Analysis and Design of the Suboptimum Class-EM/F3 Resonant Inverter

In this paper, the analysis and design method for the suboptimum Class-<inline-formula> <tex-math notation="LaTeX">$\text{E}_{\mathrm {M}}/\text{F}_{3}$ </tex-math></inline-formula> inverter (i.e., only the zero-voltage switching (ZVS), the zero-voltage-derivative switching (ZVDS), the zero-current switching (ZCS) conditions are satisfied) with a duty ratio D =0.5 are presented. In the proposed design method, a new design parameter K called the slope of switch current (when the switch turns off) is introduced for defining the suboptimum degree and the distance from the optimum condition. By using the design method of the soft switching resonant inverter, the third harmonic filter circuit is innovatively introduced to reduce the current flowing through the parallel capacitor of the main circuit, and greatly reduce the peak switching voltage of the main circuit and auxiliary circuit. And the circuit waveforms and design equations can be derived in detail. Compared with the conventional Class-<inline-formula> <tex-math notation="LaTeX">$\text{E}_{\mathrm {M}}$ </tex-math></inline-formula> inverter, the proposed inverter can offer the much lower peak switching voltage and higher efficiency. To verify the validity of the proposed method, a Class- <inline-formula> <tex-math notation="LaTeX">$\text{E}_{\mathrm {M}}/\text{F}_{3}$ </tex-math></inline-formula> inverter operating at 1MHz is designed using the IRFZ24N transistor. The experimental results show that the output power of the new inverter is 15.138W, the efficiency reaches 96.4%, and the peak switching voltage of the main circuit and auxiliary circuit is reduced by 29.3% and 15.6%, respectively. The PSpice-simulation results and the experimental measurement results of the designed inverter are agreed with the analytical results, which verified the effectiveness of the proposed design method.


I. INTRODUCTION
With the requirement for the efficient operation of the power electronic circuit at high-frequency, the design and construction of high-frequency dc/ac inverters and amplifiers are increasing significantly. The traditional switching devices work in the hard switching state, there are some defects such as low switching frequency, high loss, and too much noise, and they cannot meet the requirements of miniaturization, high efficiency, and high power of switching power supply. Therefore, a power conversion circuit with soft switching characteristics began to step on the stage of history, with its The associate editor coordinating the review of this manuscript and approving it for publication was Tae Wook Kim . simple circuit topology, theoretical 100% power transmission efficiency, and high operating frequency, which has been deeply studied by many scholars [1].
In search of converters capable of operating at higher frequencies, power electronics engineers began to develop converter topologies that could form sinusoidal current or voltage waveforms, thereby reducing switching losses [2], [3], [4].
The key idea is to use a resonant circuit with a high enough quality factor called a resonant converter. Most switching transistors and diodes in the resonant converters operate in the Soft-Switching state, i.e. Zero-Voltage-Switching (ZVS) or Zero-Current-Switching (ZCS), Such a working state can make the voltage waveform and current waveform at both ends of the switch do not overlap, reduce switching loss and electromagnetic interference. However, the traditional resonant converter can not achieve the soft switching state at both on and off time, and there is still a large switching loss in practice. Moreover, in many applications, the circuit parameters of the conventional resonant converter, such as resistance and operating frequency, are not immutable, and the circuit cannot be maintained in optimum operating condition, that is, to meet the ZVS at the same time to meet zero voltage derivative switching (ZVDS). Therefore, optimizing the circuit topology and working mode of the conventional resonant converter to reduce switching loss, improve efficiency and enhance practicability has become a research hotspot in recent years.
For power amplifiers (PAs) and resonant inverters, the main power loss usually comes from the dissipated power in the output active device, that is, the loss of transistors or vacuum tubes. Based on this, the researchers designed the Class C PA [5], [6], which reduced the current flowing through the switching device during the switching voltage duration and the voltage at both ends of the switching device during the switching current duration but did not consider the loss of the transistor at the time of on and off. In 1975, N. O. Schoal proposed the Class E high-efficiency resonant PA for the first time, which consists of a load network and a transistor operating in the switching state [7]. By adjusting the voltage and current waveform on the switching transistor, the transistor can realize zero voltage and zero voltage derivative turn-on at the on-off time. In theory, 100% efficiency is achieved. Therefrom, the class E PA has officially come into people's view, and its design method has also been applied to the design of inverter, known as Class E soft switching resonant inverter.
