IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 60, NO. 12, DECEMBER 2012 4097 PA Efficiency and Linearity Enhancement Using External Harmonic Injection Asmita Dani, Student Member, IEEE, Michael Roberg, Student Member, IEEE, and Zoya Popović, Fellow, IEEE Abstract—This paper presents analysis and experimental demonstration of a high-efficiency linear power amplifier (PA) with second-harmonic injection at the output. In this circuit, the transistor is not driven hard into compression and does not produce significant harmonic content. External injection at the output enables voltage and current wave-shaping to achieve high efficiency. Theoretical analysis of the waveforms shows that the maximal drain efficiency is 89.9% with, at most, 0.13 dB of reduction in output power compared with the class-A case. The overall PA efficiency is derived in terms of the injector circuit efficiency. A harmonically injected prototype GaN HEMT 2.45-GHz PA demonstrates over 80% efficiency with linearity improved over the class-AB PA without harmonic injection. Two-tone measurements show a reduction of the third-order intermodulation by 30 dBc in the linear region and greater than 10 dBc in saturation. Index Terms—Amplifier drain efficiency, Fourier coefficients, harmonics, linearity, microwave power amplifiers (PAs), third-order intercept, waveform shaping. I. INTRODUCTION A large portion of current research in high-power amplification of signals with carriers in the microwave range focuses on improving efficiency and linearity. There are many power amplifier (PA) topologies that achieve high efficiency by driving the active device into a nonlinear region and shaping voltage and current waveforms across the device via proper selection of the output loading network at harmonic frequencies. These techniques, such as class-F and PA topologies, rely on the nonlinear active device for harmonic current or voltage generation [1]–[4]. The concept of harmonic injection, however, refers to architectures in which power at one or more harmonics of the operating frequency is supplied externally to either the input, output, or both input and output of the active device. Analysis of efficiency improvement of tube PAs using harmonic injection into both the grid (input) and plate (output) has been presented in [5]–[7]. A harmonic-injection scheme referred to as a harmonic reaction amplifier was presented in [8]. The harmonic reaction amplifier uses two parallel devices Manuscript received July 09, 2012; revised September 19, 2012; accepted September 20, 2012. Date of publication November 15, 2012; date of current version December 13, 2012. This work was supported in part by the Berrie Hill research Corporation and the U.S. Air Force under Contract FA8650-10-D1746-0006. This paper is an expanded paper from the IEEE MTT-S International Microwave Symposium, Montreal, QC, Canada, June 17–22, 2012. A. Dani and Z. Popović are with the Department of Electrical, Computer and Energy Engineering, University of Colorado, Boulder, CO 80309-0425 USA (e-mail: dania@colorado.edu; zoya.popovic@colorado.edu). M. Roberg is with TriQuint Semiconductor, Richardson TX 75080 USA (e-mail: michael.roberg@tqs.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2012.2222918 Fig. 1. Block diagram of a harmonic-injection PA (HI-PA) with second-harmonic injection at the output. A three-port network at the output allows isolaand between ports 2 and 3, while allowing low tion between waves at between ports 1 and 2. The phase of the injected harmonic is critical loss at to obtaining high efficiency. and effectively acts as a push–pull amplifier with respect to the second harmonic. In 1992, a patent was issued for a harmonic injection amplifier in which the harmonic signal created using a frequency multiplier is injected into the transistor output [9]. A class-E VHF PA at 3.5 MHz with a secondary class-E 7-MHz PA injector is presented in [10]. An experiment demonstrating 15.2% efficiency improvement of a 2-GHz GaN PA using second-harmonic injection at the input is reported in [11]. More recently, a concept for efficiency improvement via injection of harmonics into the output of a class-B/J amplifier was demonstrated [12]. A novel scheme of efficiency improvement of a class-E amplifier using input harmonic injection via a feedback loop was shown in [13]. In this paper, we discuss injection of power at the second harmonic, as shown in the block diagram of Fig. 1 and the implications on efficiency and linearity. The approach is valid for any amplifier mode, not just for class-B/J as in [12] and [14]. The contributions of this work are organized as follows. • Section II presents, for the first time, a theoretical Fourierexpansion analysis of output harmonic injection. A relationship between the total efficiency and injected power is presented for the case of second-harmonic injection. The analysis gives an insight into the impedance that needs to be synthesized at the output of the transistor at the fundamental and harmonic frequencies. The effect of efficiency of the injector circuit on the total efficiency is calculated. • Section III discusses two prototypes used for experimental validation, one based on a packaged Cree GaN 10-W 50class-AB PA and the other based on a TriQuint GaN die with an injection network designed for a non-50- environment. • Section IV presents measurement results for the relevant amplifier parameters as a function of the phase and amplitude of the injected harmonic signal for both prototypes. 