Analysis and design of a load-independent class-E LCL resonant converter

ABSTRACT

This work brings in a load-independent Class-E LCL resonant converter. The intertwined soft-switching and load-independent functions in resonant topologies require high mathematical effort and leads to trade-off designing characteristics related to the operating frequency, output power and load range. Therefore, this paper shows the exact analysis for the Class-E LCL converter by axiomatically dealing with the second, third and fourth-order polynomials. In addition, a design method is proposed based on the implicit equations while considering the soft-switching and load-independence features. The measured built prototype operates at 1.2 MHz, which outperforms most of the state-of-the-art compared converters by ×1.2, while maintaining comparable output power and efficiency (92 %).

KEYWORDS
class-E amplifiers; load-independence; resonant converters

I. INTRODUCTION

Peerless features, like as soft-switching mechanisms and high-frequency/high-efficiency operation, places resonant converters as cutting-edge technologies in alternate current power conversions. Especially in CC/CA inverters, Class-E converters achieve zero-voltage switching (ZVS) and zero-derivative-voltage switching (ZDVS) due to the charge and discharge of the parallel switch capacitance, accomplishing low switching losses and highefficiency [¹]. Modern power electronics enjoy resonant converters in applications like as, wireless power transfer [²]–[⁵], induction heating [⁶], [⁷] and medical [⁸], [⁹].

Topologies in resonant power conversion varies in terms of number of switches/reactive elements, switch position, soft-switching mechanism and load characteristic. The most well-known examples are, Class-D [¹⁰], [¹¹], Class-E [¹²], [¹³] and Class-F converters [¹⁴]. In almost all cases, the design complexity arises from the high number of passive elements and the soft-switching conditions, leading to high mathematical effort to find the analytical solutions [¹⁵]–[¹⁷]. In addition, Class-E converters are sensitive to load variation. However, practical applications require stable output voltage or current [¹⁸], [¹⁹].

Besides the closed-loop solution, with dynamic dead-time [²⁰], hysteresis control [²¹], and frequency control [²²], [²³], recent research demonstrates the feasibility of load-independence Class-E conversion without feedback controllers. In [²⁴], a finite inductor is replaced in the standard Class-E topology to achieve load-independence with 92.7 % efficiency at 1 MHz. By creating a resonant structure with a switch capacitance and a shunt inductance, [²⁵] has provided a Class-E⁻¹ circuit operating with ZVS constant current mode. A similar topology is shown in [²⁶], however, it provides a zero-current switching (ZCS) constant voltage operation. Besides the soft-switching and the type of constant output, it is interesting to evaluate the load range. In [²⁷], a parallel Class-E resonant inverter with floating-ground switch performs improved load operating range (×40) with conversion efficiency from 37.7 − 89.3 %.

Ubiquitous, there is a trade-off among load range, input current ripple, component size, operating frequency and output power, which leads to increased analysis and design complexity [²⁸]. Furthermore, in order to achieve load-independent operation and keep the soft-switching conditions, it is necessary to find two exact analytical solutions [²⁷], [²⁹], [³⁰]: 1) A function dependent on the normalized resonant frequency and duty cycle times the input voltage; 2) A function dependent on the normalized resonant frequency, duty cycle, and output voltage/current phase times the output amplitude current times the input impedance. By ensuring that both aforementioned functions are zero in the switch transition instant, load-independence is achieved with ZVS/ZCS operation. Notwithstanding, the aforementioned functions require unique equating regarding topological configuration.

In this sense, this paper provides the exact analysis and design methodology of a Class-E LCL converter. The proposed topology has a LCL-network between the main resonant tank and the output load. The topology is interesting in inductive wireless power transfer applications because the transmitter coil inductance can be used as the output inductor of the LCL tank, reducing hardware size while ensuring the Class-E topology advantages. The analysis is performed by equating the electrical variables based on sine, cosine and exponential terms, so, it is shown how to deal with second, third and fourth-order polynomials generated in the complex frequency domain equating. By providing the exact solution, it is possible to find a governing equation for the switch voltage that reveals the soft-switching-load-independent intertwined functions. In addition, a design methodology is proposed based on implicit equations that are numerically solved in order to find the reactive component values. A 1.2 MHz prototype is experimentally validated to confirm the theoretical approach and evaluate the load-independence range.

II. MATHEMATICAL APPROACH

The proposed Class-E LCL resonant converter is composed of an input voltage V_in, input inductor L_c, switch S, inverter capacitor C_p, resonant capacitor C_o, resonant inductor L_o, LCL-network L_r1-C_r1-L_r2 and load R as depicted in Fig. 1. In addition, by considering the LCL-network frequency equals to the switching frequency, the Class-E converter drives an equivalent load Z_o.

