Ideal transistor

The theoretical load conditions for CCF and CCF⁻¹ PAs were derived by previous workers based on ideal transistors. For CCF PA, the load impedances for the fundamental frequency f ₀ and the harmonic frequencies (2f ₀ and 3f ₀) can be given as [Reference Chen and Peroulis9]

(1a)\begin{equation}{{\text{Z}}_{L1}}\left( \gamma \right) = \left( {2/\sqrt 3 } \right){R_L}^{opt} + j{R_L}^{opt}\gamma ,\end{equation}

(1b)\begin{equation}{{\text{Z}}_{L2}}\left( \gamma \right) = - j\left( {7\sqrt 3 \pi /24} \right){R_L}^{opt}{\text{ }}\gamma ,\end{equation}

(1c)\begin{equation}{{\text{Z}}_{L3}}\left( \gamma \right) = \infty ,\end{equation}

where $\gamma $ is a dimensionless parameter ( $ - 1 \leq \gamma \leq 1$) that determines the voltage waveform of the FET. The voltage waveform corresponds to that for a standard Class-F PA when $\gamma $ = 0, but it becomes distorted as $\gamma $ takes on non-zero values. However, this family of different voltage waveforms still offers the same power efficiency as the standard Class-F PA, thereby expanding the design space for Class-F PAs. ${R_L}^{opt}$ is the optimum load impedance for ideal Class-B amplifier:

(2)\begin{equation}R_L^{opt}=\frac{V_D}{I_m/2},\end{equation}

where V_D is the drain voltage and I_m is the maximum current of the FET.

As for the CCF⁻¹ PA, the theoretical load admittances for f ₀, 2f ₀, and 3f ₀ are [Reference Yang, Xia, Guo and Zhu8]

(3a)\begin{equation}{Y_{L1}}\left( \xi \right) = {G_{opt}}\left( {\sqrt 2 {i_1} - j\sqrt 2 {i_{DC}}\xi } \right),\end{equation}

(3b)\begin{equation}{Y_{L2}}\left( \xi \right) = j{G_{opt}}2\left( {{i_1} + {i_3}} \right)\xi ,\end{equation}

(3c)\begin{equation}{Y_{L3}}\left( \xi \right) = \infty ,\end{equation}

where $\xi $ ( $ - 1 \leq \xi \leq 1$) is a parameter similar to $\gamma $ in the CCF PA. However, unlike $\gamma $, $\xi $ distorts the current waveform of the FET from that of a standard CCF⁻¹ PA when $\xi $ = 0. Other parameters are given as follows: ${G_{opt}} = 1/{R_L}^{opt}$, ${i_{DC}} = 0.37$, ${i_1} = 0.43$, ${i_3} = 0.06$.

Figure 3 shows the typical theoretical load impedances (or load admittances) of (1) and (3) plotted on the Smith Chart. By relating the values of γ or ξ to different frequency points within the desired frequency band, (1) and (3) now provide frequency-dependent load conditions to sustain the high efficiency of Class-F or Class-F⁻¹ amplifiers across the band. When considering real FETs with parasitic networks, however, the situation is not as straightforward, as discussed next.

Figure 3.

Fundamental and harmonic load conditions for (a) CCF and (b) CCF⁻¹ amplifies using ideal FETs.

Transistor with parasitic network

The discussion here pertains to a CCF amplifier unless otherwise specified, and similar discussion applies to a CCF⁻¹ amplifier. In the following, simulations were conducted using Keysight ADS, and the transistor model used was the nonlinear model provided by the manufacturer. As illustrated in Fig. 1, a real FET has parasitic network associated with both the device itself and its packaging. Basically, the load impedances at the intrinsic FET plane (Z_L ^int) must satisfy the g-dependent theoretical load conditions (1a), (1b), and (1c). The load impedances at the extrinsic FET place (Z_L ^ext) are then determined as a function of frequency by projecting Z_L ^int onto the extrinsic FET plane, utilizing the FET parasitic network model along with a given γ-frequency relationship (hereafter referred to simply as γ(f). However, this approach to handling the impedances often results in Z_L ^ext with a frequency response that cannot be realized using practical circuits. Therefore, we follow the subsequent design steps for determining Z_L ^ext:

Step 1: Optimize γ(f) so that the 2^nd harmonic load Z_L ^ext(2f ₀) rotates clockwise in the Smith Chart as the frequency increases.