The class E resonant inverter has become one of the first topologies in wireless power transmission applications in recent years by virtue of its strong power transmission capacity, high efficiency and sinusoidal output current [8], [9], [10], [11]. In 2010, X. Wei from Chiba University emphasized the influence of the gate-drain parasitic capacitance of semiconductor field effect tube (MOSFET) as a switching device, gave the waveform expression and design equation that met the optimal switching conditions (ZVS and ZVDS at the same time), and designed a class E inverter with an operating frequency of 7MHz [8]. The output power is 4.06W, the efficiency is 92.8%. In the following year, the team took the influence of MOSFET drain-source nonlinear parasitic capacitance into consideration and designed a Class E resonant inverter with a working frequency of 4MHz, output power of 2.16W and efficiency of 92.2% [9]. In 2013, T. Nagashima from Chiba University proposed an induction-coupled wireless power transmission system based on a class E2 converter for the first time [10]. In the design, the class E resonant inverter was adopted as the transmitter and the class E rectifier as the receiver. Under the operating frequency of 200KHz, the output power reached 100W. The power transmission efficiency of 85.1% was achieved.
To achieve higher power conversion efficiency in the Class-E inverter, where the transistor has a long turn-off switching time, a modified Class-E inverter which is called Class-E M is presented in [12]. Commonly, it consists of a main circuit operating at the fundamental frequency (f0) and an auxiliary circuit operating at the second harmonic frequency (2f0), where the main circuit satisfies the zero-voltage switching (ZVS) and the zero-voltage-derivative switching (ZVDS) conditions at turn-on instant, and the zero-current switching (ZCS) and the zero-current-derivative switching (ZCDS) at turn-off instant simultaneously. Therefore, the Class-E M inverter can achieve higher power-conversion efficiency at high frequencies by adding the auxiliary circuit [13]. The E M class resonant PA has been widely concerned since its appearance, and its design method has also been applied to the design of the inverter. In 2010, R.Miyahara analyzed the influence of two different auxiliary loops on the overall circuit performance, proposed a more simple and accurate design method of the class E M resonant inverter, and used this method to design two class E M resonant inverters with auxiliary loops of E class inverter and E class frequency multiplier respectively [14]. The output power is 13.9 W, and the efficiency is 94.4% and 94.8%, respectively. In 2019, T.Innaba combined the class D inverter with the class E M resonant inverter, proposed a resonant inverter called the DE M class, which greatly reduced the switching voltage, and designed a DE M class resonant inverter with a working frequency of 1MHz, the output power of 4.87W and efficiency of 93.2% [15].
Nevertheless, the Class-E M topologies in [12], [13], [14], [15], [16], and [17] suffer from a rather high peak switching voltage factor of 4.3, which is about 20% higher than the Class-E commonly, i.e., 3.6, which will limit their widespread application in power electronic circuits. The Class-F inverter can solve the problem of the high drain-to-source voltage of the Class-E M inverter greatly [18]. It utilizes multipleharmonic resonators in the output network to shape the drainto-source voltage, such that the transistor loss is reduced and the efficiency is increased. The Class-F inverter has short load termination at even-order harmonics (current peaking) and open load termination at odd-order harmonics (voltage peaking), which become a representative of the high-efficiency amplifier. Very recently, the inverse class-F amplifier has started to draw attention due to its superior performance [19]. The inverse class-F amplifier has an open load at even-order harmonics (voltage peaking) and a short load at odd-order harmonics (current peaking), which can deliver higher efficiency than the Class-F operation mode. To combine the excellent switching characteristics (ZVS/ZVDS/ZCS/ZCDS) of the Class-E M inverter with the low drain-to-source voltage characteristic of the inverse class-F inverter, a novel Class-E M /F 3 amplifier topology was proposed in [20]. Meanwhile, to reduce the high switching voltage of the main circuit transistor, a new E M /F n type PA by adding an inverse class F third harmonic filter to the main circuit is designed in [21].
The PA ensured the working state of the soft switch while the peak switching voltage of the main circuit was reduced by 27.3% compared with the traditional E M class. At the operating frequency of 1.8 GHz, the output power of 42.3dBm and drain efficiency of 83% is achieved.
