**1. 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 e fficiency [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 e fficiency 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 e ffect 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 e fficiency and output power as a function of input drive phase di fference 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-o ff 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 di fference 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 e ffective approach for optimization of switching-mode power amplifiers [17–20] 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 *Lx* (*Lin*), the influence of the parallel resonant capacity *Cp* and the load *RL* on the output power, the operation e fficiency and the maximum withstand voltage of MOSFET is turned <sup>o</sup>ff, two kinds of class-E power amplifiers are optimized. The output power of the class-E power amplifier is improved on ensuring output e fficiency and the e ffectiveness 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.

#### **2. Application of Steady-State Matrix Analysis Method**

The equivalent circuit of the traditional class-E power amplifier is shown in Figure 1.

**Figure 1.** Class-E model under ideal condition.

The average value of the voltage on the MOSFET during one period is equal to the DC power supply, *VDD*:

$$V\_{DD} = \frac{1}{2\pi} \int\_0^{2\pi} V\_{DS}(\omega t) d\omega t = \frac{\pi I\_{in}}{a\mathcal{C}\_P} \tag{1}$$

In the ideal case, the e fficiency 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:

$$I\_{in}V\_{DD} = \frac{I\_{RL}^2}{2}R\_L\tag{2}$$

$$P\_{\vartheta} = 0.5768 \frac{V\_{DD}^2}{R\_L} \tag{3}$$

The voltage applied to *RL* and *Lx* is the fundamental frequency voltage, which can be obtained by Fourier analysis:

$$\begin{cases} \begin{array}{l} V\_{\mathbb{R}} = -\frac{1}{\pi} \int\_{0}^{2\pi} V\_{\text{DS}}(\omega t) \sin(\omega t) d\omega t \\\ V\_{L\mathbb{x}} = -\frac{1}{\pi} \int\_{0}^{2\pi} V\_{\text{DS}}(\omega t) \cos(\omega t) d\omega t \end{array} \end{cases} \tag{4}$$

The parameter calculation formula of the class-E power amplifier load network can be derived from (4):

$$\frac{\omega L\_x}{R\_L} = 1.1525\tag{5}$$

$$
\mu \mathbb{C}\_P R\_L = 0.1836\tag{6}
$$

The maximum withstand voltage on the MOSFET is:

$$V\_{ds\\_max} = 3.562 V\_{DD} \tag{7}$$

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-o ff state. Inductance resistance comes from choke (parallel inductor) *Lin* resonant inductor *L*0 and inductance of phase angle compensation *Lx*.

**Figure 2.** Class-E model under non-ideal condition.

The state variables in the diagram is defined as the following matrix:

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

$$q(t) = \begin{bmatrix} I\_{in}(t) & I\_{out}(t) & V\_{DS}(t) & V\_{CO}(t) \end{bmatrix}^{\mathrm{T}},\tag{8}$$

*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 o ff, the equation of state is:

$$\begin{cases} L\_{in}\frac{dl\_{in}(t)}{dt} = V\_{DD} - V\_{DS}(t) - r\_{Lin}I\_{in}(t) \\ (L\_0 + L\_d)\frac{dl\_{out}(t)}{dt} = V\_{DS}(t) - (r\_{L0} + r\_{Lx} + R\_L)I\_{out}(t) - V\_{C0}(t) \\ C\_p\frac{dV\_{DS}(t)}{dt} = I\_{in}(t) - I\_{out}(t) - I\_{lank} \\ C\_O\frac{dV\_{CO}(t)}{dt} = I\_{out}(t) \end{cases} \tag{9}$$

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 di fferential equations:

$$\frac{dq(t)}{dt} = Aq(t) + B.\tag{10}$$

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:

$$q(t) = e^{At}q\_0 + \int\_0^t e^{A(t-\tau)}B d\tau = e^{At}q\_0 + A^{-1}(e^{At} - I)B\_\prime \tag{11}$$

*q*0 is the initial value. The solution of the state equation is:

$$q\_1(t) = e^{A\_1 t} q\_{01} + A\_1^{-1} (e^{A\_1 t} - I) B\_{1\prime} \tag{12}$$

$$q\_2(t) = e^{A\_2 t} q\_{02} + A\_2^{-1} (e^{A\_2 t} - I) B\_2. \tag{13}$$

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 <sup>o</sup>ff, 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:

$$q\_{02} = q\_1(t\_1) = e^{A\_1 t\_1} q\_{01} + A\_1^{-1} (e^{A\_1 t\_1} - I) B\_{1\prime} \tag{14}$$

$$q\_{01} = q\_2(t\_2) = e^{A\_2 t\_2} q\_{02} + A\_2^{-1} (e^{A\_2 t\_2} - I) B\_2. \tag{15}$$

