Derivation of the Resonance Mechanism for Wireless Power Transfer Using Class-E Ampliﬁer

: In this study, we investigated the resonance mechanism of 6.78 MHz resonant wireless power transfer (WPT) systems. The depletion mode of a gallium nitride high-electron-mobility transistor (GaN HEMT) was used to switch the states in a class-E ampliﬁer circuit in this high frequency. The D-mode GaN HEMT without a body diode prevented current leakage from the resonant capacitor when the drain-source voltage became negative. The zero-voltage switching control was derived according to the waveform of the resonant voltage across the D-mode GaN HEMT without the use of body diode conduction. In this study, the effect of the resonant frequency and the duty cycle on the resonance mechanism was derived to achieve the highest WPT efﬁciency. The result shows that the power transfer efﬁciency (PTE) is higher than 80% in a range of 40 cm transfer distance, and the power delivered to load (PDL) is measured for different distances. It is also possible to cover different applications related to battery charging and others using the proposed design.


Introduction
Wireless power transfer (WPT) provides a convenient and flexible method of transferring power cordlessly. Several studies have investigated WPT mechanisms [1,2]. In [1], an overview of WPT techniques with an emphasis on working mechanisms, technical challenges, metamaterials, and classical applications have been introduced. Besides, challenges including efficiency, power, distance, misalignment, Electromagnetic interference (EMI), security for various applications like Electric vehicles (EVs), biomedical implants, and portable electronics have also been discussed. In [2], an overview of power converter topologies and control schemes for inductive power transfer (IPT) systems in EV charging applications has been presented. Moreover, technical considerations and future trends for EV IPT have also been introduced such as enhancing the performance in terms of efficiency, power density, scalability, and reliability, adopting soft-switching technology, using wide bandgap device technology, and applying high-performance control schemes. Among the WPT topologies, the class-E power amplifier (PA) [3,4], which is simple yet highly efficient with high operating frequencies, is critical for various devices, such as drones [5], battery chargers [6], and electronics and sensors [7,8]. As to human physiology, the resonant frequency in the range of 3.8 to 6 MHz is regarded as a safe electromagnetic frequency for the human body to be exposed to, according to the International Commission on Non-Ionizing Radiation Protection (ICNIRP) and the Institute of Electrical and Electronics Engineers (IEEE) standards [9]. Moreover, wide bandgap power devices in class-E circuits have attracted considerable attention because of characteristics such as low capacitance and high switching [10,11]. Silicon carbide (SiC) and gallium nitride (GaN) wide bandgap devices, compared with silicon devices, have superior switching performance, therefore they can improve the efficiency of the WPT systems. In particular, due to the smaller gate-to-source capacitance and output capacitance, GaN devices have outstanding switching characteristics, which helps us reduce the gate driver and Coss-related losses of WPT converters in the megahertz band, and improve efficiency over a silicon device [10]. The wireless power transfer equivalent circuit and its efficiency analyses have been provided with great detail in [11] by Infineon Technologies.
However, class-E PAs still pose some challenges. Class-E circuits are sensitive to variations in load impedance [12][13][14][15]. Small deviations from the optimum load value would cause the inverter to operate under nonoptimal and inefficient switching conditions, whereas large deviations could cause permanent damage to its switch [13]. Therefore, the optimization of the inductive link must be dynamically performed to satisfy the zero voltage switching (ZVS) and zero voltage derivative switching conditions under varying loads. Tuning methods such as replacing a capacitor and adjusting the switching frequency [12], controlling the duty cycle and the value of the DC feed inductance [13,14], and implementing a voltage-controlled capacitor and a negative impedance converter have been proposed in the literature [15]. Furthermore, the impedance matching approach based on coupling tuning was proposed to maximize transfer efficiency [16]. Moreover, control and compensation strategies, such as tracking the maximum efficiency and charge/discharge of the ultracapacitor bank [17] and regulating self/mutual inductance [18], maintain high system efficiency under various uncertainties and improve system stability. Furthermore, the effects of the transistor's parasitic elements should be considered to avoid the performance degradation of the class-E circuit [19,20]. The aforementioned literature shows there is a need for clarifying an optimization procedure to satisfy the zero voltage switching (ZVS) and so on. However, they did not introduce any general diagram regarding the optimization process, thus we attempt to provide a systematic procedure for the WPT design optimization in this paper.
In this study, first, a class-E amplifier circuit, comprising a switching power supply component and impedance loading component, was analyzed. Then, the resonant design for tuning resonant frequencies and duty cycle was presented to achieve high-efficiency WPT. The parasitic capacitances were also considered to realize the ZVS operation for WPT. Finally, the experimental results were presented to verify the proposed resonant design analysis. Using the D-mode GaN HEMT as the switching device, we derived the resonance mechanism for proper resonant frequency and the duty cycle settings based on the hypotheses of the zero voltage control (ZVS) and zero current control (ZCS) for the class-E amplifier as an extension work of [4]. In this paper, as a goal, we attempt to promote the power efficiency of WPT to above 80% as well as to extend the power transfer distance to the meter-level distance. They are done by adjusting both the duty cycle and resonant frequency to comply with the known phenomenon in GaN HEMT control including the Miller plateau and current clamp [21].

