R+C_p{C}_r^2{v}_{C_r}\left(D_{c}T\right)+C_r^3{v}_{C_r}\left(D_{c}T\right)\right]}{{\left(C_p+C_r\right)}^2}\\ \times \frac{{\alpha }_2}{2a_2{\omega }_2}\left({\mathrm{e}}^{-\left({\alpha }_2-{\omega }_2\right)t}-{\mathrm{e}}^{-\left({\alpha }_2+{\omega }_2\right)t}\right)\\ -\left[\frac{I_{\mathrm{in}}{C}_p^2{C}_r^2{L}_r+I_{\mathrm{in}}{C}_p{C}_r^3{L}_r-C_p^3{C}_r{L}_r{i}_{L_r}\left(D_{c}T\right)}{{\left(C_p+C_r\right)}^2}\right.\\ -\frac{2C_p^2{C}_r^2{L}_r{i}_{L_r}\left(D_{c}T\right)+C_p{C}_r^3{L}_r{i}_{L_r}\left(D_{c}T\right)+I_{\mathrm{in}}{C}_p^2{C}_r^3{R}^2}{{\left(C_p+C_r\right)}^2}\\ \left.+\frac{-C_p^3{C}_r^{2}R{v}_{C_r}\left(D_{c}T\right)-C_p^2{C}_r^{3}R{v}_{C_r}\left(D_{c}T\right)}{{\left(C_p+C_r\right)}^2}\right]\\ \times \frac{1}{a_2{\omega }_2}\left({\mathrm{e}}^{-\left({\alpha }_2-{\omega }_2\right)t}-{\mathrm{e}}^{-\left({\alpha }_2+{\omega }_2\right)t}\right)\mathrm{V}.\end{array}$ (36)

IV. INDUCTIVE COUPLING ANALYSIS

In order to find an equivalent circuit for the inductive coupling aiming to use it as load for the Class-E inverter, Fig. 4 shows the coupled inductor model to be analyzed. R₁ and R₂ represent the coil losses. In addition, the input voltage source v(ωt) is represented in the rectangular phasor form with real part x₁ and imaginary part y₁, i.e., v(ωt) = x₁ + jy₁.

The following equation are obtained by applying KVL in the first loop:

$-\left(x_1+j{y}_1\right)+{\mathbf{I}}_1\left(R_1+j\omega L_1\right)-{\mathbf{I}}_2\left(j\omega M\right)=0,$ (37)

which leads to

${\mathbf{I}}_1\left(R_1+j\omega L_1\right)-{\mathbf{I}}_2\left(j\omega M\right)=\left(x_1+j{y}_1\right)$ (38)

The same procedure is used in the second loop,

$-j\omega M{\mathbf{I}}_1+{\mathbf{I}}_2\left(R_2+j\omega L_2\right)+{\mathbf{I}}_2{R}_L=0$ (39)

then

${\mathbf{I}}_1(-j\omega M)+{\mathbf{I}}_2\left(R_2+R_L+j\omega L_2\right)=0$ (40)

Equations (38) and (40) provides a linear system of equations, which returns the symbolical expressions for I₁ and I₂. Loop current I₁ is found as

${\mathbf{I}}_1=\frac{\left(-x_1-j{y}_1\right)\left(R_2+R_L+j\omega L_2\right)}{-M^2{\omega }^2-\left(R_1+j\omega L_1\right)\left(R_2+R_L+j\omega L_2\right)}.$ (41)

The input impedance of the inductive coupling can be calculated by

$Z_{in }=\frac{x_1+y_{1}j}{{\mathbf{I}}_1},$ (42)

which can be used as an equivalent load for the inverter stage.

Finally, the relation between the mutual inductance M, considering two coils with diameters D₁ and D₂, and the transmitter-receiver distance d, can be mathematically expressed by [25]

$M={\mu }_o{N}_1{N}_2\sqrt{\frac{D_1{D}_2}{4}}\left[\left(\frac{2}{k}-k\right)K_1\left(k\right)-\frac{2}{k}{E}_1\left(k\right)\right],$ (43)

being µ₀ (H/m) the material permeability, N_1,2 the winding number and K₁(k) e E₁(k) are the complete elliptic integrals of the first kind and second kind, respectively.