The initial value *q*01, *q*02 can be obtained in (14) and (15), losses and output power can be defined:

$$\mathcal{W}\_1 = \int\_0^{t\_1} q\_1 q\_1^T dt\_\prime \tag{16}$$

$$\mathcal{W}\_2 = \int\_0^{t\_2} q\_2 q\_2^T dt. \tag{17}$$

Output power:

$$P\_{OUT} = \frac{R\_L}{T} \int\_0^T I\_{OUT}^2(t)dt = \frac{R\_L}{T} \{W\_1[2,2] + W\_2[2,2] \}. \tag{18}$$

Loss of the inductor resistance:

$$P\_{rL0} = \frac{r\_{L0}}{T} \{\mathcal{W}\_1[2,2] + \mathcal{W}\_2[2,2] \},\tag{19}$$

$$P\_{rLX} = \frac{r\_{LX}}{T} \{\mathcal{W}\_1[2,2] + \mathcal{W}\_2[2,2] \},\tag{20}$$

$$P\_{\rm rLin} = \frac{r\_{\rm Lin}}{T} \{\mathcal{W}\_1[1, 1] + \mathcal{W}\_2[1, 1] \}. \tag{21}$$

Loss of MOSFET in turn-on and turn-off state are shown as follows:

$$P\_{Son} = \frac{1}{T} \mathbb{C}\_{on} \left( \int\_0^{t\_2} q\_2 q\_2^T dt \right) \mathbb{C}\_{on}^T \tag{22}$$

$$P\_{Soff} = \frac{1}{T} \mathbb{C}\_{off} \left( \int\_0^{t\_1} q\_1 dt \right) \mathbb{C}\_{off'}^T \tag{23}$$

$$\mathbf{C}\_{\rm out} = \begin{bmatrix} 0 & 0 & 1/\sqrt{\mathbf{R}\_{\rm out}} & 0 \end{bmatrix} \tag{24}$$

$$\mathcal{C}\_{off} = \begin{bmatrix} 0 & 0 & \sqrt{l\_{\text{leak}}} & 0 \end{bmatrix}. \tag{25}$$

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

#### **3. 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*, *L*0*\_og*, and *C*0*\_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 *rL*0 are resistance values of the inductors.


**Table 1.** Theoretical parameter values of class-E.

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.

#### *3.1. 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.

**Figure 3.** *Cont*.

**Figure 3.** (**a**) Output power of RFC changing with *Lx* and *Cp*; (**b**) output efficiency of RFC changing with *Lx* and *Cp*; (**c**) peak voltage of RFC changing with *Lx* and *Cp*.

**Figure 4.** (**a**) Output power of RFC changing with *Lx* and *RL*; (**b**) output efficiency of RFC changing with *Lx* and *RL*; (**c**) peak voltage of RFC changing with *Lx* and *RL*.

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, the voltage peak across the MOSFET is smaller than its maximum value, ensuring that the MOSFET can work normally without being broken down.

#### *3.2. 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.

**Figure 5.** (**a**) Output power of PCL changing with *Lin* and *Cp*; (**b**) output efficiency of PCL changing with *Lin* and *Cp*; (**c**) peak voltage of PCL changing with *Lin* and *Cp*.

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 120 V, which will easily cause damage to the MOSFET.

#### *3.3. 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.

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.


**Table 2.** Parameter values of RFC before and after optimization.

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.

**Table 3.** Parameter values of PCL before and after optimization.


#### **4. 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.

**Figure 6.** Experimental platform.

#### *4.1. 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%.

**Figure 7.** (**a**) Load current and voltage waveform of RFC before optimization; (**b**) load current and voltage waveform of RFC after optimization; (**c**) drain-source voltage waveform of PCL before optimization; (**d**) drain-source voltage waveform of PCL after optimization.

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.

#### *4.2. 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.

**Figure 8.** (**a**) Load current and voltage waveform of PCL before optimization; (**b**) load current and voltage waveform of PCL after optimization; (**c**) drain-source voltage waveform of PCL before optimization; (**d**) drain-source voltage waveform of PCL after optimization.

#### **5. 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.

**Figure 9.** Simulation model of wireless power transfer system.

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.

**Figure 10.** (**a**) Simulation result of B-field strength in case 1; (**b**) simulation result of B-field strength in case 2; (**c**) simulation result of B-field strength in case 3. (**d**) simulation result of B-field strength in case 4.

**Figure 11.** The trends of the B-field strength along two lines in four cases.