Wireless Power Transfer (WPT)
The equivalent circuit of wireless power transfer was shown in Figure 1. The resonant frequency ω o was selected to closely match 1/ L T,l C T to reduce the leakage reactance effect on the power transmitting unit (PTU). The power receiving unit (PRU) was used to adjust C R so that the resonance of leakage reactance matched the resonant frequency ω o . Therefore, ω o ≈ 1/ L T,l C T ≈ 1/ L R,l C R . The load from the PTU is expressed as follows: If all the magnetic flux is transmitted through air with no low reluctance core, then L T,l + M = L T is a constant.

Class-E Amplifier Circuit Analysis
The class-E amplifier (Figure 2a) was used in the PTU to induce the resonance of the output voltage so that the flux density could increase without considerable input power. The actual load Z L for a WPT system is the mutual inductance that is parallel with the equivalent impedance of the receiver side. For simplicity of derivation, we assumed the load to be a purely resistive load and denoted it by R L . The class-E amplifier circuit is divided into the switching power supply and impedance loading. The duty cycle of switching is denoted by δ, and the pulse width is δT (Figure 2b). The conventional power Metal-Oxide-Semiconductor (MOS), Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET), and Insulated-Gate Bipolar Transistor (IGBT) have body diodes and the negative voltage of v X in the class-E amplifier will be clamped to the body diode threshold. The GaN HEMT transistor has no body diode, thus the voltage v X could be negative and we have to apply the zero current control (ZCS) condition in addition to the ZVS to eliminate this negative voltage situation for better efficiency [3,4]. As observed in Figure 3, when the switch is operated at a frequency ωo = 2π/T, the output current is assumed i2 = I2 sin(ωot + β) + I2,a cos(2ωot + β) in steady state. This assumption is made from the observation of experiments. The high order terms I2,a << I2 cause the sinusoidal output to be asymmetrical. Considering β = −180°, I2 = 1, and I2,a = 0.15 as an example, the following assumptions were made to simplify the derivation: Assumption 1: In the steady state, the output frequency is ωo = 2π/T, and the output current is i2 = I2 sin(ωot + β) + n k 1 Σ = I2,k cos(2kωot + β), as displayed in Figure 3. As observed in Figure 3, when the switch is operated at a frequency ω o = 2π/T, the output current is assumed i 2 = I 2 sin(ω o t + β) + I 2,a cos(2ω o t + β) in steady state. This assumption is made from the observation of experiments. The high order terms I 2,a << I 2 cause the sinusoidal output to be asymmetrical. Considering β = −180 • , I 2 = 1, and I 2,a = 0.15 as an example, the following assumptions were made to simplify the derivation: Assumption 1: In the steady state, the output frequency is ω o = 2π/T, and the output current is i 2 = I 2 sin(ω o t + β) + ∑ n k=1 I 2,k cos(2kω o t + β), as displayed in Figure 3.