Step 2: The function γ(f) optimized in Step 1 causes the 3^rd harmonic load Z_L ^ext(3f ₀) to rotate counterclockwise as the frequency increases. Consequently, the values of Z_L ^ext(3f ₀) are adjusted to ensure a clockwise frequency response.

Step 3: Perform a load-pull simulation to determine the fundamental load Z_L ^ext(f ₀), based on Z_L ^ext(2f ₀) and Z_L ^ext(3f ₀) established in Step1 and Step2, respectively.

In Step 1, we focus on Z_L ^ext(2f ₀) rather than Z_L ^ext(f ₀) for the following reason: As shown in Fig. 3, the load impedances for the fundamental and 2^nd harmonic frequencies tend to move in the opposite directions. This arises from the fact that, in (1a) and (1b), the terms associated with the parameter γ exhibit opposite signs. Consequently, when projecting the γ-dependent impedances to the frequency-dependent loads at the extrinsic plane, there is a possibility that either Z_L ^ext(f ₀) or Z_L ^ext(2f ₀) may rotate counterclockwise as the frequency increases. However, achieving a counterclockwise frequency response of impedance using practical circuits is challenging, especially when the impedance is on the edge of the Smith Chart, as in the case of Z_L ^ext(2f ₀). Therefore, we determine the γ-frequency relationship, in such a way that Z_L ^ext(2f ₀) rotates clockwise as the frequency increases. The method for handling Z_L ^ext(f ₀) is discussed in Step 3.

In Fig. 4, two examples of γ(f) function are illustrated, one being optimized and the other nonoptimized. The optimized γ(f) led to a clockwise rotation of Z_L ^ext(2f ₀), while the nonoptimized γ(f) caused a counterclockwise rotation. It can be seen that even a minor difference in γ(f) leads to significant variations in the frequency response of Z_L ^ext(2f ₀). Therefore, instead of relying on a trial-and-error approach to find the optimal γ(f) [Reference Aggrawal, Rawat and Roblin3, Reference Liu, Mu, Qiu, Leng and Peng5], we used design charts, as shown in Fig. 5. Each chart consists of a series of curves representing the frequency response of Z_L ^ext(2f ₀) for CCF PA and CCF⁻¹ PA. These curves are computed based on the detailed FET parasitic network model for the CGH40010F device [Reference Tuffy, Guan, Zhu and Brazil2]. As can be seen from Fig. 5, each pair of curves for the same γ value exhibits divergent behavior at a certain frequency pole. To better understand the behavior of Z_L ^ext(2f ₀), we derived an analytical expression for Z_L ^ext(2f ₀):

(4)\begin{equation}{Z_L}^{ext}\left( {2{f_0}} \right) = j \cdot \left( {\frac{{{R_L}^{opt}}}{{\omega {C_p}{R_L}^{opt} - \frac{{24}}{{7\sqrt 3 \gamma }}}} - \omega {L_p}} \right)\!,\end{equation}

which is based on a simplified FET parasitic network consisting of C_p and L_p (Fig. 1). From (4), it is evident that the parasitic elements introduce a frequency pole in Z _L^ext(2f ₀). This frequency pole, which depends on $\gamma $, complicates the relationship between the load impedances at the intrinsic and the extrinsic planes. Consequently, it becomes challenging to predict the effect of the γ(f) function on the frequency response of Z_L ^ext(2f ₀) without referring to the design chart provided in Fig. 5.

Figure 4.

Design space parameters, γ and ξ, for (a) CCF and (b) CCF-1 amplifiers as a function of frequency. The bottom illustrates the effect of γ and ξ on the frequency variation of the external 2nd harmonic load impedance.