With the deepening of research on the class E M resonant inverter in recent years, its circuit performance is constantly improving. However, because the class E M resonant inverter has two main and auxiliary loops, and there are also high-order filtering loops in the class E M /F n resonant inverter, the circuit design process is often complicated, and it is difficult to derive accurate and simple design equations. Reducing the complexity of circuit design and improving the accuracy of the calculation of circuit components has become a major difficulty to overcome. It will be further shown in this paper, through the analysis of the suboptimum Class-E M /F 3 resonant inverter, accurate analytical design equations can be derived. By using the simplified analytical method, the analysis of the main and auxiliary circuits can be performed separately, and the complexity of the analysis can be reduced while the explicit design equations and the accurate design values can be obtained. The voltage and current waveforms in different suboptimal states will be illustrated. Furthermore, a PSpice simulation and the actual circuit design example operating at 1MHz will be given to validate the proposed design method.
This paper is organized as follows. In section II, a detailed circuit analysis process for the main and auxiliary circuits is presented, and expressions of the circuit parameters varying with the free variables are given. In section III, the simulation and experimental results for the subnominal class E M /F 3 inverter are present. Finally, the conclusion is given in section IV.

II. CIRCUIT ANALYSIS OF THE SUBOPTIMUM CLASS-E M /F 3 RESONANT INVERTER
The class E soft switching resonant inverter has attracted wide attention since it was proposed with its theoretical 100% power transmission efficiency. In practical applications, although the class E soft switching resonant inverter can realize zero voltage and zero voltage slope (zero voltage derivative) turn-on, there is a switch current jump in the process of switching off, resulting in switching loss. The efficiency of Class E soft switching resonant inverter is reduced in practice. In order to overcome the above problems, the class E M soft switching resonant inverter comes into being. By injecting the second harmonic current into the main circuit, the main circuit can meet the requirements of zero voltage and zero voltage derivative turn-on, as well as zero current and zero current derivative turn-off at the same time, which further reduces the turn-off loss in the circuit, but at the same time, it has a higher peak switching voltage stress. Compared with the class E circuit, the main circuit peak switching voltage increases by 20%, which greatly limits the application scenarios of the class E M soft switching resonant inverters. In order to solve the peak switching voltage problem of the class E M soft-switched resonant inverters, it is hoped that the load network of the main circuit can reduce the high harmonic current component flowing into the parallel capacitor of the main circuit without affecting the fundamental wave and the second harmonic component, which is mainly the third harmonic component. Based on the design concept of the class E/Fn type soft switching resonant inverter to reduce the peak switching voltage, a series filtering loop L3-C3 resonant at three times the working frequency is connected in parallel in the load network of E M type resonant inverter. At this time, the load network composed of the cubic filtering loop and the output filtering loop can realize the short circuit of the odd harmonic (third) and the open circuit of the dual harmonic (second). That is, it works in the mode of class F resonant inverter. Therefore, the new hybrid inverter designed in this paper is called the class E M /F 3 soft-switch resonant inverter.
The suboptimum Class-E M /F 3 resonant inverter shown in Fig.1 consists of a main circuit, an auxiliary circuit and a third resonant circuit. Both the main and auxiliary circuits are the traditional Class-E topologies, which consist of a DC power supply (V DDi ) as energy input, a MOSFET as switch component, a choke inductor (LC i ), a capacitor (C si ) in parallel with the switch, a series resonant circuit (L i C i ) and a phase-modulated reactance (X 1 ). The choke inductor can reduce the ripple voltage of the power supply and prevent the high-frequency component in the circuit from affecting the power supply. The shunt inductance is the sum of the external linear capacitance and the output capacitance of the transistor. The series resonant circuit ensures the standard sinusoidal current output. The phase-modulated reactance adjusts the phase shift of the output current of the main and auxiliary circuits. The duty ratios of the main and auxiliary circuits are 0.5 and 0.25 as presented in [17]. The ZVS Class-E amplifier and the ZVS Class-E frequency doubler can be used in the auxiliary circuit, the latter has higher power transmission efficiency, and can enable the overall circuit to be more efficient. The auxiliary circuit can inject the second harmonic current with a specific phase and amplitude into the drain of the main circuit transistor, so the main circuit can satisfy the VOLUME 11, 2023 50015 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. ZVS and ZVDS conditions at the turn-on instant and ZCS condition at the turn-off instant. The switching current and voltage of the main circuit can achieve no overlap in the whole cycle, which greatly reduces the power loss of the transistor and improves the power transmission efficiency of the whole circuit. The third resonant circuit is a series resonant circuit with resonance frequency 3f 0 , which is in parallel with the main circuit switch. It can reduce the peak switching voltage by filtering out the third resonance current flowing through the main circuit switch.
