Parameter Analysis and Optimization of Class-E Power Ampliﬁer Used in Wireless Power Transfer System

: In this paper, a steady-state matrix analysis method is introduced to analyze the output characteristics of the class-E power ampliﬁer used in a wireless power transfer (WPT) system, which takes the inductance resistance, on-resistance and leakage current of metal-oxide-semiconductor ﬁeld e ﬀ ect transistor (MOSFET) into account so that the results can be closer to the actual value. On this basis, the parameters of the class-E power ampliﬁer are optimized, and the output power is improved under the premise of keeping the e ﬃ ciency unchanged. Finally, the output characteristics of the ampliﬁer before and after optimization are compared by an experiment, while the B-ﬁeld strength around the WPT system is studied through simulation. The experimental results verify the correctness and feasibility of the optimization method based on steady-state matrix analysis.


Introduction
The Class-E power amplifier is widely used in high-frequency power supply, wireless power transfer (WPT) and other fields because of its simple structure and high output efficiency [1,2]. In 2007, the research team of Massachusetts Institute of Technology (MIT) put forward the magnetic coupling resonant wireless power transfer technology, and the class-E power amplifier has once again become a hot research topic at home and abroad [3].
MOSFET can meet the zero-voltage switching (ZVS) and zero-derivative switching (ZDS) conditions when the class-E power amplifier works under the ideal condition and the efficiency is 100% [4]. According to the parametric characters of the class-E power amplifier, the authors studied the changes in output characteristics of class-E power amplifiers when the load deviated from the optimal value in [5]. The relationship between the DC power supply of a class-E power amplifier and MOSFET's peaks voltage is studied in [6]. The effect of various parameters of a class-E power amplifier on the output characteristics of the circuit, the voltage and the current waveform of the MOSFET are analyzed in [7]. T. Mury and his team conducted an in-depth study on the operating characteristics of class-E power amplifiers in the sub-optimal working state where the duty cycle of the MOSFET is not 50%, the mathematical modeling of the class-E power amplifier is carried out, and the influence of duty cycle on the current peak, output voltage and current is analyzed [8]. The analysis of a class-E power amplifier based on a lossless switch and ratio-frequency (RF) choke (RFC) was introduced in [9]. In [10] and [11], researchers used resonant soft-switching converters to achieve optimum switching conditions. Class-EF inverters and the equations of the voltages and currents were derived with traditional analysis in [12]. In [13], a novel topology of the Class-E M power amplifier was proposed based on a finite direct current (DC)-feed inductance and an isolation circuit. R. A. Beltran et al., proposed a simplified analysis and design of class-E outphasing transmitters that predicts efficiency and output power as a function of input drive phase difference based on classic design equations of the class-E amplifier [14]. However, the analytical processes of the traditional class-E power amplifier circuit above are ideal, generally ignoring the MOSFET's turn-on resistance and considering the turn-off resistance to be infinite. In other words, analysis of every component in load network is based on the complete ideal condition such as no impedance [15]. There is a certain difference between the theoretical result and the actual value where the performance of the class-E power amplifier can be significantly influenced by the non-ideal factor such as the inductance resistance, finite dc-feed inductance, leakage current and so on [16].
In this paper, a steady-state matrix analysis method which proved to be a fast and effective approach for optimization of switching-mode power amplifiers [17][18][19][20] is used for studying the output characteristics of two kinds of class-E power amplifiers (a traditional choke, RFC type [21], parallel load type, parallel capacitor and inductor (PCL) type [22]) under the consideration of non-ideal factors, such as the inductance resistance, leakage current and so on. On the basis of inductance of phase angle compensation L x (L in ), the influence of the parallel resonant capacity C p and the load R L on the output power, the operation efficiency and the maximum withstand voltage of MOSFET is turned off, two kinds of class-E power amplifiers are optimized. The output power of the class-E power amplifier is improved on ensuring output efficiency and the effectiveness of optimization of class-E power amplifier circuit based on steady-state matrix analysis is verified according to the experiments. The B-field strength around the WPT system is also simulated and studied.

