High-Efficiency Class-iF−1 Power Amplifier With Enhanced Linearity

This article presents a new class of power amplifier (PA), designated as Class-iF−1, that utilizes input harmonics to achieve high efficiency with enhanced linearity performance beyond the conventional Class-F−1 PA. The amplitude-to-amplitude modulation (AM/AM) profile of the conventional Class-F−1 PA is mathematically modeled as a function of the input drive level, such that the occurrence of inflection points can be investigated. Theoretical derivation shows that the appropriate utilization of input nonlinearity poses a solution to rectify the double inflection characteristics of the conventional Class-F−1 PA, which, consequently, can be realized by proper manipulation of second harmonic source impedance ( $Z_{2S}$ ). The theoretical findings were validated with load—pull results at 2.3 GHz with a 2-mm gallium nitride (GaN) device, presenting enhanced linearizable output power and efficiency for the Class-iF−1 PA, with a broad second harmonic design space over the open-circuit region. As proof of concept, a Class-iF−1 PA was designed and fabricated, obtaining 40.1–40.8 dBm output power and 71.2%–77.3% drain efficiency (DE) performance at 3-dB gain compression level operating over 2.0–2.6-GHz frequency range. When tested with a 20-MHz 8.5-dB peak-to-average-power-ratio (PAPR) long-term evolution (LTE) signal, around 32.01-dBm average output power was attained at 2.3 GHz with an average DE of 34.59% and −56.05 dBc adjacent channel power ratios (ACPRs) after digital predistortion (DPD) correction.


I. INTRODUCTION
T HE demand for highly efficient, wideband, and linear power amplifiers (PAs) has been drawing considerable attention in various industrial applications. To achieve high efficiency with optimal saturated output power, harmonic tuned Manuscript received 30  PAs are realized by minimizing the overlap between the drain current and voltage waveforms and increasing the drain voltage fundamental amplitude via a proper manipulation of harmonic load impedance [1], [2], [3], [4], [5], [6], [7], [8].
To resolve the highly nonlinear double inflection characteristics in the conventional Class-F −1 , a new class of PA, called "Class-iF −1 ," has been proposed in [29], by introducing a concept of terminating the second harmonic source impedance (Z 2S ) with open circuit instead of conventional short circuit, which results in a flat gain profile as shown in Fig. 1(b). Due to the page limit of the conference paper, no detailed analysis was given. In this article, we extend the concept with further theoretical derivations and system analysis. To explain the double inflection characteristic in the AM/AM profile of Class-F −1 PA, a comprehensive derivation with a set of new current modeling is given so that the time-domain gate-drain waveforms and the AM/AM profile can be analyzed versus different output power level and input nonlinearity parameters. It is then theoretically derived that, by varying the value of one specific input nonlinearity parameter, the first inflection point can be shifted to a higher output power level. The loadline of the PA can avoid entering the knee region at the low power level, as shown in Fig. 1(c) and (d). This explains the root cause of the double inflection characteristics of a conventional Class-F −1 PA and identifies the solution to it. To the best of the author's knowledge, this is also the first time that the drain current and voltage waveforms are modeled as a function of power driving level and input nonlinearity. The theoretical analysis and experimental validation confirm that a high-efficiency Class-iF −1 PA can be designed with the second harmonic source impedance (Z 2S ) over the open-circuit region The remainder of this article is organized as follows. Section II presents the theory of the proposed Class-iF −1 PA. Section III shows the load-pull PA performance with different input nonlinearity, namely, different second harmonic source manipulation. The design procedure of a Class-iF −1 PA and the experimental validation are presented in Section IV, with a conclusion in Section V.