In addition, k is described as

$k=\sqrt{\frac{D_1{D}_2}{d^2+{\left(\frac{D_1}{2}+\frac{D_1}{2}\right)}^2}}.$ (44)

V. DESIGN METHODOLOGY

The design of the Class-E WPT converter is performed in two parts: 1) The input impedance of the WPT system should be calculated based on the geometry and physical properties of the transmitter and receiver coils and electrical design specifications; 2) The Class-E resonant inverter component values are calculated based on a linear system of equations developed from the exact time-domain equations. The transmitter and receiver coil parameters are described in Table 1. In addition, µ_o = 4π10⁻⁷ H/m.

TABLE 1
Inductive Coil Parameters.

Parameter	Value
Primary side winding number, N1	10
Secondary side winding number, N2	5
Coil diameter D1,2	50mm
Primary side losses, R1	180mΩ
Secondary side losses, R2	145mΩ
Primary side inductance, L1	7.7µH
Secondary side inductance, L2	3.4µH

The mutual inductance can be calculated by Equation (43). However, it requires a numerical solution. Table 2 and 3 show M as function of coil distance d for distances less than 10mm and greater than 10mm, respectively.

TABLE 2
Numerical Solution: M as function of d (d < 10mm).

M (µH)	4.65	4.06	3.44	3.01	2.67	2.40	2.18	1.99	1.82
d(mm)	1	2	3	4	5	6	7	8	9

TABLE 3
Numerical Solution: M as function of d (d > 10mm).

M (µH)	1.68	0.83	0.46	0.27	0.17	0.11	0.08	0.05	0.04
d(mm)	10	20	30	40	50	60	70	80	90

In order to design the system, the highest power transfer situation is considered, which means M = 4.65µH. In addition, the electrical design specifications are considered as: operating frequency f_s = 500kHz output power P_L = 5W, inverter output RMS voltage V_LRMS = 14V. Hence, I₁ can be calculated by Equation (41), which results on I₁ = 0.622 − 2.705j A, then, by defining the transmitter coil input voltage in rectangular coordinate, for instance x₁ = 65 and y₁ = 0.0068j, the input impedance is found as Z_in = 5.25 + 23j Ω.

By calculating the input impedance of inductive coils, it can be represented by an equivalent load for the Class-E inverter. The real part represents a resistance R_in = 5.25Ω. On the other hand, the positive imaginary part denotes an inductive reactance X_in. The equivalent inductor is calculated as

$L_{\mathrm{in}}=\frac{{\mathbf{X}}_{\mathrm{in}}j}{\omega j}=7.271\mu H.$ (45)

Considering the equivalent load R_in-L_in for the ClassE inverter, it is possible to design the power stage components by means of the exact time-domain equations. By equating the solutions for stage II replacing time t for T, i.e., the end of one cycle, to the symbolical initial conditions, a linear system of equations can be built, which represents the steady-state operation. In addition, the soft-switching conditions can be incorporated. Therefore:

$\left(\begin{array}{c} II \\ i_{L_r}\left(T\right)\\ II \\ v_{C_r}\left(T\right)\\ II \\ v_{C_p}\left(T\right)\\ II \\ i_{L_r}\left(T\right)\\ I \\ i_{L_r}\left(D_{c}T\right)\\ I\\ v_{C_r}\left(D_{c}T\right)\end{array}\right)-\left(\begin{array}{c}{i}_{L_r}\left(0\right)\\ v_{C_r}\left(0\right)\\ 0\\ I_{in }\\ i_{L_r}\left(D_{c}T\right)\\ v_{C_r}\left(D_{c}T\right)\end{array}\right)=\left(\begin{array}{l}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right).$ (46)

The system in (46) considers the the steady-state operation represented by the two first lines. In the end of a cycle, switch S turns-on with ZVS and ZDVS, indicated by the third and fourth lines, respectively. In addition, the exact solutions for stage I are equated to the initial conditions of stage II. Therefore, there are 6 equations but 13 unknowns: the reactive components, L_r, C_r and C_p; the initial conditions for stage I, i_Lr(0), v_Cr(0) and v_Cp(0); the initial conditions for stage II, i_Lr(D_cT), v_Cr(D_cT) and v_Cp(D_cT); duty cycle D_c, period T, input current I_in and load R. Due to the soft-switching operating, v_Cp(0) = v_Cp(D_cT) = 0, which reduces two unknowns. In addition, T = 1/f_s, I_in and R = R_in are defined because they can be used as design specification. L_r is also defined based on L_in. Therefore, by sweeping duty cycle D_c, the values for capacitors C_p and C_r and the initial conditions i_Lr(0), v_Cr(0), i_Lr(D_cT), v_Cr(D_cT) are found. The system of linear equations in (46) was solved in a computational implementation by considering the design specifications in Table 4. The system was solved by different values of D_c and the results are shown in Table 5.