Figure 5.

Design charts for determining (a) γ-frequency relation for CCF PA and (b) ξ-frequency relation for CCF⁻¹ PA.

In Step 2, we determine the third harmonic load, Z_L ^ext(3f ₀). According to (1c), the ideal load conditions of Z_L3 exhibit no frequency dependence. However, once projected on to the external FET plane through the parasitic network, it displays a frequency response of counterclockwise rotation, as indicated by the open rectangles in Fig. 6. Therefore, we adjusted the load impedances at each frequency points to ensure clockwise rotation of the impedance, as shown by the solid rectangles. Throughout this process, there is no need to compromise on achievable efficiency because the power-added efficiency (PAE) is relatively insensitive to the load near the maximum efficiency point, as evidenced by the PAE contours, also shown in Fig. 6.

Figure 6.

Method of determining the 3^rd harmonic loads for (a) CCF PA and (b) CCF⁻¹ PA.

In Step 3, we determine the remaining Z_L ^ext(f ₀). In obtaining Z_L ^ext(f ₀), we do not rely on the theory of (1), as we did in Step 1 to obtain Z_L ^ext(2f ₀). This is because, even if we disregard the parasitic network of the transistor, the transistor still exhibits a knee voltage in its I-V characteristics, making it impossible to determine Z_L ^ext(f ₀) without accounting for this factor. Therefore, we obtain Z_L ^ext(f ₀) using load-pull simulation while maintaining Z_L ^ext(2f ₀) and Z_L ^ext(3f ₀) at values determined in Step 1 and Step 2, respectively. Figure 7 shows the trajectory of Z_L ^ext(f ₀) obtained by load-pull simulation as the frequency f ₀ is swept from 1.7 GHz to 2.3 GHz (solid circles). For comparison, the impedances of Z_L ^ext(f ₀) based on theory, i.e. using (1a) with the γ(f) established in Step 1, are also shown (open circles). It can be seen that the load-pull simulation result demonstrates an impedance trajectory close to the theoretical prediction. The difference between the two trajectories is considered to reflect the effect of the knee voltage on the optimum load line. Although the desired load Z_L ^ext(f ₀) rotates counterclockwise with increasing frequency, it is possible to approximate their trajectory using a clockwise impedance path achieved by a circuit with minimal mismatch, because the impedance varies within a relatively small area in the Smith Chart.

Figure 7.

Frequency variation of optimum fundamental loads for (a) CCF PA and (b) CCF⁻¹ PA.

Output harmonic tuning

To implement the CRLH TL in the PAs for operation in CCF and CCF⁻¹, the parameters of CRLH TL were optimized for each operating mode. Figure 9 illustrates the dispersion characteristics for these CRLH TLs. The triangle and rectangle plots correspondingly represent the optimum load impedances for the 2^nd and 3^rd harmonic frequencies, as depicted in Figs. 4 and 6. The parameter values for the CRLH TLs are listed in Table 1. It is observed that the dispersion curve for each operating mode satisfies the specified harmonic load conditions. In this manner, the dispersion curve can accommodate various kinds of frequency variations in the load impedances, showcasing the wide range of applicability of the one-port CRLH TL.

Figure 9.

Dispersion characteristics of one-port CRLH tls.

Table 1.

Parameters for CRLH transmission lines

^a Electrical length based on 2 GHz.

As another advantage of the one-port CRLH TL, its size can be nearly freely manipulated while simultaneously ensuring it meets the specified harmonic load conditions. This is explained as follows: The gradient of the dispersion for the conventional TL (with RH electromagnetic property) is inversely proportional to the length of the TL, as illustrated on the right-hand side of Fig. 8. Thus, considering that the property of CRLH TL in the RH region is predominantly determined by the RH section of the CRLH TL, the length of the RH section can be reduced by increasing the dispersion gradient of the CRLH TL in the RH region. On the other hand, the harmonic loads are determined by the dispersion curve in the LH region of the diagram. This implies that the function and the size of the CRLH TL can be optimized almost independently of each other. From Table 1, it is evident that the lengths of the RH section for the two CRLH TLs are minimized, significantly shorter than λ/12 (where λ is the wavelength for the fundamental frequency). This is a result of the design strategy aimed at enhancing the dispersion gradient as the dispersion curves extend into the RH region.