Application of Steady-State Matrix Analysis Method
The equivalent circuit of the traditional class-E power amplifier is shown in Figure 1. amplifier was proposed based on a finite direct current (DC)-feed inductance and an isolation circuit. R. A. Beltran et al., proposed a simplified analysis and design of class-E outphasing transmitters that predicts efficiency and output power as a function of input drive phase difference based on classic design equations of the class-E amplifier [14]. However, the analytical processes of the traditional class-E power amplifier circuit above are ideal, generally ignoring the MOSFET's turn-on resistance and considering the turn-off resistance to be infinite. In other words, analysis of every component in load network is based on the complete ideal condition such as no impedance [15]. There is a certain difference between the theoretical result and the actual value where the performance of the class-E power amplifier can be significantly influenced by the non-ideal factor such as the inductance resistance, finite dc-feed inductance, leakage current and so on [16].
In this paper, a steady-state matrix analysis method which proved to be a fast and effective approach for optimization of switching-mode power amplifiers [17][18][19][20] is used for studying the output characteristics of two kinds of class-E power amplifiers (a traditional choke, RFC type [21], parallel load type, parallel capacitor and inductor (PCL) type [22]) under the consideration of nonideal factors, such as the inductance resistance, leakage current and so on. On the basis of inductance of phase angle compensation Lx (Lin), the influence of the parallel resonant capacity Cp and the load RL on the output power, the operation efficiency and the maximum withstand voltage of MOSFET is turned off, two kinds of class-E power amplifiers are optimized. The output power of the class-E power amplifier is improved on ensuring output efficiency and the effectiveness of optimization of class-E power amplifier circuit based on steady-state matrix analysis is verified according to the experiments. The B-field strength around the WPT system is also simulated and studied.

Application of Steady-State Matrix Analysis Method
The equivalent circuit of the traditional class-E power amplifier is shown in Figure 1. The average value of the voltage on the MOSFET during one period is equal to the DC power supply, VDD: In the ideal case, the efficiency of the Class-E power amplifier is 100%, where the active power absorbed by the load is equal to the input provided by the DC power supply: The voltage applied to RL and Lx is the fundamental frequency voltage, which can be obtained by Fourier analysis: The average value of the voltage on the MOSFET during one period is equal to the DC power supply, V DD : In the ideal case, the efficiency of the Class-E power amplifier is 100%, where the active power absorbed by the load is equal to the input provided by the DC power supply: The voltage applied to R L and L x is the fundamental frequency voltage, which can be obtained by Fourier analysis: The parameter calculation formula of the class-E power amplifier load network can be derived from (4): The maximum withstand voltage on the MOSFET is: MOSFET's turn-on resistance, leakage current and inductance resistance in the load network are considered in the circuit diagram of a class-E power amplifier in Figure 2. MOSFET's loss can be divided into the on-resistance loss and the leakage current loss in turn-off state. Inductance resistance comes from choke (parallel inductor) L in resonant inductor L 0 and inductance of phase angle compensation L x .
The parameter calculation formula of the class-E power amplifier load network can be derived from (4): The maximum withstand voltage on the MOSFET is: MOSFET's turn-on resistance, leakage current and inductance resistance in the load network are considered in the circuit diagram of a class-E power amplifier in Figure 2. MOSFET's loss can be divided into the on-resistance loss and the leakage current loss in turn-off state. Inductance resistance comes from choke (parallel inductor) Lin resonant inductor L0 and inductance of phase angle compensation Lx. The state variables in the diagram is defined as the following matrix: Iin(t) is the input current of DC power supply, Iout(t) is the output current, VDS(t) is drain-source voltage, VC0(t) is the voltage of the resonant capacitor. They are energy storage variables in the circuit, Therefore, the switching state of the MOSFET will not cause changes immediately, which meets the requirement of steady-state analysis. The state equation of class-E power amplifier circuit can be obtained by the first derivative of the state variable. When MOSFET is turned off, the equation of state is: Changing Ileak to VDS/ron in (9), the equation of state when MOSFET is turned on is obtained. All the equations of state have the form of first-order differential equations: The state variables in the diagram is defined as the following matrix: I in (t) is the input current of DC power supply, I out (t) is the output current, V DS (t) is drain-source voltage, V C0 (t) is the voltage of the resonant capacitor. They are energy storage variables in the circuit, Therefore, the switching state of the MOSFET will not cause changes immediately, which meets the requirement of steady-state analysis.
The state equation of class-E power amplifier circuit can be obtained by the first derivative of the state variable. When MOSFET is turned off, the equation of state is: Changing I leak to V DS /r on in (9), the equation of state when MOSFET is turned on is obtained. All the equations of state have the form of first-order differential equations: The state of matrix A and matrix B have different forms when the MOSFET is turned off and turned on. For the convenience of analysis, the state matrix when MOSFET is turned off is defined as A 1 and B 1 , the off-time is t 1 . Moreover, the state matrix when MOSFET is turned on is defined as A 2 and B 2 , and the on-time is t 2 . The solution of (10) is: q 0 is the initial value. The solution of the state equation is: Because of the energy properties of the state variable, there is no immediate change of state variable when MOSFET works from turn-off to turn-on. As the MOSFET is turned off, the state variable at t 1 time is equal to the value at t = 0 that is q 01 . The following equations can be derived: The initial value q 01 , q 02 can be obtained in (14) and (15), losses and output power can be defined: Output power: Loss of the inductor resistance: Loss of MOSFET in turn-on and turn-off state are shown as follows: Energies 2019, 12, 3240

