### I. Introduction

*m*= 0.5. Then several numerical analyses were performed with the grading coefficient

*m*≠ 0.5 [4, 5]. In these articles, only a nonlinear shunt capacitance of the power transistor is considered. Suetsugu and Kazimierczuk [6] has proposed an analysis and a design procedure for the class-E PA with a shunt capacitance composed of both transistor nonlinear output capacitance and linear external capacitance. This theory has been verified in a design of a high-efficiency class-E PA for 13.56 MHz wireless power transmission application [7]. The study of frequency limitation based on the nonlinear shunt capacitance theory has been presented in article [8], which showed the nonlinear dependence of frequency upon supply voltage, output power, and transistor characteristic. The performance as well as the topology of the class-E PA depends on the maximum frequency. An optimum performance is only addressed when the operating frequency does not exceed the maximum frequency. In the event that the operating frequency is lower than the maximum frequency, an added external shunt capacitor is necessary to obtain an optimal value of the shunt capacitor, as in the topology used in Suetsugu’s work, whereas the topology used by Chudobiak [3] is optimum if the operation and maximum frequencies are equal.

### II. Analyzing and Designing Broadband Class-E PA

*V*

*is applied to the gate of the transistor to turn the transistor off. A DC-supply voltage*

_{GS}*V*

*connects to the drain of the transistor through the choke coil*

_{DD}*L*

*with the only purpose of supplying DC power to the transistor. A shunt capacitor*

_{RFC}*C*

*, in parallel with the transistor, keeps a sinusoidal current waveform to alternate flow between the transistor and the shunt capacitor*

_{p}*C*

*. In common,*

_{p}*C*

*, consisting of a nonlinear shunt capacitance of a transistor*

_{p}*C*

*and a linear external capacitance*

_{ds}*C*

*, is described as follows:*

_{e}*C*

_{j}_{0}is the shunt capacitance at the drain-to-source voltage

*v*

*= 0 V, and*

_{s}*V*

*is the built-in potential of the MOSFET body diode. A series-resonant*

_{bi}*C*

_{0}

*L*

_{0}circuit tunes to the fundamental frequency with a high unloaded quality factor

*Q*to shape the output current to be sinusoidal at the fundamental frequency. At the fundamental frequency

*f*

_{0}, a load

*Z*

*is inductive, as shown in Fig. 1(a). An optimal value of*

_{E}*Z*

*guarantees that there is no overlap between the transient drain current and the voltage.*

_{E}*f*

_{0}, the supply voltage

*V*

*, the output power*

_{DD}*P*

*, the external linear capacitance*

_{o}*C*

*, and the characteristic parameters of MOSFET,*

_{e}*C*

_{j}_{0},

*V*

*. The numerical solutions, a set of tables and figures, illustrate that relationship. For the purpose of broadband design, this work gives a practical scenario in which a specification of class-E PA is defined as containing the operating frequencies, the output power, and the model of the MOFET transistor. Based on this specification, the analysis concentrates on the dependence of the circuit parameters on the operating frequency and the DC-supply voltage. For selected values of output power*

_{bi}*P*

*and MOSFET transistor parameters*

_{o}*C*

_{j}_{0},

*V*

*, and for swept values of operating frequency*

_{bi}*f*

_{0}and DC-supply voltage

*V*

*, this work re-solves the nonlinear design equation to extract the values of external linear capacitance*

_{DD}*C*

*and fundamental load*

_{e}*Z*

*. Then the optimal values of these parameters are chosen to obtain the highest efficiency of class-E PA.*

_{E}### 1. Selecting V_{DD} to Obtain the Actual Value of C_{e}

*C*

*as a function of the operating frequency*

_{e}*f*

_{0}and the DC-supply voltage

*V*

*.*

_{DD}*C*

*= 0 at the maximum operating frequency*

_{e}*f*

*. To obtain high efficiency for a broadband design, the upper frequency*

_{max}*f*

*of the bandwidth should be less than or equal to*

_{U}*f*

*. At the same conditions of*

_{max}*P*

*and transistor, the increased*

_{o}*V*

*results in the decrement of*

_{DD}*f*

*. The theory of the maximum frequency of class-E PA with the nonlinear shunt capacitance expressing the relation between the maximum frequency and the supply voltage was presented in detail in another work [8].*

_{max}*C*

*is chosen as the average value of the optimum*

_{e}*C*

*. To approximate the idealized class-E PA mode, the deviations of the actual*

_{e}*C*

*from the design values, depending on the variation of the optimum*

_{e}*C*

*with the*

_{e}*f*

_{0}, should be minimized. The small variation can result in the decreased level of the deviation. As shown in Fig. 1(b), at a given specification, the variation at a low band is much greater than that at a high band, whereas, at a given frequency band, from

*f*

_{1}to

*f*

*, the variation will be smaller in the case of a high voltage,*

_{max1}*ΔC*

_{2}

*> ΔC*

_{1}. Hence, combined with the maximum frequency condition as mentioned above,

*V*

*corresponding to*

_{1}*f*

_{max}_{1}is the optimal supply voltage with a bandwidth of

*f*

_{1}to

*f*

_{max}_{1}.