Harmonics trap filter

The harmonic trap filter (HTF) is an essential part of the proposed PA’s harmonic tuning method. To preserve the merit of the miniature-sized one-port CRLH TL, minimizing the dimensions of the HTF is crucial. Figure 10 illustrates two different designs for the harmonic tuning networks utilized in CCF and CCF⁻¹ PAs. In both designs, two transmission zeros (TZs) at 2f ₀ and 3f ₀ are utilized to trap the harmonics. The stubs employed to generate the TZs are designed as compact as possible by loading capacitors at the end of the stubs [Reference Tanaka, Katayose, Nishizawa, Hosoya, Ishikawa and Honjo17]. For example, in CCF PA, the stub responsible for shorting at 2f ₀ comprises an 8.6-deg transmission line (TL1) and a 1.5-pF capacitor, resulting in a substantial size reduction compared to the 45-deg stub length required without the capacitor loading technique. The reason we employed double stubs for shorting the harmonics is to increase the bandwidth.

Figure 10.

Configurations and frequency responses (|S ₂₁|) of harmonic tuning networks for CCF and CCF⁻¹ PAs.

Apart from the shared features described above, different circuit topologies were employed for the HTFs in CCF and CCF⁻¹ PAs. In the case of CCF PA, a portion of the CRLH TL, specifically the branch line consisting of components C ₁, L ₂, and C ₂, is utilized for trapping 3f ₀, thus reducing the overall size of the harmonic tuning network. Here, the characteristics of the CRLH TL in the 3f ₀ band is determined by the RH section of the CRLH TL. Conversely, for CCF⁻¹ PA, the CRLH TL and the HTF are designed separately. Although the entire harmonic tuning network occupies slightly more circuit area compared to its counterpart in the CCF PA, this setup allows for individual optimization of both the CRLH TL and HTF. As a result, the harmonic tuning network for the CCF⁻¹ PA shows more ideal LPF characteristics compared to that for the CCF PA, as depicted in Fig. 10. However, both harmonic tuning networks appear to function well when actually used in the PAs, as we will see in the next section.

Input harmonic tuning

While tuning the output harmonics is effective for enhancing the efficiency of PAs, certain operation classes of the PA also require additional harmonic tuning on the input side. In [Reference Tanaka, Asami and Suzuki13], it has been shown that its significance becomes apparent if the 2^nd harmonic load impedance at the intrinsic FET plane (Z_L ^int(2f ₀)) approaches short. In such cases, the 2^nd harmonic component of the drain current is boosted and fed back to the input through the gate-to-drain parasitic capacitance, necessitating compensation for the feedback effect on the input side.

Figures 11(a) and 11(b) show the simulated PAE with varying 2^nd harmonic source impedance, Im[Z_S(2f ₀)], for CCF and CCF⁻¹ PAs, respectively. The inset in each graph illustrates the 2^nd harmonic load impedances at the intrinsic FET plane. Figure 11 illustrates that not only is the range of Im[Z_S(2f ₀)] for maximum PAE narrow but also even a slight deviation from the PAE local maxima point results in significant degradation of efficiency due to the presence of a PAE local minima point nearby. This effect is much more pronounced for CCF PA than for CCF⁻¹ PA because Z_L ^int(2f ₀) for the former is located closer to the short. This indicates that controlling Im(Z_S(2f ₀)) is highly critical in designing CCF PA. For instance, at 4.4 GHz, the achievable efficiency is low unless Im[Z_S(2f ₀)] is set in the vicinity of the local maxima point of the PAE . However, this also implies that a slight increase in the frequency will result in a collapse of efficiency. Taking all of this into consideration, the values for Im[Z_S(2f ₀)] for both CCF and CCF⁻¹ PAs were set to the triangle points depicted in Fig. 11(a) and 11(b), respectively.