To obtain a series of more comprehensible and simple analytical expressions, the circuit in this article operates in suboptimum operation, i.e., the main circuit does not satisfy the zero-current-derivative switching (ZCDS) condition. Due to the reduction of ZCDS condition limitation, the characteristics of the designed circuit can be greatly increased and the complexity of circuit design can also be greatly reduced.
The analysis of the Class-E M /F 3 resonant inverter of Fig.1 is carried out under the following assumptions.
1) The parasitic parameters of all circuit elements are ignored.
2) The transistors all are the ideal switch, i.e. its ON resistance is zero and its OFF resistance is infinity.
3) The turn-on and turn-off time of transistors is zero. 4) The resonant frequency and quality factor is Q = ωL R = 1 ωCR . 5) The filter inductance LC is high enough to make the current flowing through it approximately constant and equal to the DC input. 6) The switch-off duty ratios of main circuit D1 and auxiliary circuit D2 are 0.5 and 0.25, respectively.

A. ANALYSIS AND THEORETICAL DERIVATION OF THE MAIN CIRCUIT
The main circuit of the class E M /F 3 soft-switched resonant inverter can only satisfy zero voltage turn-on, zero voltage derivative turn-on and zero current turn-off, but not zero current derivative turn-off. For the auxiliary circuit, the working state is the same as that of the class E M soft-switched resonant inverter, which meets the zero voltage turn-on. The waveforms of the main circuit switching voltage vs1 and switching current is1 are shown in Fig. 2. Although the main circuit does not meet the optimal switching state of the four circuits, it can be clearly found from the waveforms of the switching voltage and switching current of the main circuit that the absence of the operating state of zero current derivative switching off does not lead to the overlap of switching voltage and current, so there will be no additional switching loss. The class E M /F 3 resonant inverter in sub-optimal working conditions can ensure high power transmission efficiency. A new design parameter K is introduced to represent the slope of the switching current of the main circuit at the turn-off time, that is, the degree of deviation from the optimal working state.
The main circuit of the Class-E M /F 3 resonant inverter operates with suboptimum switching conditions (ZVS/ ZVDS/ZCS), which are expressed as where v s1 (θ) and i s1 (θ) is the switch voltage and current of the main circuit, θ = 2πft is the angular frequency, and K is the slope factor of the switching current of the main circuit at the turn-off instant. The load current i 1 (θ), the injected current i 2 (θ), and the third harmonic current i 3 (θ) are all sinusoidal and can be written as: where I 1 is the amplitude of the load current, ϕ 1 is the initial phase shift at f , I 2 is the amplitude of the injected current, ϕ 2 is the phase shift of the injected current at 2f , I 3 is the amplitude of the third harmonic current, and ϕ 3 is the phase shift of the third harmonic current at 3f . Using Fourier series expansion, 0 ≤ θ ≤ π can be rewritten as When 0 ≤ θ ≤ π, the current flowing through C s1 can be obtained by using the Kirchhoff's current laws (KCL)) where i cs1 (θ) is the current flowing through C s1 . During the period, the current flowing through the main circuit switch is zero. When π ≤ θ ≤ 2π, with the conduction of the main circuit switch, the switching current can be expressed as Substituting (8) into (2), the following equations can be obtained: The charging and discharging of the shunt capacitor C s1 produce the switching voltage the of the main circuit, which is expressed as Using (1) and (11), the following equations can be obtained Solving the equations (9), (10), (12) and (13) by using the numerical methods, I 1 , I 2 , I 3 , ϕ 1 , ϕ 2 and ϕ 3 can be computed according to the K value. Fig.3 shows the changes of the switch voltage V s1 (θ), the switch current i s1 (θ), the output current i 1 (θ) and the third harmonic circuit current i 3 (θ) at different K values. As can be seen in Fig.3 (a) and (b), for any value of K, the switching voltage of the main circuit can meet the zero voltage on and zero voltage derivative on, and the switching current of the main circuit can also meet the zero current off, which is consistent with the set switching conditions of the circuit.