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The output power of class-E power amplifier, efficiency and drain-source voltage of MOSFET can be obtained according to (16)-(25).

Analysis of the Characteristic of Class-E Power Amplifier
Based on the steady-state matrix analysis method above, the effect of the inductor phase angle compensation L x (L in ) and parallel capacitance C p on the output characteristics of two kinds of class-E power amplifier can be studied. The theoretical parameter values of two types with a DC power supply of 25 V, working at 6 MHz and output power of 30 W are shown in Table 1. R L-og , L X_og , L 0_og , and C 0_og are theoretical values of load resistance, inductance of phase angle compensation, inductance and capacitance of resonant circuit. L in_og and C p_og are theoretical values of choke (parallel) inductance and capacitance. r LX , r Lin , and r L0 are resistance values of the inductors.
In order to analyze the effect of L x (L in ) and C p on the output characteristics of class-E power amplifier, the surface graph of its parametric characters is obtained based on the steady-state matrix method. In the following analysis, MRF6V2150NBR1 is used so that MOSFET's turn-on resistance R on is 0.3 Ω, the leakage current I leak is 2.5 mA, and drain source maximum withstand voltage is 110 V.

Influence of RFC Output Characteristics
The effect of phase angle compensation inductance L x which is parallel with capacitance C p and the load R L on output power, efficiency and MOSFET's peak voltage in RFC type is shown in Figures 3 and 4. The variation range of L x is from 0.27 µH to 0.47 µH, C p is from 300 pF to 500 pF and R L is from 8 Ω to 16 Ω. If MOSFET's peaks voltage exceeds the maximum withstand voltage, it will cause damage to MOSFET. Therefore, the maximum withstand voltage should also be considered in designing a class-E power amplifier.
The output power of class-E power amplifier, efficiency and drain-source voltage of MOSFET can be obtained according to (16)-(25).

Analysis of the Characteristic of Class-E Power Amplifier
Based on the steady-state matrix analysis method above, the effect of the inductor phase angle compensation Lx (Lin) and parallel capacitance Cp on the output characteristics of two kinds of class-E power amplifier can be studied. The theoretical parameter values of two types with a DC power supply of 25 V, working at 6 MHz and output power of 30 W are shown in Table 1. RL-og, LX_og, L0_og, and C0_og are theoretical values of load resistance, inductance of phase angle compensation, inductance and capacitance of resonant circuit. Lin_og and Cp_og are theoretical values of choke (parallel) inductance and capacitance. rLX, rLin, and rL0 are resistance values of the inductors.