TABLE 4
Design Specifications.

Parameter	Value
Operating frequency, fs	500kHz
Equivalent inductance, Leq	8.4µH
Resistive load, R	5.25Ω
Duty cycle, Dc	0.3 − 0.7
Input DC current, Iin	1.19A

TABLE 5
Design Results

Duty cycle	Cp	Cr	iLr(0)	vCr(0)	iLr(DcT)	vCr(DcT)
0.3	12.7nF	42.8nF	1.0A	102.06V	−4.78A	70.42V
0.4	14.8nF	21.8nF	1.0A	61.23V	−2.70A	8.92V
0.5	12.7nF	16.2nF	1.0A	43.76V	−1.33A	−19.8V
0.6	8.41nF	14.5nF	1.0A	30.96V	−0.41A	−25.8V
0.7	4.12nF	13.4nF	1.0A	22.77V	0.20A	−23.3V

The exact solution can be used in the semiconductor sizing. By evaluating the maximum value of the parallel capacitor voltage from $v_{C_p}^{\mathrm{I}}\left(t\right) \mathrm{and} v_{C_p}^{\mathrm{I}}\left(t\right)$, and the average value from $I_{\mathrm{in}}-i_{L_r}^{\mathrm{I}}\left(t\right)$, the maximum switch voltage and the average switch current can be obtained for any operating point.

FIGURE 5
Switch S maximum values as function of D_c. (a) Voltage. (b) Current.

The higher the duty cycle, higher the switch stress. On the other hand, a lower duty cycle provides higher output voltage and less distortion in the resonant current. Therefore, a trade-off must be considered while designing the converter. In this sense, the operating point was selected as D_c = 0.5. A circuit simulation was performing considering the design result for D_c = 0.5 from Table 5. Furthermore, the choke inductor was calculated as $L_c=\frac{2R}{f_s}$

The switch stress and output voltage can be also evaluated regarding the distance between the coils. In this sense, by relating the coupled inductor analysis to the exact solution, the charts in Fig. 6 are obtained. Farther the distance d, smaller is the transferred RMS voltage and maximum switch voltage, as depicted in Fig. 6(a) and in Fig. 6(b).

FIGURE 6
Evaluating over distance d(mm). (a) Output RMS voltage (V). (b) Maximum switch voltage (V). (c) Maximum switch current (A). (d) Equivalent WPT input resistance (Ω).

The simulation results are shown in Fig. 7 for gate signal v_G, switch voltage v_S, parallel capacitor current i_Cp, resonant capacitor voltage v_Cr and resonant current i_Lr. Besides demonstrating the correct behavior of the converter, the simulation results are used to select commercial capacitors. For C_p, it is taken on consideration the switch voltage maximum value and the capacitor RMS current, which are, 56.6V and 1.18A, respectively. As for C_r, it is considered the capacitor maximum voltage and the resonant RMS current, 73.3V and 2.13A.

FIGURE 7
Simulation results.

The 3D mapping that relates the output voltage (RMS), normalized frequency and duty cycle is depicted in Fig. 8. One can observe the transferred output voltage over different operating points. The normalized frequency was obtained by sweeping the equivalent input inductance of the WPT system and then using its values to calculate f_s/f based on a fixed value for C_r.

FIGURE 8
3D mapping of the relation among v_o, D_c and f_s/s.

VI. EXPERIMENTAL RESULTS

Aiming to experimentally validate the system, the Class-E resonant inverter was implemented in a printed circuit board and a WPT system was built based on transmitter

and receiver coils. The hardware components are described in Table 6. Fig. 9 shows the experimental setup, which includes the Class-E inverter and WPT system. The system was tested considering different scenarios regarding distance and output load. The measured output voltage, i.e., the received voltage in the secondary side, is shown in Fig. 10 by considering distances between transmitter and receiver of 1mm, 2.5mm, 5mm and 10mm. It can be see that, the greater the distance, lower is the output voltage. Considering d = 1mm, the RMS output voltage is 9.83V. Furthermore, aiming to validate the ZVS operation, Fig. 11 presents the measured gate signal and switch S voltage for different distances. One can see that the switch S voltage reaches zero before the gate signal changes from low to high. The ZVS is maintained from tested distances between 1mm and 10mm.