II. THEORETICAL ANALYSIS OF CLASS-IF −1 PA
The input nonlinearity of an active device is a result of the nonlinear gate to source capacitance (C GS ) profile, which varies as a function of both the gate-source voltage (V GS ) and the drain-source voltage (V DS ) [18], [19], [20], [21], [22], [23], [24], [25], [26]. For the conventional harmonically tuned PAs, an ideal sinusoidal excitation is assumed at the input of the device. However, as shown in Fig. 2(a), the nonlinear C GS -V GS profile generates out-of-phase second harmonic voltages at the intrinsic gate node of the device and results in the alternation of the input sinusoidal waveform [30], [31], [32], [33].
Although the intrinsic gate and drain waveforms have been derived versus input nonlinearity in the previously reported papers [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], the analyses were generally focused on the saturated output power level, which is limited for explaining and predicting the inflection points in AM/AM profile for the conventional Class-F −1 [19], [20], [21], [25], [27], [28]. To fully understand it, namely, the double inflection characteristics in the AM/AM profile, we revisited the time-domain drain current and voltage waveform with the consideration of input nonlinearity, while maintaining the same load terminations with conventional Class-F −1 , where second harmonic load (Z 2L ) is open circuit and third harmonic load (Z 3L ) is short circuit. The time-domain waveforms modeling is presented in detail in Sections II-A-II-C.

A. Gate Waveform Shaping
Assuming the device is excited by a sinusoidal input signal and considering up to second harmonic source impedance, the generalized gate voltage waveform can be expressed as where V GS0 is the dc gate bias voltage, and V GS1 and V GS2 are the fundamental and second harmonic gate voltage components, respectively. Two nonlinearity parameters, φ 2 and γ , are defined to describe the input nonlinear characteristics, where φ 2 is the phase difference between V GS2 and V GS1 , and γ = V GS2 /V GS1 denotes the normalized magnitude of the second harmonic gate voltage with respect to the fundamental voltage. Fig. 2 shows the input nonlinearity parameters γ and φ 2 for a 2-mm GaN device [25]. With respect to the second harmonic source phase 2S , the value of γ varies from 0 to 0.7. While the value of φ 2 changes over (170 • , 320 • ), φ 2 keeps that value of 180 • in most of cases. For simplicity, the analysis is under the circumstance of φ 2 = 180 • , with which the γ value varies from 0.15 to 0.25, while Z 2S sweeps over Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
Generally, the class of PA operation can be categorized based on the device conduction angle α, namely, the bias condition. Herein, α is the value with the assumption of a second harmonic source short circuit. However, the input nonlinearity changes the conduction angle from the original value of α [21], [23]. The modified conduction angle, β, for a Class-B bias condition (α = π) can be found by solving (2)

B. Drain Current Waveform Modeling
To quantify the PA performance at different output power levels, herein, we define a new parameter, ρ, as the input voltage scaling factor. To analyze the double inflection characteristic, a comprehensive derivation is conducted with the drain current waveform with respect to power driving level: the low-power (preinflection) region with 0 ≤ ρ ≤ ρ 1 , and the high-power (post-inflection) region with ρ 1 < ρ ≤ 1. Hence, drain current waveform modeling is derived as According to (2), for an active device with constant transconductance (g m ), the generalized bias-dependent drain current waveform can be defined as where I max is the maximum current limit of the device. Under the previous Class-B biasing, by substituting α = π to (2) and (4), the generalized drain current before the first inflection point, namely 0 ≤ ρ ≤ ρ 1 , can be expressed as To further evaluate the inflection point and find a solution for the conventional Class-F −1 , the drain current waveform is different from (5) after the power driving level ρ 1 . Load impedance conditions with the second harmonic (Z 2L ) open circuit and third harmonic (Z 3L ) need to be considered for proper drain current modeling. Although the intrinsic current waveform i DS,F −1 attempts to generate the second harmonic component i 2 , it cannot be achieved due to infinite Z 2L . Hence, under the input nonlinearity, the drain current waveform within power level, ρ 1 < ρ ≤ 1, can be expressed as where i 2 = r cos 2θ − q sin 2θ (7) herein, r and q denote the coefficients for real and reactive terms of second harmonic drain current, respectively. In both (4) and (6), there is no surprise that the coefficient of reactive term q appears to be a function of γ and φ 2 due to the input second harmonic nonlinearity. By using the Fourier transformation for (4) and (5), the dc (I dc ), fundamental (I 1 ), second harmonic (I 2 ), and third harmonic (I 3 ) components can be calculated as functions of ρ, β, γ , and φ 2 as (8)-(14), shown at the bottom of the next page, where I nr and I nq (n = 1, 2, 3) represent the real and reactive current components, respectively. Fig. 3(a)-(c) shows the normalized intrinsic drain current waveforms versus power driving level with γ = 0, 0.2, 0.35, respectively. The waveforms for γ = 0.35 have lower ripple, whereas clipping happens at ρ = 0.2 for γ = 0.