0018-9480/$31.00 © 2012 IEEE 4098 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 60, NO. 12, DECEMBER 2012 The experimental results demonstrate an improvement from 58% to 80% in efficiency with an output power of 37 dBm and over 20-dB gain from a single TriQuint 6-W device. In addition, the amplifier linearity is shown to be improved with harmonic injection w.r.t. the class-AB case, which implies significant linearity improvement compared with the same device used in harmonically terminated compressed nonlinear PAs. II. THEORETICAL ANALYSIS The theoretical analysis of a harmonically injected PA (HI-PA) shown in Fig. 1 is developed based on Fourier expansions of the voltage and current waveforms at the current source of the transistor, expanding on the approach in [12] and [15]. First, expressions are derived for injected voltage which optimizes efficiency, followed by the analysis of total dissipated power and a discussion of linearity. To obtain an HI-PA, a linear three-port network is cascaded at the output of a linear PA, and the required -parameters of that network are given in [15]. Fig. 2. Optimal drain current and voltage waveforms for second-harmonic injection amplifier, normalized to 1-W output power. where corresponds to a point at which the first derivative of the voltage waveform w.r.t. is equal to zero. Note that (5) is only valid when (7) A. Waveform Analysis Consider the normalized drain voltage and current waveforms at the virtual drain of a linear field-effect transistor (FET) PA, which are given by (1) (2) where , and the bar indicates a normalized quantity. For instance, when , the normalized class-A output power is 1 W, and the waveforms result in 50% efficiency. If the drain waveforms can be shaped by harmonic content in a manner such that the overlap of the voltage and current is minimized for a given fundamental frequency output power, then drain efficiency will be maximized. Consider the addition of only the second-harmonic term in (1) and (2). In order to maintain waveform symmetry, only cosinusoidal components are added. Such a condition will result in the voltage waveform of the same shape but 180 out of phase with the current waveform. The waveforms following addition of a second-harmonic term become Therefore, the range of over which (5) is valid is limited to (8) It remains to be proven which critical points correspond to the global minima and global maxima. Substituting the critical point in (5) into the second partial derivative of (3) results in (9) If is negative in sign, the critical point corresponds to a minimum, while, if it is positive in sign, then it corresponds to a maximum. Applying the second derivative test to the critical point described by (6) results in (10) Therefore, the critical point described by (6) will be an extremum as follows: is a minimum (11) is a maximum (12) (3) (4) as shown in Fig. 2, showing minimal overlap of the voltage and current waveforms. From (3) and (4), it can be concluded that the impedance at is the negative of that at . Effectively, this requires that power is delivered to the transistor at the second harmonic. An optimal value of that maximizes the efficiency can be found. First, the critical points of the drain current and voltage waveforms are expressed as (5) (6) B. Efficiency Analysis The normalized total DC power consumed by the amplifier is expressed as (13) is the efficiency of the injector circuit. Note that, due where to the symmetry of the current and voltage waveforms, , the optimal dc supply voltage is that which results in a drain voltage waveform with minimum of zero. Therefore, from (3), we have (14) DANI et al.: PA EFFICIENCY AND LINEARITY ENHANCEMENT USING EXTERNAL HARMONIC INJECTION Fig. 3. Contour plot for . as a function of and injector circuit efficiency 4099 Fig. 4. Optimal solution for Fourier coefficient and second-harmonic delivered power relative to fundamental frequency output power versus second . harmonic injector circuit efficiency The total dc power may now be expanded to the form (15) (16) Since we use a normalization that sets the fundamental output power to 1 W, the total efficiency is calculated as the inverse of the normalized dc power Fig. 5. Total efficiency versus injector circuit efficiency . (17) power , also shown in Fig. 4. The PA efficiency is determined by inserting into (15) and (16) to yield Fig. 3 shows the total efficiency plotted as a function of both and normalized magnitude of secondinjector efficiency . harmonic The value of is optimized by setting the partial derivative of w.r.t. the Fourier coefficient to zero and solving for as follows: (20) (21) (18) (22) (19) , the optimal These values minimize . Given . A plot of , Fourier coefficient reduces to which corresponds to the amplitude of the required injected second-harmonic versus , is shown in Fig. 4. As one would expect, the magnitude of the Fourier coefficient decreases as the injector efficiency decreases. Another interesting parameter to investigate is the ratio of the delivered fundamental output power to the required delivered second-harmonic injected (23) A plot of the total efficiency versus injector circuit efficiency is shown in Fig. 5 for optimized solution at and a plot for as a function of , and is shown in the total efficiency Fig. 3. The maximum value is 89.9%, and it rolls off reasonably slowly with decreasing injector efficiency. This is intuitive because the power required from the injector is significantly lower 4100 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 60, NO. 12, DECEMBER 2012 relative to 1 W. When , the output power is only reduced by only 0.13 dB relative to the class-A amplifier. Also, it is of practical interest to find the supply voltage and current normalized to class-A: (30) Fig. 6. Power reduction and normalized supply voltage relative to class-A versus injector efficiency . than the fundamental output power of the amplifier, as shown in Fig. 4. As expected, when the amplifier efficiency reaches 50%, the injector circuit is turned off. In this case, the amplifier degenerates to the canonical class-A mode. As previously mentioned, the load presented to the transistor at the second harmonic is the negative of that presented at the fundamental, so the load resistance normalized to the class-A fundamental load is 1. To find the output power of the PA normalized to class-A output power, normalization conditions corresponding to peak voltage and current constraints are enforced. and are now found, enabling determination of the maximum instantaneous normalized voltage and cur, and the output power normalized to the class-A amrent plifier output power is determined. The normalized dc voltage is expressed as (24) (25) Note that, due to the symmetry of the current and voltage wave, the maximum normalized voltage is calcuforms lated as (26) (27) The output power normalized to class-A is then given by (28) (29) Fig. 6 depicts the fundamental frequency output power reduction relative to a class-A amplifier versus the injector efficiency. This was calculated by computing as a function of , then and determining the ratio computing the output power from Fig. 6 shows the normalized supply voltage which is approxi. mately 0.7107 for A similar analysis can be performed for third-harmonic injection at the output, since symmetric square waveforms can be achieved using odd harmonics only. In the case of third-harmonic injection, the impedance at the third harmonic is positive rather than negative, so the ideal waveforms can be realized with a passive set of output terminations. The analysis shows, however, that the total efficiency given by (22) and (23) is around 65% for injector efficiencies above 40% and does not reach the high efficiencies of the second-harmonic injection case. Details of the analysis can be found in [16]. C. Linearity Transistors exhibit nonlinearities due to various factors such as input and output device capacitance, transconductance, and drain–source resistance resulting in a characteristic between and , which can be represented using the power series (31) where . As seen in (31), the second-order nonlinearity causes an additional dc component and a signal at twice the fundamental frequency to appear in the output voltage. For a two-tone signal, the second-order nonlinearity can be easily filtered out and does not cause any in-band distortion of the signal. However, the third-order nonlinearity results in in-band distortion products. The gain of the fundamental component under nonlinear operation can be expressed in terms of the fundamental gain and the third-order gain and amplitude, and the derivation is given in [17]. The analysis in [17] also shows that the amplitude of the second-harmonic output signal is inversely proportional to the magnitude of the transfer characteristic of the amplifier at the third harmonic, which is referred to in Section IV. A good discussion on extracting linearity information from a continuous-wave (CW)-fed amplifier by measuring the third-harmonic output content is presented in [18]. Based on this theory, in this paper, a CW signal is used for harmonic injection analysis as the device enters saturation. In particular, we measure second and third harmonic as a function of the injected power and phase in order to assess the linearity. III. PROTOTYPE PA DESIGN Two prototype PAs were used to demonstrate the HI-PA concept. A packaged device in a demo board with class-AB broadband PA configuration is injected through