Analysis of OMN

The extended continuous Class-GF mode has been presented in paper [Reference Xuan, Cheng, Gong, Liu and Le11]. The output fundamental and second harmonic impedances are expressed as

(1)\begin{align} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad {Z_{{\rm{ECCGF}}, 1}}\left( {\alpha , \beta , \gamma , {I_{\rm{m}}}, \delta , \xi } \right) = \left( {\frac{{2 - \sqrt 3 \delta }}{{\sqrt 3 {i_{{\rm{ECCGF}}, 1}}}}} \right) + j \cdot \left[ {\frac{{36 - 7\sqrt 3 \cdot \delta }}{{36{i_{{\rm{ECCGF}}, 1}}}}} \right] \cdot \xi \end{align}

(2)\begin{equation} \qquad\qquad\qquad\qquad\qquad\qquad\qquad {Z_{{\rm{ECCGF}}, 2}}\left( {\alpha , \beta , \gamma , {I_{\rm{m}}}, \delta , \xi } \right) = \frac{{5\sqrt 3 }}{{18{i_{{\rm{ECCGF}}, 2}}}} \cdot \delta + j \cdot \frac{{9\delta - 7\sqrt 3 }}{{18{i_{{\rm{ECCGF}}, 2}}}} \cdot \xi \\ \end{equation}

in which i _ECCGF,1 is the fundamental drain current, i _ECCGF,2 is the drain current of the second harmonic, and γ is the voltage ratio of the second harmonic and fundamental voltage component at the input. In addition, δ and ξ are two parameters respectively to maintain a positive drain voltage. The remaining parameters are drain peak current (I_m), zero-crossing cutoff angle (β), and conduction angle (α).

With the aiming of achieving impedance matching, the intermediate frequency of the passband is selected as the fundamental frequency f ₀, so that the fundamental impedance and the second harmonic impedance can satisfy equations (1) and (2) respectively. Since the impedance within the bandwidth varies with frequency, a rectangular double transmission line structure proposed here can be used to make the impedance of the entire passband satisfy equations (1) and (2).

In order to satisfy the impedances desired by equations (1) and (2), the harmonic control network (HCN) and the fundamental matching network (FMN) are proposed as shown in Fig. 1. To realize a wideband PA, this paper employs a rectangular double transmission line structure [Reference Li, An, Wang and Zhang12, Reference Doan, Che and Nguyen13] for the fundamental impedance matching. As shown in Fig. 1, transmission lines TL1, TL2, and TL3 are employed to adjust harmonic impedances expressed in equation (2). Simultaneously, these three transmission lines also play a role in fundamental impedance matching. The entire OMN is the cascade connection of HCN and FMN.

Figure 1. Topological structure of the proposed output matching network.

It is observed that the rectangular double transmission line structure is symmetric about the plane A-A′. The odd–even-mode analysis [Reference Roberg and Campbell14, Reference Gao, Cao, Gao, Li, Yang and Li15] is introduced to theoretically derive the impedance of the OMN. Figure 2 depicts the equivalent circuit diagrams for odd and even modes, respectively.

Figure 2. Equivalent circuits of the output matching network: (a) odd-mode and (b) even-mode.

As displayed in Fig. 2(a), based on the transmission line theory [Reference Pozar16], the theoretical input odd-mode impedance Z _in,o can be calculated as

(3)\begin{equation}{Z_{\rm{in,o}}} = {Z_{\rm{o1}}}//{Z_{\rm{o2}}} = \frac{{{Z_{\rm{o1}}} \times {Z_{\rm{o2}}}}}{{{Z_{\rm{o1}}} + {Z_{\rm{o2}}}}}\end{equation}

where

(4)\begin{equation}{Z_{\rm{o1}}} = j{Z_1}{\rm{tan}}(\frac{1}{2}{\theta _1})\end{equation}

(5)\begin{equation}{Z_{\rm{o2}}} = j{Z_2}{\rm{tan}}(\frac{1}{2}{\theta _2})\end{equation}

where θ ₁, θ ₂ refer to the electrical length of the transmission lines as depicted in Fig. 1. Meanwhile, Z ₁, Z ₂ represent the characteristic impedance of the transmission lines as depicted in Fig. 1.