TABLE 6
Hardware Components

Component/Parameter	Value	Part number/Manufacturer
Choke inductor, Lc	180µH	Handcrafted
Parallel capacitor, Cp	12nF	KEMET
Resonant inductor, Lr	1.2µH	Handcrafted
Resonant capacitor, Cr	15nF	KEMET
Switch, S	-	IPP530N15N3G
Switch gate-driver	-	IRS2011PBF
PCB size	8mm × 6mm	-
Transmitter coil, L1	7.7µH	Handcrafted
Receiver coil, L2	3.4µH	Handcrafted
Coil diameter, D1,2	50mm	-
Nominal input voltage, Vin	15.5V	-
Operating frequency, fs	500kHz	-
Duty cycle, Dc	0.5	-
Load, RL	4.7 − 24Ω	-

FIGURE 9
Experimental setup.

FIGURE 10
Output voltage v_o. (a) d < 1mm. (b) d = 2.5mm. (c) d = 5mm. (d) d = 10mm.

FIGURE 11
Gate signal v_G and switch voltage v_S. (a) d < 1mm. (b) d = 2.5mm. (c) d = 5mm. (d) d = 7.5mm.

The Class-E WPT converter was tested also by varying the output power. In this regard, Fig. 12 depicts the efficiency as function of the output power. This evaluation was performed by considering the minimum distance between coils, which means, d = 1mm. The highest achieved efficiency was 74%.

FIGURE 12
Efficiency as function of the output power P_o.

CONCLUSION

In this work, the exact solution of the Class-E WPT resonant inverter has been introduced. The inductive coupling of the WPT system was represented as a RL equivalent load for the Class-E inverter, which allows to analyze the converter in its standard configuration. The exact time-domain solution was obtained for the resonant inductor current, resonant capacitor voltage and parallel capacitor voltage. By using the proposed approach, it is possible to design the hardware components without restraining the quality factor. An experimental setup was built to verify the theory. A 500kHz ZVS Class-E inverter was designed to drive a WPT system with 50mm coil diameter. The conclusions are drawn based on the comparison to related works shown in Table 7.

TABLE 7
Comparison to Related Works

Work	Topology	Frequency	Rating Power	Efficiency @distance	Nº of switches	Leq	Output waveform
[17]	Class-E−1	1000kHz	14.2W	84.1%@8mm	1	13.96µH	Sinusoid
[20]	Class-E	1000kHz	9.87W	73%@100mm	1	23.1µH	Sinusoid
[26]	Half-Bridge Class-D	98kHz	48.16W	85.57%@100mm	2	52.7µH	Sinusoid
[27]	Full-Bridge Class-D	13560kHz	40W	91%@0mm	4	1.3µH	Sinusoid
[22]	Class-E	500kHz	10W	61%@30mm	1	119µH	Sinusoid
Present	Class-E	500kHz	9.7W	74%@1mm	1	8.9µH	Exact

Class-E topologies have only one switch, in contrast to half-bridge Class-D and full-bridge Class-D, which have 2 and 4 switches, respectively. In the present work, the highest achieved efficiency was 74% at 1mm coil distance.

Reference [26] shows a Class-D topology that achieves 85.57% efficiency at 100mm distance. However, it operates at lower frequency, which leads to bigger hardware size as emphasized by the equivalent inductance of52.7µH. The equivalent inductance, i.e., the resonant inductor added to the WPT system equivalent inductor, is reduced in the proposed work. This can be achieved by allowing distortion in the output waveforms, which is predicted by the proposed exact solution. It was shown that the converter maintains ZVS operation over distance variation from tested 1mm to 7.5mm. The proposed work contributes to the design of resonant WPT converters by providing the exact solution of the main electrical variables, ultimately, unfolding the theoretical results into hardware design.

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The data used in this research is available in the body of the document.

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Edited by

• Associate Editor
  Filipe P. Scalcon https://orcid.org/0000-0002-8976-2188

• Editor-in-Chief
  Heverton A. Pereira https://orcid.org/0000-0003-0710-7815

Publication Dates

• Publication in this collection
  21 Nov 2025
• Date of issue
  2025

History

• Received
  10 Mar 2025
• Accepted
  01 July 2025
• Published
  30 July 2025