FIGURE 1
Class-E LCL resonant converter.

Axiom 1.The inductors in the LCL-network are equal: L_r1 = L_r2.

Axiom 2.The LCL-network angular frequency is equal to the angular switching frequency: ω_LCL = ω.

Lemma 1.The LCL-network can be represented by an equivalent load Z_ofor the Class-E inverter described by Z_o = L_r1/RC_r1.

Proof:

The LCL-network can be described in the s-domain frequency as:

(1) $Z_o=(R+s{L}_{\mathrm{r}2})//\left(\frac{1}{s{C}_{\mathrm{r}1}}\right)+s{L}_{\mathrm{r}1},$

then

(2) $Z_o=\frac{s^3{L}_{\mathrm{r}1}{L}_{\mathrm{r}1}{C}_{\mathrm{r}1}+s^{2}R{L}_{\mathrm{r}1}{C}_{\mathrm{r}1}+s{L}_{\mathrm{r}1}+s{L}_2+R}{s^2{L}_2{C}_{\mathrm{r}1}+sR{C}_{\mathrm{r}1}+1},$

by taking into account Axioms 1 and 2:

(3) $Z_o=\frac{-s{L}_{\mathrm{r}1}-R+s{L}_{\mathrm{r}1}+s{L}_{\mathrm{r}1}+R}{sR{C}_{\mathrm{r}1}+1-1}=\frac{L_{\mathrm{r}1}}{R{C}_{\mathrm{r}1}}.$

A. Exact Time-Domain Solutions

In order to analyze the converter, two operating stages are considered: switch S on/off ruled by duty cycle D_c. In addition, the output current is described as a pure sinusoid waveform.

Axiom 3. The load quality factor is sufficiently high, therefore, the output current is

(4) $i_o\left(t\right)=I_o\sin (\omega t+\varphi )$

in which, I_o is the amplitude and ϕ the phase. In this regard, the equivalent circuits to be analyzed are shown in Fig. 2(a) and in Fig. 2(b). I₁ and I₂ are the loop currents, which describe the input inductor current and the output current, respectively.

FIGURE 2
Class-E LCL topology. (a) Switch S on. (b) Switch S off.

When switch S is on, the governing equation in first loop is given as

(5) $-V_{\mathrm{in}}+v_{L_c}\left(t\right)=0$

which is described in the s-domain by

(6) $-\frac{V_{\mathrm{in}}}{s}+s{L}_c{I}_1\left(s\right)-L_c{i}_1\left(0\right)=0$

therefore,

(7) $I_1\left(s\right)=\frac{V_{\mathrm{in}}}{s^2{L}_c}+\frac{i_1\left(0\right)}{s}$

which can be converted to the time-domain by using the Laplace Inverse Transform by means of the known Laplace pairs:

(8) $I_1\left(s\right)=\frac{V_{\mathrm{in}}}{s^2{L}_c}+\frac{i_1\left(0\right)}{s}$

Due to the parallel connection to the switch S, the capacitor C_p voltage is:

(9) $\stackrel{\mathrm{I}}{v_{C_p}}\left(t\right)=0\mathrm{V}.$

When switch S is off, the governing equation related to the V_in − L_c − C_p loop is

(10) $-V_{\mathrm{in}}+v_{L_c}\left(t\right)+v_{C_p}\left(t\right)=0$

which is rewritten in the complex frequency domain by

(11) $\frac{-V_{\mathrm{in}}}{s}+s{L}_c{I}_1\left(s\right)-L_c{i}_1\left(D_{c}2\pi \right)+\frac{I_1\left(s\right)-I_2\left(s\right)}{s{C}_p}+\frac{v_{C_p}\left(D_{c}2\pi \right)}{s}=0,$

being i₁(D_c2π) and v_Cp(D_c2π) the initial conditions and I₂ = i_o. Further rearranging leads to:

(12) $\begin{array}{rl}{I}_1\left(s\right)& =V_{\mathrm{in}}\left(\frac{C_p}{s^2{L}_c{C}_p+1}\right)+L_c{i}_1\left(Dc2\pi \right)\left(\frac{s{C}_p}{s^2{L}_c{C}_p+1}\right)\\ & -v_{C_p}\left(D_{c}2\pi \right)\left(\frac{C_p}{s^2{L}_c{C}_p+1}\right)+I_o\left(\frac{s\sin \left(\varphi \right)+\omega \cos \left(\varphi \right)}{s^2+{\omega }^2}\right)\\ & \times \left(\frac{1}{s^2{L}_c{C}_p+1}\right)\end{array}$

Mathematical operations should be conducted to adequate the equation into the well-known Laplace pairs. Second-order polynomials are modified by the completing the square method. On the other hand, fourth-order polynomials need the expansion in partial fractions.