C. Drain Voltage Modeling
To analyze the drain voltage components, the fundamental and harmonic load impedance conditions are needed and can be expressed as where R opt is the optimal fundamental load impedance at Class-B operation and is defined as Herein, V DD denotes the drain voltage dc component, and V k is the knee voltage. Hence, the drain voltage can be evaluated by using the impedance values in (15). At the first inflection point ρ 1 , the drain voltage hits the knee voltage region (V K ). Assuming knee voltage to be zero (V K = 0), the voltage swing at the intrinsic drain node can be expressed as a function of drain voltage fundamental (V 1 ), second harmonic (V 2 ), and third harmonic ( where For simplicity, the load terminations are considered only resistive in the following analysis. When the voltage swing V DS,swing equals to V DD , the driving level reaches the inflection point ρ 1 . The value of Z 2L can be chosen as ten times the value of fundamental impedance Z 1L or higher based on the practicality of realization for Class-F −1 mode of operation. On further increasing the driving level ρ, the drain voltage swing V DS,swing remains fixed as V DD . This is due to the fact that the loadline keeps hitting the knee region with the drain voltage swing of V DD . The second harmonic current component should be maintained at zero until the saturated output power level with ρ = 1. As such, the second harmonic current i 2 coefficients r and q can be calculated by solving the expressions given as Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. By substituting (9)- (12) to (18), the coefficient r and q of i 2 can be solved and calculated as where σ 1 = 3β + 3 sin β + 2γ cos φ 2 3 sin β 2 + sin 3β 2 As φ 2 equals to 180 • , the value of q becomes 0. By using (15)-(17), the generalized intrinsic drain voltage waveform can be expressed as Hence, the intrinsic drain voltage waveforms are plotted in Fig. 3(d)-(f). Similar to the drain current waveforms in Section II-B, when γ increases, the voltage waveforms become flatter, which provides a hint of how input nonlinearity changes the drain waveforms at different output power levels. Yet, it is still not enough to explain the occurrence and predict the locations of the first inflection point in the AM/AM profile.
The normalized drain current components versus output power for γ = 0, 0.2, 0.35 are shown in Fig. 4(a)-(c), respectively. Similarly, the normalized drain voltage components are also plotted in Fig. 4(d)-(f). When the output power rises, the dc (I dc ) and fundamental (I 1r ) drain current components increase monotonically. It is interesting to see that the position of the drain current second harmonic (I 2r ) first peak moves forward to a higher power level when the γ value increases. At the first inflection point, drain voltage hits the knee voltage and voltage swing equals V DD at low input power driving level. The reason for this is the high second harmonic drain impedance causing V 2 to increase at a much faster rate than V 1 .

D. Proposed Class-iF −1 PA
It is of the utmost interest to see the theoretically estimated PA performance such as the AM/AM profile and DE. Assume the source impedance is Z 0 , the input power can be calculated as The output power can be calculated as Accordingly, the DE and power gain can be calculated as  As shown in Fig. 5(a), when γ increases, the AM/AM profile achieves better flatness as the first inflection point moves to the higher output power level with decreased gain compression, leading to enhanced performance in terms of linearity, while maintaining almost the same efficiency profile, as shown in Fig. 5(b). Herein, γ = 0 indicates the second harmonic source with short-circuit termination, referring to the conventional Class-F −1 . It should be noted that a continuous increase of γ helps push the inflection point toward a higher power region and corrects the flattens of the AM/AM profile. However, after a certain value of γ , over-correction will not bring further benefits. This is because the inflection point from the harmonic is pushed away to the higher power region where the compression resulting from the fundamental drain voltage and current components dominates.
As shown in Fig. 5(c), the loadline for γ = 0 starts entering the knee region at a low power level, resulting in early gain compression, hence, the first inflection of AM/AM profile. Conversely, when the value of γ becomes 0.2 or 0.35, the loadline stays far from the knee region at low power and only enters the region at saturated output power level, as shown in Fig. 5(c) and (d). This phenomenon, in return, results in significant advantages of delaying the first inflection, and ultimately, eliminating the double inflection characteristic in the AM/AM profile. To practically realize the higher γ value, the second harmonic source (Z 2S ) manipulation is required as shown in Fig. 2(b). As has been analyzed in Section II-A, when utilizing the γ value from 0. 15  The AM/AM profile is then expressed by normalizing the power gain versus output power to a small-signal gain AM AM = G T, dB − G S,max dB (25) where G S,max dB is the maximum value of the small-signal gain. By substituting (9), (15), and (22)- (24) to (25), the AM/AM profile can be predicted as a function of input drive level, ρ, and input nonlinearity parameter, γ . Therefore, the PA class with the γ value higher than 0.15 is designated as Class-iF −1 PA, which is realized by terminating Z 2S along the open-circuit termination at the edge of Smith chart. Hence, high saturated efficiency, low gain