Switching Power Supply Side
For 0 < t < (1 − δ)T, when the transistor is switched off, the inductor current is expressed as follows: On the basis of previous assumptions, the capacitor voltage can be expressed as follows: Equation (1) becomes where The aforementioned equation may be approximated using the following expression: The ω 3 and ϕ are the parameters of input current waveform during the switch-off time derived in [3], and are the most fundamental parameters for the class-E amplifier circuit design. Both the current phase angle ϕ and frequency ω 3 are functions of the duty ratio which will be derived in the later sequels.
According to Figure 4, the initial conditions of the switch-off time can be calculated from the previous charging time, namely the switch-on time, as follows: The first derivative of the current i 1 must be continuous at the start of the switch-off time. From Equations (6) and (7), we obtain the following expression: The boundary condition for the inductor current should also satisfy the following equation: Therefore, The zero-voltage condition of v c1 in which the transistor is turned on again from Equation (2) yields the following equation: Comparing Equations (9) and (13), we obtained the following equation: Thus, Equation (11) yields the following expression: Combining this equation with Equation (9), we obtained the following expression: The maximum voltage in the capacitor is derived from dv x (t)/dt = 0 that Substituting the equation into Equation (12), we obtained the following expression: and Table 1 displays the lookup table for some variables versus the duty cycle. The 50% duty cycle is a well-known setting to obtain the nominal switch-voltage waveform [4]. In this paper, we attempt to transcend this conventional setting by varying the duty cycle as shown in Table 1 and not to limit to the 50% duty cycle. As a result, we can achieve better power efficiency for WPT. By adjusting both the duty cycle and resonant frequency, the class-E amplifier can comply with the known phenomenon in GaN HEMT control including the Miller plateau and current clamp [21]. When the maximum voltage in the capacitor is reached, then the following expression is applicable: that is,  Here, η 1 is the distortion factor of i 2 from the perfect sinusoidal function.
Here, the estimation of I 1 in terms of I 2,max can be expressed as follows: Combining Equations (22) and (23), we obtained the following equation:

Impedance Loading Side
The impedance side has some unknown parameters, namely L 2 and R L , to be identified during WPT. In the following derivations, we assumed that they are known.
Due to We obtain The input of the aforementioned equation is approximated using the following equation and presented in Figure 5. The steady-state output current is expressed as follows: where Assuming ω 0 = ω 2 + ∆ω, 1 >> ∆ω ω 2 , we obtain the following equation: and From Equation (22), the maximum voltage in the capacitor V X,max occurs in the middle of the switch-off time. Therefore, we obtain the following expression: Then, we have the following expression: which follows that ∆ω > 0. Thus, ω 0 > ω 2 . Here, we define the proper resonant state, which indicates that Equation (37) (22) and (24), we obtain the following equation: Therefore, a successful ZVS must follow that Assumption 3: To ensure proper resonance occurs, ω 0 > ω 2 . Thus, ∆ω > 0 (see Equation (38)).
Assumption 4: To guarantee a resonance, ω 1 << ω 0 . Assumption 5: In a proper resonant state, A monotonically increases with the duty cycle δ when the switching frequency ω 0 remains constant (see Table 1).
The coefficient η introduced in Equation (22) is the ratio between the input current and the output current. The condition of η = 0.99 implies that the input current is equal to the output current so that there is no other current loss, neither the loss from the power line, the inductor, nor the transistor. Simplified class-E amplifier circuit for the SPICE was shown in Figure 6. In the SPICE analysis, different from the experiment, we had simplified that circuit model, thus the equations used in Table 2 are assuming η = 0.99. Table 2 presents the benchmark of SPICE analyses.