Axiom 3. Second-order polynomials can be rewritten as:

(13) $\frac{1}{a_1{s}^2+b_{1}s+c_1}=\frac{1}{a_1[{(s+{\alpha }_1)}^2+{\beta }_1]}$

in which, α₁ = b₁/2a₁ and β₁ = c₁/a₁ − α₁².

Axiom 4. Third and fourth-order polynomials are rewritten as:

(14) $\left(\frac{s\sin \left(\varphi \right)+\omega \cos \left(\varphi \right)}{s^2+{\omega }^2}\right)\left(\frac{1}{a_1{s}^2+c_1}\right)=\frac{A_{1}s+B_1}{s^2+{\omega }_2}+\frac{C_{1}s+D_1}{a_{1}s+c_1},$

with constants: $A_1=-\frac{\sin {\displaystyle }\left(\varphi \right)}{-c_1+a_1{\omega }^2},B_1=-\frac{\omega \cos \left(\varphi \right)}{-c_1+a_1{\omega }^2},C_1=\frac{a_1\sin \left(phi\right)}{-1+a_1{\omega }_2}and{D}_1=\frac{a_1\omega \cos \left(\varphi \right)}{-1+a_1{\omega }^2}.$

Therefore, Equation (12) is expressed as

(15) $\begin{array}{rl}{I}_1\left(s\right)& =\frac{V_{\mathrm{in}}{C}_p}{a_1\sqrt{{\beta }_1}}\left[\frac{\sqrt{{\beta }_1}}{{(s+{\alpha }_1)}^2+{\beta }_1}\right]+\frac{L_c{C}_p{i}_1\left(D_{c}2\pi \right)}{a_1}\\ & \times \left[\frac{s+{\alpha }_1}{{(s+{\alpha }_1)}^2+{\beta }_1}\right]-\frac{L_c{C}_p{i}_1\left(D_{c}2\pi \right){\alpha }_1}{a_1\sqrt{{\beta }_1}}\\ & \times \left[\frac{\sqrt{{\beta }_1}}{{(s+{\alpha }_1)}^2+{\beta }_1}\right]-\frac{C_p{v}_{C_p}\left(D_{c}2\pi \right)}{a_1\sqrt{{\beta }_1}}\\ & \times \left[\frac{\sqrt{{\beta }_1}}{{(s+{\alpha }_1)}^2+{\beta }_1}\right]+I_o[A_1\left(\frac{s}{s^2+{\omega }^2}\right)\\ & +\frac{B_1}{\omega }\left(\frac{\omega }{s^2+{\omega }^2}\right)+\frac{C_1}{L_c{C}_p}\left(\frac{s}{s^2+\frac{1}{L_c{C}_p}}\right)\\ & +\frac{D_1}{L_c{C}_p}\left(\frac{1}{s^2+\frac{1}{L_c{C}_p}}\right)],\end{array}$

with a₁ = L_cC_p, b₁ = 0 and c₁ = 1. By applying the inverse Laplace Transform, the time-domain solution is obtained as:

(16) $\begin{array}{rl}{i}_1\left(t\right)& =\frac{V_{\mathrm{in}}{C}_p}{a_1\sqrt{{\beta }_1}}[e^{-{\alpha }_{1}t}\sin \left(\sqrt{{\beta }_1}t\right)]+\frac{L_c{C}_p{i}_1\left(D_{c}2\pi \right)}{a_1}\\ & \times [e^{-{\alpha }_{1}t}\cos \left(\sqrt{{\beta }_1}t\right)]-\frac{L_c{C}_p{i}_1\left(D_{c}2\pi \right){\alpha }_1}{a_1\sqrt{{\beta }_1}}\\ & \times [e^{-{\alpha }_{1}t}\sin \left(\sqrt{{\beta }_1}t\right)]-\frac{C_p{v}_{C_p}\left(D_{c}2\pi \right)}{a_1\sqrt{{\beta }_1}}\\ & \times [e^{-{\alpha }_{1}t}\sin \left(\sqrt{{\beta }_1}t\right)]+I_o[A_1\cos (\omega t)\\ & +\frac{B_1}{\omega }\sin \left(\omega t\right)+\frac{C_1}{L_c{C}_p}\cos \left(\sqrt{\frac{1}{L_c{C}_p}}t\right)\\ & +\frac{D_1}{L_c{C}_p}\sin \left(\sqrt{\frac{1}{L_c{C}_p}}t\right)]\mathrm{A}.\end{array}$

Equating for v_Cp(t) in the complex frequency domain leads to:

(17) $V_{C_p}\left(s\right)=\frac{1}{s{C}_p}{I}_1\left(s\right)-\frac{1}{s{C}_p}\left(s\right)I_2\left(s\right)+\frac{1}{s}{v}_{C_p}\left(D_{c}T\right),$

which is re-written considering (12) as

(18) $\begin{array}{rl}{V}_{C_p}\left(s\right)& =\frac{V_{\mathrm{in}}}{s}\left(\frac{1}{s^2{L}_c{C}_p+1}\right)+L_c{i}_1\left(Dc2\pi \right)\left(\frac{1}{s^2{L}_c{C}_p+1}\right)\\ & -\frac{v_{C_p}\left(D_{c}2\pi \right)}{s}\left(\frac{1}{s^2{L}_c{C}_p+1}\right)+\frac{I_o}{C_p}\frac{1}{s}\\ & \times \left(\frac{s\sin \left(\varphi \right)+\omega \cos \left(\varphi \right)}{s^2+{\omega }^2}\right)\left(\frac{1}{s^2{L}_c{C}_p+1}\right)\\ & -\frac{I_o}{C_p}\frac{1}{s}\left(\frac{s\sin \left(\varphi \right)+\omega \cos \left(\varphi \right)}{s^2+{\omega }^2}\right)+\frac{1}{s}{v}_{C_p}\left(D_{c}T\right).\end{array}$

By taking advantage of Axioms 4 and 5, the following is achieved:

(19) $\begin{array}{rl}{V}_{C_p}\left(s\right)& =\frac{E_1}{L_c{C}_p}\left(\frac{s}{s^2+\frac{1}{L_c{C}_p}}\right)+\frac{F_1\sqrt{L_c{C}_p}}{L_c{C}_p}\left(\frac{\sqrt{1/L_c{C}_p}}{s^2+\frac{1}{L_c{C}_p}}\right)\\ & +\frac{G_1}{s}+\frac{i_1\left(D_{c}T\right)\sqrt{C_p}}{L_c{C}_p}\left(\frac{\sqrt{1/L_c{C}_p}}{s^2+\frac{1}{L_c{C}_p}}\right)-\frac{L_1}{L_c{C}_p}\\ & \times \left(\frac{s}{s^2+\frac{1}{L_c{C}_p}}\right)-\frac{M_1\sqrt{L_c{C}_p}}{L_c{C}_p}\left(\frac{\sqrt{1/L_c{C}_p}}{s^2+\frac{1}{L_c{C}_p}}\right)\\ & -\frac{N_1}{s}+O_1\left(\frac{s}{s^2+{\omega }^2}\right)+\frac{P_1}{\omega }\left(\frac{\omega }{s^2+{\omega }^2}\right)\\ & \frac{Q_1}{L_c{C}_p}\left(\frac{s}{s^2+\frac{1}{L_c{C}_p}}\right)+\frac{R_1\sqrt{L_c{C}_p}}{L_c{C}_p}\left(\frac{\sqrt{1/L_c{C}_p}}{s^2+\frac{1}{L_c{C}_p}}\right)\\ & +\frac{S_1}{s}-T_1\left(\frac{s}{s^2+{\omega }^2}\right)-\frac{U_1}{\omega }\left(\frac{\omega }{s^2+{\omega }^2}\right)\\ & -\frac{V_1}{s}+\frac{v_{C_p}\left(D_{c}T\right)}{s},\end{array}$

with constants: $E_1=-C_p{L}_c{V}_{\mathrm{in}},F_1=0,G_1=V_{\mathrm{in}},H_1=-C_p{L}_c^2{i}_1\left(D_{c}2\pi \right),J_1=0,K_1=L_c{i}_1\left(D_{c}2\pi \right),L_c=-C_p{L}_c{v}_{C_p}\left(D_{c}2\pi \right),M_1=0,N_1=v_{C_p}\left(D_{c}2\pi \right),O_1=\frac{I_o\cos \left(\varphi \right)}{C_p\omega \left(-1+C_p{L}_c{\omega }^2\right)},P_1=-\frac{I_o\sin \left(\varphi \right)}{C_p\left(-1+C_p{L}_c{\omega }^2\right)},Q_1=-\frac{C_p{I}_o{L}_c^2\omega \cos \left(\varphi \right)}{-1+C_p{L}_c{\omega }^2},R_1=\frac{I_o{L}_c\sin \left(\varphi \right)}{-1+C_p{L}_c{\omega }^2},S_1=\frac{I_o\cos \left(\varphi \right)}{C_p\omega },T_1=-\frac{I_o\cos \left(\varphi \right)}{C_p\omega },U_1=\frac{I_o\sin \left(\varphi \right)}{C_p}\mathrm{and}{V}_1=\frac{I_o\cos \left(\varphi \right)}{C_p\omega }.$. Therefore, the time-domain solution is written by