It is shown in Fig. 3 that V s1 (θ) decreases as K value changes from 1.4 to 1.6 and increases as K value changes from 1.6 to 1.8. The i s1 (θ), i 1 (θ) and i 3 (θ) increase with the increasing of K. Therefore, the following conclusions can be drawn, with the change of K value, the normalized peak switching voltage of the main circuit firstly decreases and then increases, and compared with the normalized peak switching voltage 4.3 of the traditional class E M resonant inverter, the peak switching voltage of class E M /F 3 resonant inverter decreases greatly. For all K values, the ZVS, ZVDS and ZCS conditions are all implemented.
The DC voltage V DD1 of the main circuit is the average value of v s1 (θ),which is given as Substituting (11) into (14), the analytical expression of C s1 is expressed as: + 18πI 1 cos(ϕ 1 ) − 9πI 2 cos(ϕ 2 ) + 6πI 3 cos(ϕ 3 )] Fig. 4 shows the equivalent circuit diagram of the fundamental wave and second harmonic of the class E M /F 3 resonant inverter under sub-optimal working conditions. As shown in Fig.4(a), the impedance of the series resonant circuit at the resonant frequency f 0 is equal to zero. According to Kirchhoff's voltage law, the component of  V s1f 1 (θ) can be expressed as: where V R (θ) is the fundamental component of the voltage across R and V Lx1 (θ) is the fundamental component of the voltage across L x1 , where V R (θ) and V Lx1 (θ) are the fundamental components of the voltage across R and L x1 , which are given as follows: where R is load impedance and X 1 is the inductive reactance of L x1 . By using Fourier series expansion, the V R (θ ) and V Lx1 (θ) can be rewritten as By substituting (11) into (19) and (16), the load impedance R and inductive reactance X 1 can be expressed as Curves of load resistance R and phase-modulated reactance X 1 at different K values. + 4I 2 cos(ϕ 1 ) sin(ϕ 2 )−8I 2 cos(ϕ 2 ) sin(ϕ 1 ) + 4I 3 cos(ϕ 3 ) sin(ϕ 1 )+12I 1 cos(ϕ 1 ) sin(ϕ 1 ) − 6πI DD1 sin(ϕ 1 )] where [6I DD1 sin(ϕ 1 ) + 6I 1 cos(ϕ 1 ) 2 − 4I 2 cos(ϕ 1 ) cos(ϕ 1 )] (23) From (21) and (22), R and X 1 as the function of K are shown in Fig.5. It is seen that R and X 1 decrease as K value changes from 1.3 to 1.7, and increase as K value changes from 1.7 to 1.9. It can be seen from Fig.5 that, under any value of K, the value of the inductance X 1 of the phase-modulated inductance of the main circuit is negative, that is, the phase-modulated inductance L x1 is negative. The negative phase-modulated inductance L x1 means that the resonant inductor L 1 minus L x1 resonates with the resonant capacitor C 1 at the fundamental frequency, which is very important for the calculation of the resonant loop in circuit design.

B. ANALYSIS AND THEORETICAL DERIVATION OF THE AUXILIARY CIRCUIT
The auxiliary circuit can satisfy the ZVS condition when the switch of the auxiliary circuit turns on, which is expressed as v s2 (θ) θ= π 2 = 0 (25) 50018 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
When 0 ≤ θ ≤ π 2, the switch of the auxiliary circuit is in the off-state, so the current flows C s2 can be expressed as The switch voltage of the auxiliary circuit V s2 (θ) can be obtained as Substituting (26) into (27), V s2 (θ) can be rewritten as (28) Fig.6 shows the V s2 (θ) changing curve as the function of K. It is seen that V s2 (θ ) decreases as the increase of K. And for any K value, the V s2 (θ) satisfies ZVS switching condition. In order to use MOSFETs with lower drain voltage tolerance, a larger value is the best choice.
Substituting (19) into (21), the following equation is expressed as Therefore, the DC input current of the auxiliary circuit I DD2 is calculated below.
The input power of the main circuit P in and the total circuit output power P out are expressed as: Substituting (21) into (32), the total output power P out can be rewritten as The output power of the auxiliary circuit P ′ out can be obtained as Then, the DC voltage of the auxiliary circuit V DD2 is expressed as follows and V DD2 is also the average value of v s2 (θ), thus, the expression of V DD2 can be rewritten as where Fig. 7 shows the K value curve of the DC input voltage V DD2 of the auxiliary circuit. As shown in Fig. 7, the input voltage V DD2 shows a trend of first decreasing and then increasing with the increase of K value, and reaches the minimum value of 5.14 V when K =1.7.