ZVS Control
Without considerable loss of generality, the capacitor current during the transistor switch-off time is as follows: In a proper resonant state, the transistor switch-off time was typically D << I 2 A . Thus, the current ripple of the switching power supply inductor A is small. β can affect the capacitor i c1 during the transistor switch-off time more than the current ripple A. In this case, ω o is tuned correctly [∠G 2 (jω o )]. However, if β is too large, we must decrease δ to obtain a favorable ZVS condition. The result is displayed in Figure 7a. If β is too negative, δ should be increased to obtain appropriate ZVS. The result is displayed in Figure 7b.
When i c1 = 0, the following stationary condition for v X is obtained: In a proper resonant state, A/I 1 << 1 so that A/I 1 sin(ω 3 t + ϕ) + 1 > 0 and sin(ω o t + β) = D = sin((1 − δ)π + β) yields the maximum capacitor voltage, which is consistent with the solution in Equation (18). If β is too small, then we may have multiple stationary values during the switch-off time. In this case, β should be increased by adding the switching power supply inductor charging time. Thus, δ should be increased to obtain a satisfactory ZVS. Furthermore, ω o should be increased and the equivalent switch-off time should be reduced to prevent a multiple stationary value situation. The result is displayed in Figure 7c. In other cases, no stationary value exists in the transistor switch-off time, as displayed in Figure 7d. Therefore, ω o should be increased to avoid an improper resonance state.
In a resonant state, at time t = (1 − δ)T for D = sin((1 − δ)π + β), we have the following expression: The coherence of ZVS and ZCS depend on the current ripple percentage (A/I 1 %) in the switching power supply inductor. ZCS facilitates smooth turning on of the transistor without disturbing the impedance loading side. The following condition holds for ZCS: The ZCS condition is provided in Equation (46). In addition, Figure 7b shows an extra ZVS control case with a negative voltage v x other than those found in [4], which needs to satisfy the ZCS condition. Achieving ZCS during ZVS, the resonant frequency ω o could be slightly decreased so that |G 2 (jω o )| and β increase I 1 without affecting ZVS, as displayed in Figure 7e.
The ZVS control depicted in Figure 7a-e is applied to find the best operating condition for resonant frequency and duty cycle, and to control the gate of GaN HEMT. In this research, we used the function generator Tektronix AFG 3104, which is dual channel output equipment that sends 0 to 5V square wave signal to the gate drive [21]. The output of vx is taken from the class-E amplifier to show on the oscilloscope Tektronix MDO 3054. The demo of the ZVS manual operation is shown in Figure 7f.

D-Mode GaN HEMT
The experiment was based on a D-mode GaN HEMT. The characteristics of the GaN HEMT are displayed in Table 3. A charge pump-type gate drive was applied to switch the GaN HEMT. The parasitic capacitance measurement data are presented in Figure 8. The C OSS is the most critical limiting factor for realizing ZVS operation for WPT. Completely eliminating hard switching is difficult for a large increase in C OSS at a low drain-source voltage [9]. Here, C OSS = C GD + C DS of the D-mode GaN HEMT is only five times that of the low to high drain-source voltages. Furthermore, the capacitances are not sensitive to the switching frequencies in the range 4-5 MHz. Here, C ISS = C GD + C GS is three times that of asymmetric effects on the Miller plateau when the transistor is turned on and off.