Figure 11.

Simulated PAE with varying second harmonic source impedance for (a) CCF PA and (b) CCF⁻¹ PA.

PA fabrication

Figure 12 illustrates the configuration of the proposed PAs implemented in microstrip line (MSL) technology. The substrate utilized was Panasonic Megtron-6 R-5775 laminate (ε_r = 3.6, h = 0.63 mm, tanδ = 0.002). For the CRLH TL, MSL inductors and 1005-size high-Q type chip capacitors (Murata, GJM series) were employed. The drain bias was provided via the inductor of the CRLH TL [Reference Tanaka, Oda and Saiki12]. The LPF for the CCF PA was designed utilizing a stepped-impedance MSL. Since the LPF only needs to manage the fundamental frequency, a 2-stage filter design using TL sections with modestly low impedance (Z₀ > 25 Ω) was deemed sufficient. In contrast, conventional continuous mode PAs typically utilized 3-stage filters using TL sections with very low impedance (Z₀ < 10 Ω) [Reference Sun and Zhu7, Reference Chen and Peroulis9], as depicted by an image in Fig. 1(a). Regarding the CCF⁻¹ PA, an attempt was made to further reduce the total circuit size, by substituting lumped element capacitors of modest values (1.2 pF and 1.4 pF) for the low-impedance TL sections. Once again, this was made feasible due to the less stringent demand imposed on the LPF design compared to conventional continuous mode PAs.

Figure 12.

Configurations of the fabricated (a) CCF PA and (b) CCF⁻¹ PA.

Figure 13 shows the pictures of the fabricated PAs. The circuit layout was optimized using EM simulation. As can be seen from the picture, the footprints of the one-port CRLH TLs are remarkably small, comparable to that of the ceramic hermetic package of the FET. In this study, we define the effective circuit area of the PA as determined by the rectangles on both the input and output sides, encompassing only the essential part of the circuit (excluding components such as biasing circuits and DC-cut capacitors). As can be seen from Fig. 2, the effective circuit areas for the PAs of this work are significantly reduced compared to the conventional CCF and CCF⁻¹ PAs.

Figure 13.

Photographs of the fabricated (a) CCF PA and (b) CCF⁻¹ PA.

S-parameter measurement

Prior to assembling the PAs, the circuit boards on the input and output side were measured for S-parameters to verify if they meet the desired source/load conditions. Figure 14 compares the measured and simulated impedances for CCF and CCF⁻¹ PAs. The measurements reasonably align with the simulations. The fact that the measured load impedances in the 2f ₀ band (3.4-4.6 GHz) and those in the 3f ₀ band (5.1-6.9 GHz) precisely fall on the outer edge of the Smith Chart indicates that the harmonics trap filters for both CCF and CCF⁻¹ PAs are effectively functioning. As the frequency transitions from the high end of the f ₀ band (2.3 GHz) to the low end of the 2f ₀ band (3.4 GHz), the desired load impedance shifts from the inner region to the outer edge of the Smith Chart. Since the load impedances at 2.3 GHz and 3.4 GHz for CCF⁻¹ PA are closely situated, designing the OMN for the CCF⁻¹ PA appears to be more challenging compared to the case of the CCF PA. However, despite the demanding conditions for CCF⁻¹ PA, the load impedance trajectory realized by the proposed circuit precisely adhered to the intended path from 2.3 GHz to 3.4 GHz. The rapid transition of the impedance from the inner region to the outer edge of the Smith Chart is realized if the harmonic trap filter is adequately designed.

Figure 14.

Measured and simulated impedances for IMNs and OMNs of the proposed (a) CCF PA and (b) CCF⁻¹ PA.