Parameter
RFC Type PCL Type In order to analyze the effect of Lx (Lin) and Cp on the output characteristics of class-E power amplifier, the surface graph of its parametric characters is obtained based on the steady-state matrix method. In the following analysis, MRF6V2150NBR1 is used so that MOSFET's turn-on resistance Ron is 0.3 Ω, the leakage current Ileak is 2.5 mA, and drain source maximum withstand voltage is 110 V.

Influence of RFC Output Characteristics
The effect of phase angle compensation inductance Lx which is parallel with capacitance Cp and the load RL on output power, efficiency and MOSFET's peak voltage in RFC type is shown in Figures  3 and 4. The variation range of Lx is from 0.27 μH to 0.47 μH, Cp is from 300 pF to 500 pF and RL is from 8 Ω to 16 Ω. If MOSFET's peaks voltage exceeds the maximum withstand voltage, it will cause damage to MOSFET. Therefore, the maximum withstand voltage should also be considered in designing a class-E power amplifier.  With the increase of Lx and Cp, the output power of the RFC type decreases significantly, the efficiency increases firstly and then decreases. The trend of output efficiency is opposite to the trend of output power, and the influence of the two factors should be considered comprehensively in the design of circuit parameters. Moreover, Lx has little effect on the maximum voltage of MOSFET, and its value decreases mainly with the increase of Cp, which can be obtained from Figure 3.  It can be seen from Figure 4 that the output power of the class-E power amplifier decreases while the efficiency increases with the increase of RL. The effect of load RL on output power is obvious, while the trend of the output efficiency is opposite to that of the output power. On the other hand, Lx has less influence on the maximum voltage of MOSFET than RL. However, within the variation range, With the increase of Lx and Cp, the output power of the RFC type decreases significantly, the efficiency increases firstly and then decreases. The trend of output efficiency is opposite to the trend of output power, and the influence of the two factors should be considered comprehensively in the design of circuit parameters. Moreover, Lx has little effect on the maximum voltage of MOSFET, and its value decreases mainly with the increase of Cp, which can be obtained from Figure 3.  It can be seen from Figure 4 that the output power of the class-E power amplifier decreases while the efficiency increases with the increase of RL. The effect of load RL on output power is obvious, while the trend of the output efficiency is opposite to that of the output power. On the other hand, Lx has less influence on the maximum voltage of MOSFET than RL. However, within the variation range, With the increase of L x and C p , the output power of the RFC type decreases significantly, the efficiency increases firstly and then decreases. The trend of output efficiency is opposite to the trend of output power, and the influence of the two factors should be considered comprehensively in the design of circuit parameters. Moreover, L x has little effect on the maximum voltage of MOSFET, and its value decreases mainly with the increase of C p , which can be obtained from Figure 3.
It can be seen from Figure 4 that the output power of the class-E power amplifier decreases while the efficiency increases with the increase of R L . The effect of load R L on output power is obvious, while the trend of the output efficiency is opposite to that of the output power. On the other hand, L x has less influence on the maximum voltage of MOSFET than R L . However, within the variation range, Energies 2019, 12, 3240 7 of 13 the voltage peak across the MOSFET is smaller than its maximum value, ensuring that the MOSFET can work normally without being broken down.

Influence of PCL Characteristics
The effect of phase angle compensation inductance L in on output power, efficiency and MOSFET peaks voltage in PCL type is shown in Figure 5. The variation range of L in is from 0.45 µH to 0.65 µH, C p is from 550 pF to 750 pF.
Energies 2019, 12, x FOR PEER REVIEW 7 of 13 the voltage peak across the MOSFET is smaller than its maximum value, ensuring that the MOSFET can work normally without being broken down.