(20) $\begin{array}{r}{v}_{C_p}^{\mathrm{II}}\left(t\right)=\frac{E_1}{L_c{C}_p}\cos \left(\sqrt{\frac{1}{L_c{C}_p}}t\right)+\frac{F_1\sqrt{L_c{C}_p}}{L_c{C}_p}\sin \left(\sqrt{\frac{1}{L_c{C}_p}}t\right)\\ +G_1+\frac{i_1\left(D_{c}T\right)\sqrt{L_c{C}_p}}{C_p}\sin \left(\sqrt{\frac{1}{L_c{C}_p}}t\right)-\frac{L_1}{L_c{C}_p}\\ \times \cos \left(\sqrt{\frac{1}{L_c{C}_p}}t\right)-\frac{M_1\sqrt{L_c{C}_p}}{L_c{C}_p}\sin \left(\sqrt{\frac{1}{L_c{C}_p}}t\right)\\ -N_1+O_1\cos \left(\omega t\right)+\frac{P_1}{\omega }\sin \left(\omega t\right)\\ \frac{Q_1}{L_c{C}_p}\cos \left(\sqrt{\frac{1}{L_c{C}_p}}t\right)+\frac{R_1\sqrt{L_c{C}_p}}{L_c{C}_p}\sin \left(\sqrt{\frac{1}{L_c{C}_p}}t\right)\\ +S_1-T_1\cos \left(\omega t\right)-\frac{U_1}{\omega }\sin \left(\omega t\right)-V_1+v_{C_p}\left(D_{c}T\right)\mathrm{V}\end{array}$

B. Load-Independent Conditions

In order to find the load-independent operating point, the switch voltage v_S governing equation is considered as

(21) $v_S\left(\omega t\right)={\mathrm{\Psi }}_1(q,D_c)\times V_{\mathrm{in}}+{\mathrm{\Psi }}_2(q,D_c,\varphi )\times I_o{Z}_{\mathrm{in}},$

in which, Ψ₁ is a function dependent on D_c and the normalized resonant frequency $q=1/\omega \sqrt{L_c{C}_p}$ and Ψ₂ is a function dependent on I_o and the input impedance $Z_{\mathrm{in}}=\sqrt{L_c/C_p},$, described in detail as (D^¯_c = 1 − D_c):

(22) $\begin{array}{rl}{\mathrm{\Psi }}_1(q,D_c)& =[\frac{1}{q^2}-\frac{1}{q^4}-\frac{2\cos \left(q2\pi {\overline{D}}_c\right)}{q^2}+\frac{2\cos \left(q2\pi {\overline{D}}_c\right)}{q^4}\\ & +\frac{\cos {\left(q2\pi {\overline{D}}_c\right)}^2}{q^2}-\frac{\cos {\left(q2\pi {\overline{D}}_c\right)}^2}{q^4}+\frac{2\pi D_c\sin \left(q2\pi \overline{D}c\right)}{q}\\ & -\frac{2\pi D_c\sin \left(q2\pi \overline{D}c\right)}{q^3}+\frac{\sin {\left(q2\pi {\overline{D}}_c\right)}^2}{q^2}-\frac{\sin {\left(q2\pi {\overline{D}}_c\right)}^2}{q^4}]\\ & /\left[(\frac{-1}{q^2}+\frac{1}{q^4})(-1+\cos \left(q2\pi {\overline{D}}_c\right))\right]\end{array}$

and

(23) $\begin{array}{rl}{\mathrm{\Psi }}_2(q,D_c,\varphi )& =[\frac{\cos \left(q2\pi {\overline{D}}_c\right)\cos \left(\varphi \right)}{q^3}-\frac{\cos {\left(q2\pi {\overline{D}}_c\right)}^2\cos \left(\varphi \right)}{q^3}\\ & -\frac{\cos \left(2\pi {\overline{D}}_c\right)\cos \left(\varphi \right)}{q^3}+\frac{\cos \left(q2\pi {\overline{D}}_c\right)\cos (2\pi {\overline{D}}_c+\varphi )}{q^3}\\ & -\frac{\sin {\left(q2\pi {\overline{D}}_c\right)}^2\cos \left(\varphi \right)}{q^3}-\frac{\sin \left(q2\pi {\overline{D}}_c\right)\sin \left(\varphi \right)}{q^2}\\ & +\frac{\cos \left(2\pi {\overline{D}}_c\right)\sin \left(q2\pi {\overline{D}}_c\right)\sin \left(\varphi \right)}{q^2}\\ & +\frac{\sin \left(2\pi {\overline{D}}_c\right)\sin \left(q2\pi {\overline{D}}_c\right)\cos \left(\varphi \right)}{q^2}\\ & +\frac{\sin \left(2\pi {\overline{D}}_c\right)\sin \left(\varphi \right)}{q^3}]\\ & /\left[(\frac{-1}{q^2}+\frac{1}{q^4})(-1+\cos \left(q2\pi {\overline{D}}_c\right))\right].\end{array}$

Lemma 2.The load-independent operation is achieved by satisfying Ψ₁ (q, D_c) = 0 and Ψ₂ (q, D_c, ϕ) = 0.