III. LOAD-PULL ANALYSIS
To validate the performance in terms of output power, efficiency, and linearity, load-pull analyses were conducted with a 2-mm GaN device at 2.3 GHz with controlled load harmonic terminations, second harmonic load impedance (Z 2L ) as open, third harmonic load impedance (Z 3L ) as short. The fundamental load impedance (Z 1L ) was terminated at the maximum efficiency (MXE) point, while the fundamental source termination (Z 1S ) was under complex conjugate match. The drain voltage supply of the device was at 28 V and the dc quiescent current was set to 20 mA/mm.

A. Load-Pull With CW Signal Stimulation
The load-pull was first conducted under the CW signal stimulation. The second harmonic source impedance Z 2S is varied by sweeping the second harmonic source phase 2S from 0 • to 360 • with | 2S | ≈ 0.9. Although a higher reflection coefficient is desirable, | 2S | ≈ 0.9 is enough to extract the important source-pull data.
The parasitic parameters were de-embedded to extract the design impedance and intrinsic voltage and current waveforms at the intrinsic drain plane of the active device. The fundamental load impedance for power added efficiency (PAE) contours  Fig. 6(b), (e), and (h) shows the magnitude of the extracted intrinsic fundamental, second, and third harmonic drain voltage components, respectively, for the corresponding γ values. We can observe the variation of the second harmonic drain voltage component (V 2 ) and how it is suppressed with different levels of input nonlinearity (γ ). For a conventional Class-F −1 PA (γ = 0), the drain voltage waveform swing (V 1 + V 2 + V 3 ) equates to the swing value of V DD and enters knee region at an early power level, as shown in Fig. 6(c). At the same time, the second harmonic component V 2 hits its peak and starts to drop versus the further increased output power level. When the device loadline enters the knee region due to voltage saturation, its gain starts dropping and results in a double inflection characteristic in a conventional Class-F −1 PA. When the γ value increases, the peak value of V 2 decreases and pushes to the higher power level, which in turn results in the elimination of the first inflection point in AM/AM profile, as shown in Fig. 6(c), (f), and (i). It is interesting to see that the conventional Class-F −1 achieves 37.2 dBm P 3 dB (output power with 3-dB gain compression)  Fig. 7. The proposed Class-iF −1 PA shows a monotonous AM/PM profile which is highly desirable.
Compared to Class-F −1 , the proposed Class-iF −1 (γ ≥ 0.15) has much less compression at the same saturation power level, showing enhanced linearity performance. These load-pull validations provide a comprehensive insight into the mathematical derivation versus power driving level in Section II.
In consistency to [29], a broad design space with source second harmonic around the open-circuit region is confirmed, where excellent gain flatness can be achieved. Based on the comprehensive analysis and load-pull simulation, a generalized second harmonic source design space is concluded and shown in Fig. 8