As displayed in Fig. 2(b), the theoretical input even-mode impedance Z _in,e can be derived as

(6)\begin{equation}{Z_{\rm{in,e}}} = {Z_{\rm{e1}}}//{Z_{\rm{e2}}} = \frac{{{Z_{\rm{e1}}} \times {Z_{\rm{e2}}}}}{{{Z_{\rm{e1}}} + {Z_{\rm{e2}}}}}\end{equation}

where

(7)\begin{equation}{Z_{\rm{e1}}} = - j{Z_1}{\rm{cot}}(\frac{1}{2}{\theta _1})\end{equation}

(8)\begin{equation}{Z_{\rm{e2}}} = - j{Z_2}{\rm{cot}}(\frac{1}{2}{\theta _2})\end{equation}

According to transmission line theory [Reference Pozar16], the impedance matrix at the point of M in the Fig. 1 can be calculated as follows:

(9)\begin{equation}{\textbf{[Z]}} = \frac{1}{2}\left[ {\begin{array}{*{20}{c}} {{Z_{\rm{in,e}}} + {Z_{\rm{in,o}}}}&{{Z_{\rm{in,e}}} - {Z_{\rm{in,o}}}} \\ {{Z_{\rm{in,e}}} - {Z_{\rm{in,o}}}}&{{Z_{\rm{in,e}}} + {Z_{\rm{in,o}}}} \end{array}} \right]\end{equation}

Then, based on the load impedance Z _f of the OMN, the impedance matrix can be converted into a scattering matrix, which is expressed as follows:

(10)\begin{equation}{\textbf{[S]}} = \frac{1}{2}\left[ {\begin{array}{*{20}{c}} {\frac{{{Z_{\rm{in,e}}} \cdot {Z_{\rm{in,o}}} - {Z_f}}}{{({Z_{\rm{in,e}}} + {Z_f}) \cdot ({Z_{\rm{in,o}}} + {Z_f})}}}&{\frac{{{Z_f} \cdot ({Z_{\rm{in,e}}} - {Z_{\rm{in,o}}})}}{{({Z_{\rm{in,e}}} + {Z_f}) \cdot ({Z_{\rm{in,o}}} + {Z_f})}}} \\ {\frac{{{Z_f} \cdot ({Z_{\rm{in,e}}} - {Z_{\rm{in,o}}})}}{{({Z_{\rm{in,e}}} + {Z_f}) \cdot ({Z_{\rm{in,o}}} + {Z_f})}}}&{\frac{{{Z_{\rm{in,e}}} \cdot {Z_{\rm{in,o}}} - {Z_f}}}{{({Z_{\rm{in,e}}} + {Z_f}) \cdot ({Z_{\rm{in,o}}} + {Z_f})}}} \end{array}} \right]\end{equation}

Based on this, the input impedance of the rectangular double transmission line structure, Z _IN, is expressed as

(11)\begin{equation}{Z_{\rm{IN}}} = \frac{{{Z_{{f}}}({Z_{\rm{in,e}}} + {Z_{\rm{in,o}}}) + 2{Z_{\rm{in,e}}}{Z_{\rm{in,o}}}}}{{2{Z_{{f}}} + {Z_{\rm{in,e}}} + {Z_{\rm{in,o}}}}}\end{equation}

where Z _f represents the load impedance of the fundamental rectangular double transmission line structure.

Then, according to transmission line theory, the input impedance of the fundamental matching network Z _IN3 at the point of N is derived as

(12)\begin{equation}{Z_{\rm{IN3}}} = {Z_3}\frac{{{Z_{\rm{IN}}} + j{Z_3}{\rm{tan}}({\theta _3})}}{{{Z_3} + j{Z_{\rm{IN}}}{\rm{tan}}({\theta _3})}}\end{equation}

(13)\begin{equation}{Z_{\rm{IN4}}} = - j{Z_4}{\rm{cot}}({\theta _4})\end{equation}

(14)\begin{equation}{Z_{\rm{IN5}}} = j{Z_5}{\rm{tan}}({\theta _5})\end{equation}

where θ ₃, θ ₄, θ ₅, refer to the electrical length of the transmission line as depicted in Fig. 1. The Z ₃, Z ₄, Z ₅ refer to the characteristic impedance of the transmission lines as depicted in Fig. 1.

At the point Q as shown in Fig. 1, the input impedance Z _IN,Sum is equivalent to Z _IN3, Z _IN4, Z _IN5 in parallel.