### 2. Output Matching Network

*Z*

*with a standard load resistance*

_{E}*R*

*. In broadband design, the conventional topology of a class-E power amplifier is not proper for obtaining broadband design, due to the limitation of the series resonator, which requires a high Q factor to perfectly prevent harmonic frequency.*

_{L}*Z*

*. In the conventional method [1, 9, 10],*

_{E}*Z*

*is expressed as follows:*

_{E}*Z*

*is expressed by a fundamental inductive load that consists of*

_{E}*R*and

*X*in series:

*φ*

_{1}is the phase of

*v*

*. Because the reactance of the resonator is zero at the operating frequency, the relation between*

_{s}*φ*

_{1}and

*v*

*is described by:*

_{s}*R*is only a function of

*V*

*and*

_{DD}*P*

*,*

_{o}*X*depends on

*V*

*,*

_{DD}*P*

*,*

_{o}*f*

_{0}, and the characteristic of the transistor, different from the conventional formula. Similar to the method of extraction of

*C*

*, the MATLAB program is composed to solve the nonlinear equation of*

_{e}*X*. Due to the dependence of the output impedance on the frequency, the LMN is required to match the various

*Z*

*to the standard load. The simple method to design the various output impedance LMNs is to use the Impedance Matching Utility Tool of the Advanced Design System (ADS).*

_{E}### 3. Design Procedure

Based on specifications of the operating frequencies and expected output power, select the type of transistor.

Using the MATLAB program to solve the nonlinear de sign equations, plot

*C*as a function of_{e}*f*_{0}and*V*, as shown in Fig. 1(b)._{DD}Select

*V*to obtain the actual value of_{DD}*C*. The rule of_{e}*V*selection is as follows: supposing the bandwidth of design class-E PA from_{DD}*f*_{1}to*f*_{2}, the optimal*V*has a value whose corresponding maximum frequency_{DD}*f*is equal to_{max}*f*_{2}. Then, the actual value of*C*is chosen as an average value of the optimal_{e}*C*at every frequency of bandwidth._{e}Use Eqs. (5)–(6) to extract

*Z*. Select the order of LMN based on the specification of bandwidth. Synthesize LMN by using the Impedance Matching Utility Tool of ADS._{E}

### III. Simulation and Experiment Results

*P*

*of 34 dBm over a bandwidth of 140–170 MHz. The MOSFET used in this design was MRF282SR1, with*

_{o}*C*

_{j}_{0}described by the following:

*C*

*and*

_{oss}*C*

*, respectively, are the output and reverse transfer capacitances of the transistor, (often provided in catalogs by power MOSFET manufacturers), and*

_{rss}*V*

*is a special value of supply voltage used to measure the values of*

_{spec}*C*

*and*

_{oss}*C*

*. Using MATLAB, the*

_{rss}*C*

*as a function of*

_{e}*f*

_{0}for selected values of

*V*

*was plotted in Fig. 2. From Fig. 2, the actual*

_{DD}*V*

*and*

_{DD}*C*

*were 8 V and 1.2 pF, respectively. Eqs. (5)–(6) were solved using MATLAB to determine the output impedance at various frequencies. Fig. 3 plots the output impedance versus the operating frequency. By setting the output impedances at corresponding frequencies, the ADS Impedance Matching Tool synthesized the matching network. A three-stage LMN with a bandwidth of 20% was extracted, illustrated in Fig. 4. The simulated drain minimized power dissipation on the transistor. A voltage and current waveforms depicted in Fig. 5 under the optimum operation slightly overlap, resulting in a prototype of class-E PA circuit was realized on a Taconic TLC PCB material with a thickness of 0.787 mm and a dielectric constant of 3.2. A photograph of the prototype is shown in Fig. 6. The output power was measured using an Agilent 85665EC spectrum analyzer with a maximum measurement output of 30 dBm. An attenuator of − 39.67 dB was added at the end of the circuit. The PAE and output power versus the frequency and input power were plotted in Fig. 7. The lumped components are available only in discrete values provided by the manufacturers, which is the main reason for the differences between the simulated and measured results. A summary of the experimental results compared with other articles on high efficiency proving this method’s advantages is shown in Table 1.*

_{e}