Proof:

Considering ωt = 0, the switch S voltage is zero, under ZVS operation. In order to satisfy v_S = 0 for any load while ensuring power transfer, V_in ≠ 0 and I_oZ_in ≠ 0. Therefore, Ψ₁(q,D_c) = 0 and Ψ₂(q,D_c,ϕ) = 0.

Taking Ψ₁(q,D_c) = 0 and Ψ₂(q,D_c,ϕ) = 0 and applying a parametric sweep into D_c in (22), q can be numerically found for any operating point. Subsequently, (23) is solved in order to find ϕ. Therefore, the loadindependent operation requires specific values for q and ϕ dependent on the duty cycle. The normalized resonant frequency q as function of D_c and phase ϕ as function of D_c are depicted in Fig. 3. It can be seen that, load independence is only achieved if q > 1.

Aiming to relate the minimum load R_omin to the input impedance Z_in, the ZDVS condition is considered: i_Cp(T) = 0, which is written as

(24) ${\mathrm{\Psi }}_3(L_c,C_p,\omega ,\varphi ,D_c,T)\times I_o+{\mathrm{\Psi }}_4(L_c,C_p,\omega ,D_c,T)\times \frac{V_{\mathrm{in}}}{Z_{\mathrm{in}}}=0$

being:

(25) $\begin{array}{rl}{\mathrm{\Psi }}_3& =\left[C_p\right(\frac{1}{C_p{L}_c}{)}^{3/2}[2C_p{L}_c^2\omega \cos (\varphi )\\ & \times \sin \left((-1+D_c)\sqrt{\frac{1}{C_p{L}_c}}T\right)-2L_c\sqrt{C_p{L}_c}\\ & \times \cos \left(\sqrt{\frac{1}{C_p{L}_c}}(T-D_{c}T)\right)\sin \left(\varphi \right)\\ & +2C_p^2\sqrt{\frac{1}{C_p{L}_c}}{L}_c^3{\omega }^2\sin (\varphi -(-1+D_c)T\omega )\\ & +2L_c\sqrt{C_p{L}_c}\cos \left(\sqrt{\frac{1}{C_p{L}_c}}(T-D_{c}T)\right)\\ & \times \sin (\varphi +T\omega -D_{c}T\omega )-2C_p^2\sqrt{\frac{1}{C_p{L}_c}}{L}_c^3{\omega }^2\\ & \times \cos \left(\sqrt{\frac{1}{C_p{L}_c}}(T-D_{c}T)\right)\sin (\varphi +T\omega -D_{c}T\omega )]\\ & /2(-1+C_p{L}_c{\omega }^2)(-1+\cos \left(\sqrt{\frac{1}{C_p{L}_c}}(T-D_{c}T)\right))]\end{array}$

and

(26) $\begin{array}{rl}{\mathrm{\Psi }}_4& =[\frac{1}{C_p{L}_c}[2D_c\sqrt{C_p{L}_c}T\cos \left(\sqrt{\frac{1}{C_p{L}_c}}(T-D_{c}T)\right)\\ & -2C_p{D}_c{L}_c\sqrt{C_p{L}_c}T{\omega }^2\cos \left(\sqrt{\frac{1}{C_p{L}_c}}(T-D_{c}T)\right)\\ & -2C_p{L}_c\sin \left((-1+D_c)\sqrt{\frac{1}{C_p{L}_c}}T\right)+2C_p^2{L}_c^2{\omega }^2\\ & \times \sin \left((-1+D_c)\sqrt{\frac{1}{C_p{L}_c}}T\right)+C_p{L}_c\\ & \times \sin \left(2(-1+D_c)\sqrt{\frac{1}{C_p{L}_c}}T\right)-C_p{L}_c\\ & \times \sin \left(2(-1+D_c)\sqrt{\frac{1}{C_p{L}_c}}T\right)-C_p^2{L}_c^2{\omega }^2\\ & \times \sin \left(2(-1+D_c)\sqrt{\frac{1}{C_p{L}_c}}T\right)\\ & /2(-1+C_p{L}_c{\omega }^2)(-1+\cos \left(\sqrt{\frac{1}{C_p{L}_c}}(T-D_{c}T)\right))].\end{array}$

FIGURE 3
Load-independent conditions. (a) q as function of Dc. (b) ϕ as function of D_c.