Let us now shift our focus in Fig. 14 to the source impedances. In the Smith Chart, the PAE contours of the 2nd harmonic frequency of 4.0 GHz are shown as a reference. The local minima and maxima points exhibit minimal change with frequency variation, as depicted in Fig. 11. It can be observed that, for the CCF PA, the 2^nd harmonic source impedance, Z_S(2f ₀), is controlled to avoid crossing the local minima at the high end of the targeted frequency band (2f ₀ = 4.6 GHz). In other words, the bandwidth of the CCF PA is restricted by the design of the input matching network (IMN) considering the 2nd harmonic source impedance. In contrast, it is relatively straightforward to design the IMN for the CCF⁻¹ PA, because the efficiency decreases only minimally even if the impedance is apart from the local maxima, as demonstrated in Fig. 11(b).

Figure 15 shows the typical frequency responses of |S₂₁| and |S₁₁| measured for the CCF PA. While |S₂₁| exceeds − 20 dB at certain points within the frequency ranges corresponding to the 2^nd and 3^rd harmonic bands, specifically 3.4–4.6 GHz and 5.1–6.9 GHz, it remains predominantly below − 20 dB. This result indicates that the HTFs are functioning as intended and confirms that the CRLH TLs effectively operate as one-port TLs, as anticipated.

Figure 15.

Measured S-parameters for CCF PA.

Power measurement

Figure 16 shows the measured CW power performance for the fabricated CCF PA and CCF⁻¹ PA. For the CCF PA, at 1.8 GHz, the output power, drain efficiency (DE) and PAE reached 40.1 dBm, 79.5% and 75.6%, respectively. Meanwhile, the CCF⁻¹ PA achieved 41.2 dBm output power, 79.6% DE, and 74.0% PAE at 1.9 GHz. Figure 17 plots the maximum DE and PAE for the two types of PAs as a function of the frequency. Both types of the PAs exhibited a similar high-efficiency operational bandwidth. The bandwidth within which the PA operates with DE > 70% were 630 MHz and 620 MHz for CCF and CCF⁻¹ PA, respectively. Table 2 and Fig. 2 demonstrate that the bandwidths achieved by the proposed CCF and CCF⁻¹ PAs are comparable to those of the state-of-the-art PAs, despite the remarkable reduction in circuit size.

Figure 16.

Measured power performance for proposed (a) CCF PA and (b) CCF⁻¹ PA.

Figure 17.

Frequency variation of efficiency for (a) CCF PA and (b) CCF⁻¹ PA.

Table 2.

Comparison of state-of-the-art broadband high-efficiency PAs

* Estimated from graph.

However, notwithstanding the similarity regarding the bandwidth between the two PAs, there is a distinct difference in the efficiency behavior. While the efficiency of the CCF⁻¹ PA gradually decreased with frequency above 2.2 GHz, the efficiency of the CCF PA collapsed at a frequency of around 2.2 GHz. The behavior of the CCF PA was expected because it was anticipated that the 2^nd harmonic source impedance would intersect the point of local efficiency minima as the frequency is increased, as depicted in Fig. 11(a). Thus, this is the main reason why we targeted a high-efficiency operational bandwidth of 1.7–2.3 GHz—it was challenging to ensure that Im(Z_S(2f ₀)) did not intersect with the local efficiency minima before reaching a frequency of 2.4 GHz. In this regard, the CCF⁻¹ PA had the potential to achieve a wider bandwidth, as the management of Im(Z_S(2f ₀)) was less stringent. However, for the CCF⁻¹ PA investigated in this study, the fundamental load at 2.3 GHz (the high end of the f ₀ band) was very close to the 2^nd harmonic load at 3.4 GHz (the low end of the 2f ₀ band). Consequently, this led to constraints for designing the LPF for fundamental frequency matching, resulting in decreased efficiency for frequencies less than 1.7 GHz (Fig. 17(b)). The issue may be alleviated by adjusting the ξ(f) function so that the 2^nd harmonic loads are placed similarly to the case of CCF PA (see Fig. 15), but addressing this issue remains a subject of our future work. Overall, while both the CCF PA and CCF⁻¹ PA ultimately exhibited nearly the same high-efficiency operational bandwidths, it was found that the factors fundamentally limiting the bandwidth are different.