By combining formulas (35) and (36), a system of equations is established, in which there is only one unknown parameter C s2 . By solving the equation, the expression of parallel capacitor C s2 of the auxiliary circuit can be obtained: Fig.4(b) illustrates that the series resonant impedance of the auxiliary circuit is equal to zero at the resonant frequency 2f 0 . Using the Kirchhoff's voltage laws (KVL) for the switch voltage of the main circuit and the auxiliary circuit, which is given as follows where V Lx2 (θ) is the voltage across L x2 ,and V Lx2 (θ), and can be rewritten as Integrating both sides of equations (39) at the same time Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. The integral in (41) is calculated as follows where Therefore, L x2 can be calculated as The power efficiency of the whole circuit can be written as follows The output power capability of the designed circuit is given as where V S1 max and V S2 max are the peak switch voltage of the main and auxiliary circuit, i S1 max and i S2 max are the peak switch current of the main and auxiliary circuit. Fig. 8 shows the curves of the normalized peak switching voltage of the main circuit V S1 max V DD1 and the normalized peak switching voltage of the auxiliary circuit concerning the different K values. As can be seen from Fig. 8, with the increase of K value, V S1 max V DD1 and V S2 max V DD1 show a trend of first decreasing and then increasing, and V S1 max gets the minimum between K equals 1.6 and 1.7, and V S2 max gets the minimum between K equals 1.7 and 1.8. Therefore, in order to reduce the voltage stress of the switching transistor of the main circuit and the auxiliary circuit, an appropriate K value should be chosen between 1.5 and 1.8. In addition, compared with the traditional class E M resonant inverter, the class E M /F 3 resonant inverter designed in this paper not only reduces the peak switching voltage of the main circuit but also reduces the peak switching voltage of the auxiliary circuit, further reducing the on-off loss of transistors in the actual circuit. Fig.9 the variation of C p as a function of K , where the waveform of Cp is an obvious quadratic function. When the K value is set from 1.3 to 1.6, the Cp value increases continuously. When the K value is set from 1.6 to 1.9, the Cp value decreases continuously. When K =1.6, the Cp value reaches the maximum value of 0.0605. And the larger the C p , the greater the maximum output power of the inverter.

III. DESIGN AND EXPERIMENTAL RESULTS
In order to verify the analysis derived in Section II, a low peak switching voltage and high-efficiency Class-E M /F 3 inverter was designed and simulated in Cadence OrCAD. The design specifications were set as V DD1 =10 V, I DD1 =1.0 A, K = 1.6 and the fundamental frequency of the proposed inverter was 1.0 MHz. According to the given circuit design parameters, it can be obtained that the peak switching voltage of the main circuit and the auxiliary circuit is 36.48 V and 36.72 V respectively. Therefore, the maximum drain-source voltage that the selected switching transistor should be greater than 36.72 V least, and it has better operating characteristics at the operating frequency of 1MHz. Therefore, this design selects the IRFZ24N model MOSFET of IR Company, which can withstand the maximum drain-source voltage of 55 V. In addition, the drain-source conduction resistance of the IRFZ24N transistor at the working frequency of 1 MHz is only 70 milliohms, and the on-off loss generated in the actual circuit is very small, so it is very suitable for this design. The gate driver chip is ADP3624 from ADI company. The ADP3624 is a high-current, dual-channel high-speed driver capable of producing drive voltages from 4.5 V to 18 V, up and down in just 10ns, and can drive two MOSFETs simultaneously. Normally, the gate-source voltage of the IRFZ24N is about 10V, so the driver chip ADP3624 can effectively drive the used transistor.
As the values of components need to be adjusted slightly in the test process, especially in the resonant loop, the inductance part of this design adopts a self-made hand-wound iron powder core inductance coil, in which the iron powder core is the annular iron powder core of Micrometals Company. In addition to the convenience of adjusting the inductance value, the parasitic resistance of the ring inductance coil with an iron powder core is smaller and the power loss is lower. The frequency characteristic of the inductance coil was measured by the vector network analyzer, and the number of turns of the copper wire wound on the iron core was adjusted, so that the inductance coil reached the required sensing value and had a low parasitic resistance at the working frequency of 1MHz. The capacitor part selects the ATC100B series chip capacitor, which can be fine-tuned to the resonant loop and parallel capacitor by paralleling several small capacitances.