III. DESIGN METHODOLOGY

In order to design the converter, the exact time-domain solutions and the load-independent conditions are evaluated in an iterative method that calculates the reactive element values, component stress and load range for any operating point. The time-domain equations are dependent on 11 variables: V_in, ω, I_o, D_c, L_c, C_p, ϕ, i₁(0), i₁(D_cT), v_Cp(0) and v_Cp(D_cT). Thus, V_in, ω = 2πf and I_o are specifications, with f as the operating frequency. In addition, v_Cp(0) = v_Cp(D_cT) = 0 is the ZVS condition. By parameter sweeping D_c, calculate q and ϕ based on the load-independent conditions and using the exact time-domain solutions as a linear system:

(27) $\left(\begin{array}{c}\stackrel{\mathrm{II}}{i_1\left(T\right)}\\ \stackrel{\mathrm{II}}{v_{C_p}\left(T\right)}\\ \stackrel{\mathrm{I}}{\mathrm{I}}\\ i_{L_1\left(D_{c}T\right)}^{\mathrm{II}}\\ {\frac{d_1\left(t\right)}{dt}|}_{t=T}\end{array}\right)-\left(\begin{array}{c}{i}_1\left(0\right)\\ v_{C_p}\left(0\right)\\ i_1\left(D_{c}T\right)\\ \frac{V_{\mathrm{in}}}{L_c}\end{array}\right)=\left(\begin{array}{l}0\\ 0\\ 0\\ 0\end{array}\right),$

the remaining variables, L_c, C_p, i₁(0) and i₁(D_cT) are found for any operating point. The output load is calculated as R = V_o/I_o, where the amplitude output voltage V_o is

(28) $V_o=\frac{2}{T}{{\int }_{D_{c}T}^T}_{V_p}^{\mathrm{II}}\left(t\right)\sin (\omega t+\varphi )dt$

and the minimum and maximum output load are

(29) $\begin{array}{rl}{R}_{o_{min}}& =-\frac{{\mathrm{\Psi }}_3}{{\mathrm{\Psi }}_4}\frac{V_o}{V_{\mathrm{in}}}{Z}_{\mathrm{in}}\end{array}$

and

(30) $\begin{array}{rl}{R}_{o_{max}}& =\frac{L_{\mathrm{r}1}}{C_{\mathrm{r}1}}\frac{1}{R_{o_{min}}}\end{array}$

with LCL circuit designed as

(31) $L_{\mathrm{r}1}=L_{\mathrm{r}2}=\frac{V_o}{I_o\omega }$

and

(32) $C_{\mathrm{r}1}=\frac{1}{{\omega }^2{L}_{\mathrm{r}1}}$

Finally, with quality factor Q_L, the main resonant circuit is found as follows:

(33) $\begin{array}{rl}{L}_o& =\frac{Q_{\mathrm{L}}{R}_{o_{min}}}{\omega }\end{array}$

and

(34) $\begin{array}{rl}{C}_o& =\frac{1}{{\omega }^2{L}_o}\end{array}$

The design specifications are shown in Table 1. The approximations for the initial conditions and reactive elements, required to numerically solve (27), are detailed in Table 2. The obtained theoretical waveforms for different duty cycles are depicted in Fig. 4. The switch stresses are drawn in Fig. 5. If D_c > 0.5, higher is both the peak switch current I_Speak and peak switch voltage V_Speak. At lower D_c values, I_Speak is reduced, however, V_Speak increases. The designed L_c and C_p as function of D_c are highlighted in Fig. 6. The described results are used to design the converter aiming the experimental validation.

TABLE 1
Design Specifications.

Parameter	Value
Input DC voltage, Vin	11V
Operating frequency, fs	1.2MHz
Output current, Io	1.3A
Duty cycle, Dc	0.2 − 08

TABLE 2
Approximations to Numerically Solve the Linear System.

Quantity	Value
Initial input current, i1(0)	2.00913A
On-off transition input current, i1(DcT)	3.03560A
Inductance, Lco	1.78605µH
Capacitance, Cpo	9.41566nF

FIGURE 4
Theoretical waveforms for different duty cycles. (a) i₁ for D_c =0.25. (b) v_Cp for D_c =0.25. (c) i₁ for D_c =0.35. (d) v_Cp for D_c =0.35. (e) i₁ for D_c =0.50. (f) v_Cp for D_c =0.50. (g) i₁ for D_c =0.75. (h) v_Cp for D_c =0.75. ^I_Speak(A) ^V_Speak(V)

FIGURE 5
Switch stresses. (a) I_Speak as function of D_c. (b) V_Speak as function of D_c.

FIGURE 6
Component values. (a) L_c as function of D_c. (b) C_p as function of D_c.