Influence of PCL Characteristics
The effect of phase angle compensation inductance Lin on output power, efficiency and MOSFET peaks voltage in PCL type is shown in Figure 5. The variation range of Lin is from 0.45 μH to 0.65 μH, Cp is from 550 pF to 750 pF. With the increase of Lin and Cp, the output power of the PCL type also decreases as shown in Figure 5a; when the Cp value is fixed, the efficiency increases firstly and then decreases with the increase of Lin, when the Lin value is small, the efficiency increases with the increase of Cp, while the Lin value becomes larger, the efficiency gradually turns to decrease with the increase of Cp as shown in Figure 5b; the maximum withstand voltage of PCL type decreases significantly with the increase of Lin and Cp as shown in Figure 5c. When the Lin and Cp values are too small, the maximum withstand voltage will exceed 120V, which will easily cause damage to the MOSFET.

Optimization Strategy of Two Kinds of Class-E Power Amplifier
According to the analysis above, due to the existence of inductance resistance and non-ideal MOSFET, the designed circuit parameters are no longer optimal under ideal condition. The model of class-E power amplifiers using state equations is closer to the real situation. For the design of a broadband class-E power amplifier, if the reactance corresponding to the phase shifting inductor Lx is regarded as part of the load impedance, the circuit can be optimized overall by adjusting the phase shifting inductor Lx (Lin), Cp and RL under the condition of working at 6 MHz and output power of 30 W. With the increase of L in and C p , the output power of the PCL type also decreases as shown in Figure 5a; when the C p value is fixed, the efficiency increases firstly and then decreases with the increase of L in , when the L in value is small, the efficiency increases with the increase of C p , while the L in value becomes larger, the efficiency gradually turns to decrease with the increase of C p as shown in Figure 5b; the maximum withstand voltage of PCL type decreases significantly with the increase of L in and C p as shown in Figure 5c. When the L in and C p values are too small, the maximum withstand voltage will exceed 120 V, which will easily cause damage to the MOSFET.

Optimization Strategy of Two Kinds of Class-E Power Amplifier
According to the analysis above, due to the existence of inductance resistance and non-ideal MOSFET, the designed circuit parameters are no longer optimal under ideal condition. The model of class-E power amplifiers using state equations is closer to the real situation. For the design of a broadband class-E power amplifier, if the reactance corresponding to the phase shifting inductor L x is regarded as part of the load impedance, the circuit can be optimized overall by adjusting the phase shifting inductor L x (L in ), C p and R L under the condition of working at 6 MHz and output power of 30 W. The RFC type parameters before and after optimization are shown in Table 2. The ideal values are calculated according to (1)- (7). As we select L x = 0.34 µH, C p = 0.38 nF, R L = 9.12 Ω according to Figures 3 and 4, the output efficiency of the optimized class-E power amplifier has no obvious change, the output power is increased by 9.25% (2.6 W). Moreover, the maximum withstand voltage of MOSFET is slightly increased in turn-off stage. The PCL type parameters before and after optimization are shown in Table 3. Select L in = 0.54 µH, C p = 0.6 nF. Under the condition that the output efficiency of the optimized class-E power amplifier has no obvious change, the output power is increased by 7.61% (2.02 W). Moreover, the maximum withstand voltage of MOSFET in turn-off stage is increased significantly, still within the safe range.

Experimental Results and Analysis
Comparing the output character of two kinds of class-E power amplifier before and after optimization, the validity of the optimization method is verified using the parameters obtained from the analysis. The experimental platform is built as shown in Figure 6.
The RFC type parameters before and after optimization are shown in Table 2. The ideal values are calculated according to (1)-(7). As we select Lx = 0.34 μH, Cp = 0.38 nF, RL = 9.12 Ω according to Figures 3 and 4, the output efficiency of the optimized class-E power amplifier has no obvious change, the output power is increased by 9.25% (2.6 W). Moreover, the maximum withstand voltage of MOSFET is slightly increased in turn-off stage. The PCL type parameters before and after optimization are shown in Table 3. Select Lin = 0.54 μH, Cp = 0.6 nF. Under the condition that the output efficiency of the optimized class-E power amplifier has no obvious change, the output power is increased by 7.61% (2.02 W). Moreover, the maximum withstand voltage of MOSFET in turn-off stage is increased significantly, still within the safe range.