(15)\begin{equation}{Z_{\rm{IN,Sum}}} = {Z_{\rm{IN3}}}//{Z_{\rm{IN4}}}//{Z_{\rm{IN5}}}\end{equation}

When Z _IN,Sum is obtained, the input impedance Z _OMN at the point of P can be obtained as

(16)\begin{equation}{Z_{\rm{OMN}}} = {Z_6}\frac{{{Z_{\rm{IN,Sum}}} + j{Z_6}{\rm{tan}}({\theta _6})}}{{{Z_6} + j{Z_{\rm{IN,Sum}}}{\rm{tan}}({\theta _6})}}\end{equation}

where θ ₆ refers to the electrical length of the transmission line as depicted in Fig. 1. The Z ₆ refer to the characteristic impedance of the transmission line as depicted in Fig. 1.

The driving point input impedance Z _OMN of the OMN encompasses all the parameters of Fig. 1 to be determined or selected by the designer. For the sake of simplicity, the rectangular double transmission lines are selected to have equal characteristic impedances, denoted as Z ₁ = Z ₂ in Fig. 1. The HCN has an equivalent electrical length of 90° at the second harmonic frequency 2f ₀ to cause a short circuit for the second harmonic. Therefore, the variables θ ₄, θ ₅, Z ₄, and Z ₅ are determined as 20°, 70°, 60 Ω and 70 Ω, respectively. Besides, the HCN has an equivalent electrical length of 60° at the third harmonic frequency 3f ₀ to lead to an open circuit for the third harmonic. Thus the variables Z ₆ and θ ₆ are taken as 50 Ω and 40°. Consequently, the electrical length of TL6, TL5, and TL4, θ ₆, θ ₅, and θ ₄ are set as 40°, 70°, and 20°. The characteristic impedances Z ₆, Z ₅, and Z ₄ are taken as 50 Ω, 70 Ω, and 60 Ω, respectively.

Figures 3–6 illustrate the impact of characteristic impedance and electrical length of the rectangular double transmission line structure on the impedance Z _OMN.

Figure 3. (a) Real part of Z _OMN. (b) Imaginary part of Z _OMN, when θ ₁ = 90°, θ ₂ = 50°, θ ₃ = 40°.

Figure 3 illustrates that the real part of the OMN impedance increases with a larger Z ₃ and a smaller Z ₁, Z ₂. Meanwhile, the imaginary part of the impedance of the OMN decreases with increasing Z ₃. As shown in Fig. 4, the impedance of the output matching network Z _OMN increases dramatically when θ ₁ is between 120° and 160° and θ ₂ varies between 0° and 80°. Outside this interval, the impedance Z _OMN varies smoothly. Figure 5 reveals a substantial increase in the OMN impedance when θ ₁ varies between 120° and 160°. The variation in θ ₃ minimally influences the real part of the OMN impedance, yet there is a slight increase in the imaginary part as θ ₃ rises. From Fig. 6, it can be seen that the real part of the OMN impedance spikes for both small and large θ ₂ values. As θ ₂ decreases and θ ₃ increases, the imaginary part of the OMN impedance grows larger.

Figure 4. (a) Real part of Z _OMN. (b) Imaginary part of Z _OMN, when Z ₁ = Z ₂ = 100 Ω, Z ₃ = 35 Ω, θ ₃ = 40°.

Figure 5. (a) Real part of Z _OMN. (b) Imaginary part of Z _OMN, when Z ₁ = Z ₂ = 100 Ω, Z ₃ = 35 Ω, θ ₂ = 50°.

Figure 6. (a) Real part of Z _OMN. (b) Imaginary part of Z _OMN, when Z ₁ = Z ₂ = 100 Ω, Z ₃ = 35 Ω, θ ₁ = 90°.

Based on the above analysis, it is evident that the relationship between the variables has been established. Figure 3 indicates that when Z ₁, Z ₂, and Z ₃ are within the ranges of 90–110 Ω, 90–110 Ω, and 20–40 Ω respectively, the impedance Z _OMN generally typically falls within the impedance solution space of equation (1). Similarly, Figs. 4–6 demonstrate that when associated with angles θ ₁, θ ₂, and θ ₃ in the range of 40°–100°, 40°–120°, and 20°–50° respectively, the impedance Z _OMN typically falls within the impedance solution space of equation (1). For example, when variables Z ₁, Z ₂, Z ₃, θ ₁, θ ₂, and θ ₃ are respectively 100 Ω, 100 Ω, 35 Ω, 90°, 50°, and 40°, the impedance Z _OMN is 34.6 + j × 11.8 Ω, satisfying equation (1).