Results
The experimental layout is displayed in Figure 9. The layout consists of a function generator that generates a rectangular pulse train with adjustable frequency and duty cycle in the output. The oscilloscope measures the output current and voltage on the PRU as well as the gate-source voltage and the resonant voltage of the PTU. The oscilloscope (Figure 10a) displays the result when V DD = 110 V such that the i 2,PRU (green curve) and v R,PRU (blue curve) of the PRU differ by almost 90 • under the resonant WPT condition; the angle β of Equation (33) can be estimated from the value of v GS,ON in Table 3 and the zero current point of i 2,PRU . The ZVS control according to Figure 7a-e is applied to manually obtain the drain-source voltage v X (v DS ; purple curve). The result is displayed in Figure 10a. Figure 10b reveals the other experiment result of i D,PRU (green curve) and i 1 (blue curve) relative to v gs (cyan curve), in which the angle ϕ of Equation (17) can be estimated. In the first experiment, we fixed the values of the parameters (Table 4) and measured the input power with no PRU in front of the PTU. Figure 10a and b show two different operating conditions. Table 4 shows only the nominal parameters for the ZVS control used in this research. We have summarized more operating conditions into figures from Figures 11-14 to prove the validity of this research. The ZVS control results for various frequencies are displayed in Figure 11. Most of the input power is subjected to the switching loss, which depends on the transistor characteristics, namely the input capacitance C ISS = C GD + C GS and also the output capacitance C OSS = C GD + C DS . As displayed in Figure 10, the blue curve indicates that the gate-source voltage is not symmetric because of a 30% rectangular gate signal. This is due to the difference in the drain-source voltage at various gate-source voltage states. When the gate is turned on, a successful ZVS occurs on the drain source such that the parasitic capacitor C GD accumulates limited charges. However, a tremendous amount of charge should flow through the gate drive from the ground when the gate turns off because of the high voltage on the drain source when the gate is turned off. The switching loss is the product of the drain current i D and v GS , and almost all of the switching power loss occurs during the gate turning-off time. With the same value of the resonant voltage v X , the higher the switching frequency ω o is, the higher the switching power loss is because it results in a larger ratio of Miller plateau time to the switching time period T. The switching power loss determines the power transfer efficiency. In the experiment displayed in Figure 11, the ZVS control is successful in the range of 3.8-6 MHz. The minimum power loss is at 4.0 MHz when the duty cycle is 30%. Thus, we set the switching frequency to 4.0 MHz for this GaN HEMT for the experiments.

Symbol
Value Unit     The switching loss is a complex function of the gate-drive parameters, switching frequency, duty cycle, GaN HEMT characteristics, and V DD . We verified the validity of the results in Table 1, which form a basis for ZVS control and a resonant mechanism; a comparison between the results of the experiment and values derived using Equation (17) is presented in Figure 12. The validity of Equation (17) is confirmed even when the switching loss is not included in the derivation. We verified the resonant voltage amplification V X,max /V DD and illustrated the comparison in Figure 13 when the resonant frequency was set to 4.0 MHz. The comparisons indicate that the experiment exhibited slightly higher voltage amplification than the analysis in Equation (19) because the turningoff time of the GaN HEMT was extended. Thus, the equivalent duty ratio percentage in the experiment was higher than the actual input duty ratio percentage. Furthermore, according to Table 1, the voltage amplification V X,max /V DD monotonically increases with an increase in duty ratio. However, the experiment revealed a maximum voltage amplification of approximately 28%. Furthermore, Table 1 provides a guideline for ZVS control.
Power transfer efficiency (PTE (%)) is defined as follows: The values of V R,PRU and I 2,PRU were obtained from the PRU directly. The experiment was performed with a single PTU, which transferred power to a single PRU. The power loss is calculated from the power difference between the input power sent to the PTU by the power supply CHROMA 62012P-600-8 and the output power measured from the current probe and voltage probe on the PRU taken from the oscilloscope Tektronix MDO3054. The results ( Figure 14) revealed that for 1 W of WPT, the efficiency can be as high as 99% when the distance is 20 cm. The PTE (%) can be as high as 80% when the PRU is 40 cm away from PTU. The maximum distance of 1 W power transfer can be as long as 140 cm when V X is ≥700 V, when the corresponding input voltage V DD is around 300 V. The power delivered to load (PDL) is also depicted in Figure 14. It shows that the PDL from a single PTU to a single PRU varies with the distance apart from one another. In order to reach the constant power delivered to the PRU, a closed-loop feedback control mechanism shall be considered [8]; for example, controlling V DD to maintain the constant power is a feasible option. A live photo demonstrating the experiment of multiple PRUs charging is displayed in Figure 15. The design of PTU coils based on Biot-Savart's law [22] was presented in Appendix A. Multiple PRUs for multiple-input multiple-output (MIMO) wireless power transfer can yield high PTE and the maximum amount of power delivered to the load (PDL) [23,24]. The 99% PTE obtained in our experiment is comparable to the experimental result shown in [16] which uses the coupling tuning-based impedance matching for maximum wireless power transfer efficiency. For 1 W power transfer application, the corresponding I 1 on L 1 inductor is 20 mA. The class-E amplifier in this paper uses only one GaN HEMT transistor with no body diode, thus no rectifier loss. The GaN HEMT with 100 mΩ R D,on loss is less than 0.1% of the total power input. The switching loss under the zero-voltage switching condition is also less than 1 mW. The high-frequency inductor L 1 coil loss is 0.2 mW loss. The coil loss of L 2 is a function of distance, which is not directly measurable, therefore Equation (47) is used instead to calculate the PTE. The systematic procedure for WPT design is shown in Figure 16. Following the systematic design procedure, the comparison through analyses, simulations, and experiments can be verified and the operating conditions can be adjusted according to the best power efficiency for various applications.