Experimental Results and Analysis
Comparing the output character of two kinds of class-E power amplifier before and after optimization, the validity of the optimization method is verified using the parameters obtained from the analysis. The experimental platform is built as shown in Figure 6.

Experiments of RFC Type under Non-Ideal Condition
The experimental waveforms of RFC type before and after optimization is shown in Figure 7. Before optimization, the RMS of the output current, the output voltage and the output power of RFC

Experiments of RFC Type under Non-Ideal Condition
The experimental waveforms of RFC type before and after optimization is shown in Figure 7. Before optimization, the RMS of the output current, the output voltage and the output power of RFC type were 1.48 A, 18.8 V and 27.82 W. The operational efficiency was 85.24%. After optimization, the RMS of the output current, the output voltage and the output power became 1.56 A, 19.3 V and 30.11 W. The operational efficiency was 84.76%.
Under the condition that the efficiency has no obvious change, the output power is increased by 2.29 W (8.23%). The maximum withstand voltage of MOSFET in turn-off stage increases from 90.4 V to 92.8 V, which is still within the safe range. Moreover, due to the small voltage oscillation in turn-on stage which increases the loss of the MOSFET, the experimental output efficiency is smaller than the theoretical value.

Experiments of PCL Type under Non-Ideal Condition
The experimental waveforms of PCL type before and after optimization are shown in Figure 8. Before optimization, the RMS of the output current of PCL type was 0.97 A, the output voltage was 26.8 V and the output power was 25.99 W; the efficiency was 85.57%. After optimization, the RMS of the output current of RFC type was 1.0 A, the output voltage was 27.9 V, and the output power was 27.90 W; the efficiency was 85.84%. Under the condition that the output efficiency had no obvious change, the output power increased by 1.91 W (7.35%). The maximum withstand voltage of MOSFET was increased from 88.8 V to 95.2 V, but still within the safe range. Under the condition that the efficiency has no obvious change, the output power is increased by 2.29 W (8.23%).
The maximum withstand voltage of MOSFET in turn-off stage increases from 90.4 V to 92.8 V, which is still within the safe range. Moreover, due to the small voltage oscillation in turn-on stage which increases the loss of the MOSFET, the experimental output efficiency is smaller than the theoretical value.

Experiments of PCL Type under Non-Ideal Condition
The experimental waveforms of PCL type before and after optimization are shown in Figure 8. Before optimization, the RMS of the output current of PCL type was 0.97 A, the output voltage was 26.8 V and the output power was 25.99 W; the efficiency was 85.57%. After optimization, the RMS of the output current of RFC type was 1.0 A, the output voltage was 27.9 V, and the output power was 27.90 W; the efficiency was 85.84%. Under the condition that the output efficiency had no obvious change, the output power increased by 1.91 W (7.35%). The maximum withstand voltage of MOSFET was increased from 88.8 V to 95.2 V, but still within the safe range.

Simulation Result of B-Field Strength
A simulation model of wireless power transfer system was established according to the experimental platform as shown in Figure 9. Four cases are considered in this paper: wireless power transfer system with RFC-type before optimization (Case 1); wireless power transfer system with RFC-type after optimization (Case 2); wireless power transfer system with PCL-type before optimization (Case 3); wireless power transfer system with PCL-type after optimization (Case 4). B-field strength results of the four cases are shown in Figure 10.

Simulation Result of B-Field Strength
A simulation model of wireless power transfer system was established according to the experimental platform as shown in Figure 9.