B. Load-Pull With Modulated Signal Stimulation
To further evaluate the performance in terms of output power, efficiency, and linearity in the practical wireless communication system, load-pull simulation was also conducted under the stimulation of modulated signal. The comparison was performed between the conventional Class-F −1 (γ = 0) and Class-iF −1 (γ = 0.2). A 20-MHz long-termevolution (LTE) signal with 8.5-dB peak-to-average power   Table I, compared with the Class-F −1 PA, the Class-iF −1 PA has improved average output power (∼+1.8 dBm) and average efficiency (∼+3.6%) while maintaining the ACPR lower than −55 dBc after DPD correction. Therefore, the utilization of open-circuit second harmonic source termination significantly improved the performance in terms of average output power, average efficiency, and linearity under the modulated signal stimulation for the proposed Class-iF −1 PA.
IV. PROTOTYPE AND EXPERIMENTAL RESULTS As a proof of concept, an RF-input Class-iF −1 PA was designed and fabricated from 2.0 to 2.6 GHz, using a GaN HEMT (CG2H40010F) from Wolfspeed. The designed PA was realized on a 31-mil-thick Taconic TLY-5 substrate with a 2.2 dielectric constant. Fig. 10(a) and (c) shows the output matching network and input matching network. To extract the intrinsic source/load impedance, the de-embedded network for the CG2H40010F transistor was utilized. The intrinsic trajectories shown in Fig. 10(b) indicate the typical Class-F −1 load conditions. As shown in Fig. 10(d), it can be seen that the intrinsic second harmonic source impedance Z 2S is over the opencircuit region, which follows the theoretical design space of the proposed Class-iF −1 . Fig. 11 presents the photograph of the fabricated PA. The drain supply voltage of 28 V and total quiescent current of 5 mA were set during all the measurements. The CW and modulated signals were both generated by a vector signal generator, and the output power was measured with a spectrum analyzer. A broadband linear driver amplifier was used to drive the PA with enough input power. A broadband circulator was added between the driver and the PA to improve isolation.

A. Measurement Results With CW Signal
The PA was first measured under CW signal stimulation. Fig. 12(a) presents the measured DE and gain versus output power at different operating frequencies. The measured DE, PAE, output power, and gain at 3-dB gain compression (P 3 dB ) level versus operation frequencies are presented in Fig. 12(b). From 2.0 to 2.6 GHz, the proposed Class-iF −1 PA can achieve 40.1-40.8-dBm output power, 71.2%-77.3% DE, and 67.4%-74.1% PAE at P 3 dB level. Moreover, Fig. 12(b) also shows 58.7%-66.7% DE and 57.2%-64.5% PAE at 1-dB gain compression (P 1 dB ). Therefore, high saturation efficiency and flat gain response over the operation bandwidth are obtained.

B. Measurement Results With Modulated Signals
To evaluate the linearity and efficiency performance under modulated signal stimulation, 20-and 100-MHz LTE signals with 8.5-dB PAPR were used to test the PA. Fig. 13(a) presents the output spectrum with and without DPD linearization at 2.3 GHz with the 20-MHz signal. The measured ACPRs were −28.03/−27.61 dBc without DPD. After applying DPD, the ACPRs were improved to −56.05/−56.10 dBc, with around 32.01 dBm average output power and 34.59% average DE. The AM/AM and AM/PM characteristics with and without DPD are given in Fig. 13(b). Under the stimulation of the 100-MHz signal, the measured ACPRs were −22.99/−21.66 dBc without DPD and −48.76/−48.21 dBc with DPD correction at   Fig. 14(a). Around 31.80 dBm average output power and 34.21% average DE were obtained after applying DPD. The AM/AM and AM/PM characteristics with and without DPD are also given here in Fig. 14(b).     [35].

C. Performance Comparison
V. CONCLUSION In this article, we presented a new class of PA, designated as Class-iF −1 , with enhanced linearity compared to the conventional Class-F −1 PA. For the first time, the drain current/voltage waveforms and AM/AM profile are modeled as a function of power driving level and input nonlinearity. Theoretical derivation and load-pull results show that different levels of input nonlinearity result in the alternation of the AM/AM inflection points, which can be ultimately controlled with second harmonic source (Z 2S ) manipulation. A Class-iF −1 PA was designed and fabricated using packaged GaN device. Measured 40.1-40.8-dBm output power, 71.2%-77.3% DE, and 67.4%-74.1% PAE were achieved at 3-dB gain compression level over 2.0-2.6 GHz.