Conclusions
The resonance mechanism of the 6.78 MHz resonant WPT was analyzed. The D-mode GaN HEMT with no body diode can prevent the current leakage from C 1 and enhance the voltage resonance. The ZVS control was derived in terms of the resonant frequency and duty cycle. The experiments agreed with the derivation of the ZVS control. For 1 W of WPT, the efficiency can be as high as 99% when the transfer distance is within 20 cm. The PTE can be as high as 80% when the PRU is 40 cm. The maximum distance of 1 W power transfer can be 140 cm when V X is ≥ 700 V. The GaN HEMT made in National Chiao Tung University (NCTU) has a breakdown voltage higher than 1000 V. The critical problem for the WPT is the switching loss of the GaN HEMT transistor. In the future, a detailed SPICE circuit analysis of the class-E amplifier should also consolidate the precise GaN HEMT transistor model and its gate-drive circuit to improve the power efficiency. In order to reach a constant PDL while the distance of the PRU is varying, the V DD may follow the corresponding distance change. In the closed control loop, the required feedback of the actual power reading of the PRU via a 2.4 GHz communication network is recommended by the Airfuel Alliance.  Theoretically, the inductance of circular loops may be as though the magnetic field is everywhere in the interior plane of a single circular loop of wire carrying current I, where the loop has radius R = D/2 and the wire has radius a. According to Figure A1, we put the origin at the center of the loop and let the z-axis point normal to the plane of the loop. We can also orient the x-axis so that the point at which we want to find the magnetic field → B is located at coordinates 0 ≤ x < R − a, y = 0, and z = 0 as sketched below.
The following equation is obtained based on Biot-Savart's law [22].
On the other hand, the inductance can also be derived from Faraday's law that From an experiment using the RLC meter GWINSTEK LCR819, we measured three inductors using the AWG22 copper wire whose wire diameter was 0.643 mm with different arrangements. The result is shown in Table A1. The following equation is obtained based on Biot-Savart's law [22]. From an experiment using the RLC meter GWINSTEK LCR819, we measured thre inductors using the AWG22 copper wire whose wire diameter was 0.643 mm with differ ent arrangements. The result is shown in Table A1. The coefficient α is calculated using  The coefficient α is calculated using Comparing Equation (A1) with Equation (A2), we have α = π 2(ln D − ln a − 0.6) (A4) Figure A2 shows the comparison between the result from empirical Equation (A3) and that from theoretical Equation (A4). The result shows that the theoretical equation can only be valid for a range between 20 cm and 30 cm in the wireless power transfer application with flat coils. (A2) substituted with Equation (A3) is adopted in this paper for the coil design.  Table A2.