The input impedance of the rectangular double transmission line structure, Z _IN, is related to four parameter variables. The wider design space for these parameter variables allows for greater flexibility in achieving broadband matching. Therefore, the load impedance requirements of different PA operating modes can be satisfied by selecting the appropriate combination of parameter values, so that high efficiency can be maintained in a wide frequency band. The proposed structure is capable of satisfying the impedance requirements of various PA operating modes, such as continuous Class-F/F⁻¹ mode, continuous Class-F mode, Class-B/J mode, etc. This extended continuous Class-GF mode here is just an example.

The PA circuit design and simulation

To validate the effectiveness of the proposed OMN, this work designs a high-efficiency PA operating in the frequency range of 0.8–3 GHz. The PA employs the commercial Cree CGH40010F GaN device and Rogers 4350B substrate, featuring a thickness of 0.762 mm and a dielectric constant of 3.66. The DC bias voltage is set to V _GS = −3 V, and the drain DC bias voltage is set to V _DS = 28 V, enabling operating in the Class AB operation condition. Based on Figs. 3–6, the Z ₁, Z ₂, Z ₃, θ ₁, θ ₂, θ ₃ are chosen as 98 Ω, 98 Ω, 33 Ω, 89°, 54°, 37° to satisfy the load impedances required by equations (1) and (2). Optimal values for these variables are determined to ensure a suitable match between the OMN and the load. Subsequently, these chosen values are integrated into the OMN and fine-tuned using ADS simulation software. Following the optimization process in ADS, the sum of the electrical lengths of TL1 and TL3 microstrip lines is less than 60°. This deviation may arise due to the small value of electrical length, which has a better effect on the impedance of third harmonics. The designed OMN is shown in Fig. 7(a). The multibranch stepped matching technique is used to design the broadband input network. The resulting comprehensive circuit schematic diagram is depicted in the Fig. 7(b).

Figure 7. (a) The designed output matching network and (b) circuit schematic of the broadband high-efficiency PA.

Figure 8 displays the simulated voltage and current waveforms at several representative frequencies (0.8 GHz, 1 GHz, 2 GHz, and 3 GHz). The voltage and current waveforms have less overlap in the time domain, aligning with the expected waveform of an ideal Class-GF PA.

Figure 8. Simulated drain voltage and current waveforms at several frequencies: (a) 0.8 GHz, (b) 1.0 GHz, (c) 2.0 GHz, and (d) 3.0 GHz.

This simulated load impedance trajectory of the PA at the I-gen plane is illustrated in Fig. 9, the solid shadow part is the fundamental impedance region, and the hollow shadow part is the second harmonic impedance region. As observed from Fig. 9, the fundamental impedance resides within the impedance solution space of the extended continuous Class-GF mode. In addition, most of the second harmonic impedance falls in the impedance solution space, and the part that does not fall in the impedance solution space may be caused by frequency deviation.

Figure 9. Simulated impedance trajectory at the I-gen plane.

PA fabrication and measurement

The designed PA was fabricated based on the schematic shown in Fig. 7. A photograph of the realized PA is shown in Fig. 10.

Figure 10. Photograph of the fabricated PA.

The manufactured PA underwent large-signal testing. In Fig. 11, the measurement results depict the drain efficiency (DE), output power, and gain versus working frequency of from 0.8 to 3.0 GHz. Within the operational band from 0.8 to 3.0 GHz, the PA has a DE ranging from 63.2% to 71.9%, a power-added efficiency ranging from 61.4% to 68.4%, an output power ranging from 40.2 to 42.2 dBm, and a gain ranging from 9.4 to 11.3 dB. Figure 12 illustrates the relationship between gain, DE, and output power at several representative frequencies.

Figure 11. Simulated and measured DE, PAE, output power, and gain across the entire bandwidth.

Figure 12. (a) Simulated and (b) measured DE and gain versus output power at different frequency points.

As shown in Fig. 13, the PA was simulated and measured for S-parameters. Within the working bandwidth, the PA has the S11 ranging from −20.7 to −7.7 dB and the S21 ranging from 14.2 to 19.3 dB. The small signal performance of the designed PA meets the requirements.

Figure 13. Simulated and measured S-parameter of the designed PA.

In order to better reflect the performance of this PA, Table 1 lists several PA works. It can be seen that the designed PA exhibits characteristics of large bandwidth and high efficiency, thus verifying the effectiveness of the output-matching network structure.

Table 1. Performance comparison with PAs

ATSTLBLPMS = a three-stage transmission-line-based low-pass matching structure; ALPMS = a low-pass matching structure; ATTLS = a traditional transmission line structure; ARDTLS = a rectangular double transmission line structure.