Simulation Result of B-Field Strength
A simulation model of wireless power transfer system was established according to the experimental platform as shown in Figure 9. Four cases are considered in this paper: wireless power transfer system with RFC-type before optimization (Case 1); wireless power transfer system with RFC-type after optimization (Case 2); wireless power transfer system with PCL-type before optimization (Case 3); wireless power transfer system with PCL-type after optimization (Case 4). B-field strength results of the four cases are shown in Figure 10. Four cases are considered in this paper: wireless power transfer system with RFC-type before optimization (Case 1); wireless power transfer system with RFC-type after optimization (Case 2); wireless power transfer system with PCL-type before optimization (Case 3); wireless power transfer system with PCL-type after optimization (Case 4). B-field strength results of the four cases are shown in Figure 10.
For further research on the B-field strength around the wireless power transfer system, we studied the B-field strength along the y-axis and the z-axis both from the specified point (0.0.3 m, 0.25 m) which was 0.1 m to the edge of the coils using the coordinate system in Figure 9.
The trends of the B-field strength along two lines in four cases are shown in Figure 11. 'I' represents B-field strength in case 1 along the y-axis; 'II' represents B-field strength in case 1 along the z-axis; 'III' represents B-field strength in case 2 along the y-axis; 'IV' represents B-field strength in case 2 along the z-axis; 'V' represents B-field strength in case 3 along the y-axis; 'VI' represents B-field strength in case 3 along the z-axis; 'VII' represents B-field strength in case 4 along the y-axis; 'VIII' represents B-field strength in case 4 along the z-axis. B-field strength around the system with the RFC-type is stronger than that with the PCL-type, while the B-field strength after optimization was stronger than that before optimization. In general, the B-field strength in the four cases was always less than 27 µT which is the reference level for general public exposure formulated by ICNIRP-2010. For further research on the B-field strength around the wireless power transfer system, we studied the B-field strength along the y-axis and the z-axis both from the specified point (0.0.3 m, 0.25 m) which was 0.1 m to the edge of the coils using the coordinate system in Figure 9.
The trends of the B-field strength along two lines in four cases are shown in Figure 11. 'I' represents B-field strength in case 1 along the y-axis; 'II' represents B-field strength in case 1 along the z-axis; 'III' represents B-field strength in case 2 along the y-axis; 'IV' represents B-field strength in case 2 along the z-axis; 'V' represents B-field strength in case 3 along the y-axis; 'VI' represents B-field strength in case 3 along the z-axis; 'VII' represents B-field strength in case 4 along the y-axis; 'VIII' represents B-field strength in case 4 along the z-axis. B-field strength around the system with the RFCtype is stronger than that with the PCL-type, while the B-field strength after optimization was stronger than that before optimization. In general, the B-field strength in the four cases was always less than 27 μT which is the reference level for general public exposure formulated by ICNIRP-2010. For further research on the B-field strength around the wireless power transfer system, we studied the B-field strength along the y-axis and the z-axis both from the specified point (0.0.3 m, 0.25 m) which was 0.1 m to the edge of the coils using the coordinate system in Figure 9.
The trends of the B-field strength along two lines in four cases are shown in Figure 11. 'I' represents B-field strength in case 1 along the y-axis; 'II' represents B-field strength in case 1 along the z-axis; 'III' represents B-field strength in case 2 along the y-axis; 'IV' represents B-field strength in case 2 along the z-axis; 'V' represents B-field strength in case 3 along the y-axis; 'VI' represents Bfield strength in case 3 along the z-axis; 'VII' represents B-field strength in case 4 along the y-axis; 'VIII' represents B-field strength in case 4 along the z-axis. B-field strength around the system with the RFC-type is stronger than that with the PCL-type, while the B-field strength after optimization was stronger than that before optimization. In general, the B-field strength in the four cases was always less than 27 μT which is the reference level for general public exposure formulated by ICNIRP-2010. Figure 11. The trends of the B-field strength along two lines in four cases. Figure 11. The trends of the B-field strength along two lines in four cases.

Conclusions
In this paper, a steady-state matrix analysis method suitable for class-E power amplifier was introduced. Compared with the analysis process of the working principle of a class-E power amplifier, non-ideal factors are considered in this method, such as inductance resistance and leakage current, etc. The experimental results are more accurate. Based on steady-state matrix analysis, the output characteristic of two kinds of class-E power amplifier circuits were analyzed and optimized. The experimental results show that the output power of the two types of class-E power amplifier increased by 2.29 W (8.23%) and 1.91 W (7.35%), respectively, while the output efficiency had no obvious change. The B-field strength of the systems with two types of class-E power amplifier before and after optimization meets the ICNIRP-2010 standard.