IV. EXPERIMENTAL RESULTS

By considering D_c = 0.5, the specifications and components for the designed converter are detailed in Table 3. The IRF510 MOSFET was used as main switch activated by the IRS2011PBF gate-driver. Results were obtained by a Tektronix MSO2024 Oscilloscope with a TCP0020 current probe (1:2 attenuation). Component inductance and capacitance were measured by means of a R&S HM8118 LCR Bridge.

The experimental results are shown in Fig. 7 for gate signal voltage v_G, switch voltage v_S, output voltage v_o and input inductor current i_Lc, considering different loads. ZVS is maintained in all cases with small or negligible delay in contrast to the off-on gate signal transition. In order to evaluate the load-independence, the load was varied from 2.2Ω to 25Ω, and the amplitude output current I_o was measured. Fig. 8 compares the measured I_o to the setpoint 1.3A and the theoretical constant current (CC) range, which confirms the designed span from 7.9 to 14.7Ω (10Ω nominal), ensuring load-independence. Fig. 9 shows the efficiency as function of output power. The calculated power density is 3.16W/mm³. A comparison to related works is detailed in Table 4. With the exception of [²⁶], all compared works use finite input inductor, which leads to reduced inductance value.

TABLE 3
Designed Converter.

Component	Value	Specification
Input inductor, Lc	1.09µH	Handcrafted (AWG 19)
Parallel capacitor, Cp	10nF	Ceramic disc
Resonant inductor, Lo	3.16µH	Handcrafted (AWG 19)
Resonant capacitor, Co	5.6nF	Ceramic disc
LCL input inductor, Lr1	1.78µH	Handcrafted (AWG 19)
LCL output inductor, Lr2	1.78µH	Handcrafted (AWG 19)
LCL capacitor, Cr1	10nF	Ceramic disc

TABLE 4
Comparison to Related Works.

Work	Topology	Input inductor	Switching	Output mode	Reactive elements	Power	Load range	Efficiency	Frequency
[24]	Class-E	Finite	ZVS	CC	4	5.85 W	20 − 40 Ω	89.3%	1 MHz
[25]	Class-E	Finite	ZVS	CC	6	10.8 W	1 − 10 Ω	92.1%	1 MHz
[26]	Class-E	Infinite	ZCS	CV	5	6.96 W	5 − 100 Ω	94.4%	1 MHz
[27]	Floating-gnd Class-E	Finite	ZVS	CC	4	6.72 W	5 − 20 Ω	89.3%	1 MHz
[31]	Class-E	Finite	ZVS	CV	4	3.64 W	30 − 50 Ω	93.17%	1 MHz
[32]	Isolated Class-E	Finite	ZVS	CC	5	12 W	1 − 30 Ω	87%	6.78 MHz
This work	Class-E LCL	Finite	ZVS	CC	6	7.2 W	7.9 − 14.7 Ω	92%	1.2 MHz

FIGURE 7
Experimental results for gate signal voltage v_G, switch voltage v_S, output voltage v_o and input inductor current i_Lc . (a) R = 3.2 Ω. (b) R = 10 Ω. (c) R = 12.9 Ω. (d) R = 14.2 Ω. (e) R = 15 Ω. (f) R = 25 Ω.

FIGURE 8
Load-independence characteristic.

FIGURE 9
Efficiency as function of output power.

Due to the LCL-network, the proposed work has 6 reactive elements, being a drawback in contrast to the standard Class-E topology (4 elements) [²⁴], [³¹]. The proposed work surpasses by ×1.2 operating frequency all compared works, with exception of [³²], which has the lower efficiency. The implemented converter is depicted in Fig. 10. Four loop mitigation barriers were added into the PCB layout to avoid parasitic inductance.

FIGURE 10
Implemented converter.

V. CONCLUSION

A Class-E LCL has been proposed, analyzed, designed and experimentally verified. Due to the high number of components, it was shown how to deal with the highorder polynomials in order to find explicit exact solutions for the electrical variables. The load-independent condition was described based on the normalized resonant frequency, duty cycle and output phase. The converter was designed by a parametric sweep into equations that include the soft-switching conditions. The designed converter outperforms most of the compared converters by ×1.2 operating frequency and shows commensurate efficiency. The load-independence was ensured from 7.9 to 14.7Ω while keeping ZVS.

• PLAGIARISM POLICY
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DATA AVAILABILITY

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

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Edited by

• Associate Editor
  Fabrício Bradaschia https://orcid.org/0000-0002-2086-7862

• Editor-in-Chief
  Allan F. Cupertino https://orcid.org/0000-0001-8418-1985

Publication Dates

• Publication in this collection
  17 Apr 2026
• Date of issue
  2026

History

• Received
  07 Nov 2025
• Reviewed
  16 Dec 2025
• Accepted
  19 Jan 2025
• Published
  30 Jan 2025