By drawing PCB schematic diagram, processing the PCB board, winding the inductor coil, welding components and other processes, the production of the E M /F 3 soft-switching resonant inverter is completed. The finished E M /F 3 softswitching resonant inverter and its test platform are shown in Fig. 10. And in Fig. 11, the photographic of the measured circuit is presented. The test platform mainly includes the IDP-3305SLU DC power supply produced by Interrock Company, the DG4102 signal generator produced by Puyuan Company and the TDS3052C oscilloscope produced by Tektronek Company. The IDP-3305SLU DC power supply is a dual output power supply, which can provide DC input signals for the main circuit and the auxiliary circuit at the same time, and can directly read the amplitude of the input  voltage and current through the LCD screen on the power supply. The DG4102 signal generator has two independent output channels, and the phase between the two channels can be accurately adjusted, so the signal generator can provide the two gate driver chips with a turn-off duty ratio of 50% and 25% pulse signals and keep the phase consistent. The TDS3052C oscilloscope has 500MHz bandwidth and up to 5GS/s sampling rate, and can accurately collect the voltage and current signals in the circuit; The oscilloscope has two sampling channels, which can verify whether the main circuit and the auxiliary circuit realize the same phase. In addition, the oscilloscope also has a built-in floppy drive for data storage, which can easily export the measured waveform data.
The theoretically calculated values, PSpice circuit simulation values and experimental test values for the proposed circuit are shown in Table 1. It can be clearly seen from Table 1 that the theoretical normalized peak switching voltage, simulated normalized peak switching voltage and tested normalized peak switching voltage of the main circuit of the E M /F 3 soft-switching resonant inverter are 3.04 V, 3.03 V and 3.04 V respectively. Compared with the EM inverter, the peak switching voltage is reduced by 29.3%. The theoretical, simulation and test normalized peak switching voltage of the auxiliary circuit is 3.06 V, 3.08 V and 3.12 V, respectively, which is reduced by 15.6% compared with the class E M inverter. In addition, because the designed class E M /F 3 resonant inverter can realize the zero voltage turn-on, zero voltage derivative turn-on, zero current turn-on of the main circuit and zero voltage turn-on of the auxiliary circuit, the theoretical efficiency, simulation efficiency and test efficiency of the circuit reach 99.6%, 99.3% and 96.4% respectively, which realizes the low loss working state. Finally, through Cp value obtained by the circuit theory, simulation and test results, it can be found that the circuit has a large power output capacity. As shown in Table 1, the theoretical calculation results of each parameter are in good agreement with the simulation results and test results, which proves the rationality and correctness of the theoretical derivation. Fig. 12 shows the voltage waveform comparison of theoretical expression, Pspice simulation and waveform obtained from experimental tests of the designed class E M /F 3 softswitched resonant inverter, including grid voltage V gs , main circuit switching voltage V s1 , auxiliary circuit switching voltage V s2 and load voltage V R . As can be seen from the figure, both the simulation results and test results of the main circuit switching voltage V s1 achieve zero voltage switching and zero voltage derivative switching, the auxiliary circuit switching voltage V s2 also achieve zero voltage switching, and the output voltage V R is the standard sinusoidal signal. The experimental results confirm the theoretical results. Table 2 shows the comparison between the presented circuit with those given in some published references. By comparison, the peak switching voltage of the main circuit for the presented Class-E M /F 3 inverter is lower than other proposed Class-E M inverter and the efficiency for the Class-E M /F 3 inverter is higher than other proposed Class-E M inverter. In [21], the simulated results are given in Fig. 10, which shows that the normalized peak transistor voltage is 3.25 and 3.6. The suboptimum operation of the class E M /F 3 can  provide a better performance. In overall, it is seen that the proposed circuit has better peak switching voltage and efficiency compared with those in the references.

IV. CONCLUSION
In this paper, the complete design method and accurate analytical expressions for the suboptimum Class-E M /F 3 resonant inverter are presented. Compared with the conventional Class-E M inverter, the advantages of the proposed inverter lie in its lower peak switching voltage and higher efficiency while achieving the soft-switching characteristics. The peak switching voltage of the main circuit is only 3.04 times of V DD1 which was 41.4% lower than the conventional Class-E M circuit. By using the proposed design method, not only the ZVS/ZVDS/ZCS conditions of the main circuit, but also the ZVS condition of the auxiliary circuit can be satisfied without using any tuning process. The analytical prediction results agree well with the PSpice simulation and experimental results, which verify the validity of the proposed design method in this paper.