a 50- three-port injection circuit described in more detail in [12], [14], and [15]. In order to have more design freedom and lower matching network loss, a TriQuint GaN die was used in the second narrowband prototype with a non-50- three-port injector circuit. DANI et al.: PA EFFICIENCY AND LINEARITY ENHANCEMENT USING EXTERNAL HARMONIC INJECTION 4101 Fig. 8. Measured drain efficiency at PA shown in Fig. 7 without injection. and and gain for the class-AB Fig. 7. Hybrid HI-PA with a 6-W TriQuint TGF2023-01 die. The output at network integrates the harmonic injection three-port network with matched to 65 and at matched to 71 . The input network does an impedance transformation from 50 to 10 in order to achieve high gain and at the fundamental. A. Packaged Device Prototype PA is used in a broadA Cree GaN HEMT with 10-W band (DC-6 GHz) demo board provided by the manufacturer (CGH40006P-TB) with a packaged device and matched to 50 as the first prototype. This PA gives 40 dBm at 2.45 GHz with and 12 dBm [15]. The input power to the fundamental PA is swept from 22 to 34 dBm (linear to saturation) with the drain bias at 22 and 28 V. The gate bias was set to 1.6, 1.8, and 2 V for a class AB mode. The fundamental PA starts compressing at an input power level of 27 dBm when no harmonic injection is present. B. Discrete Die Prototype PA A hybrid class-AB PA is designed using load-pull measurements on the TriQuint TGF2023-01 device at 2.45 GHz, as shown in Fig. 7. The reference plane for all measurements on this PA is the virtual drain of the transistor, i.e., the current source behind the output capacitance of the device. The HI-PA is designed with the three-port injection network integrated into the output matching circuit of the amplifier in order to minimize loss. Design and performance of the injection network is similar to the one presented in [12] and [15] with the fundamental and second-harmonic impedances matched close to 65 as explained in (3) and (4). Due to fabrication tolerances, the fundamental impedance at the virtual drain of the device was found to be matched to 65 and the impedance to 71 (10% higher). This class-AB PA has 58% drain efficiency with an output power of 37 dBm without any harmonic injection at a drain bias voltage of 28 V. Fig. 8 shows the measured drain efficiency , output power at fundamental , third harmonic , and the gain as a function of fundamental . input drive power IV. HI-PA MEASUREMENTS AND ANALYSIS The block diagram shown in Fig. 9 shows the measurement setup for HI-PA prototypes. A portion of the fundamental input is frequency doubled to create the second harmonic for injection. A voltage-controlled phase shifter and variable gain Fig. 9. Block diagram of the HI-PA measurement setup. The input signal is split . A voltage and frequency doubled to create the injected harmonic, controlled phase shifter and variable gain amplifier are used to control the am. plitude and phase of amplifier (VGA) are used to control the amplitude and phase at . All of the measurements are de-embedded to the virtual drain of the transistor by calibrating the loss in the output network and taking into account the intrinsic transistor parasitics, i.e., output capacitance of the device. A bondwire model in Ansoft High Frequency Structure Simulator (HFSS) was simulated to consider the inductance loss in the bondwire transition for the hybrid PA design. The drain efficiency for an HI-PA takes into account the amount of second-harmonic power injected into the virtual drain of the device assuming as follows: (32) where . A. Packaged Device Prototype HI-PA Fig. 10 compares the measured efficiency, output power, and gain for the PA with and without harmonic injection. It is seen that the HI-PA saturates at a higher input power (32 dBm) as compared with the class-AB PA (27 dBm), resulting in higher 4102 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 60, NO. 12, DECEMBER 2012 Fig. 11. Comparison of power levels for single-tone and third-order IMD products for HI-PA and class-AB PA without harmonic injection. phase adjustment. The measured results in Fig. 10 show that the HI-PA (marked line) saturates at a higher input power than the PA with no harmonic injection (solid line). At lower input powers, the IMD3 level is over 30 dB lower for the HI-PA and remains 10 dB lower after the PA saturates. In Fig. 11, only the frequency is injected, resulting in a decrease in the IMD while the remains unchanged. Symmetrically, is injected, the will decrease. Both of the when IMD products will be reduced equally for a signal injected at . This is consistent with measurements obtained with tube amplifiers in [19] and harmonic injection at input of solid-state amplifiers in [20], [21]. B. Discrete Die Prototype HI-PA Fig. 10. Comparison of measured (a) drain efficiency, (b) , and (c) 22 V, 28 V gain for the HI-PA to the PA with no harmonic injection at 1.6 V (class-AB). The dashed green line indicates input power at and which the PA becomes nonlinear. linearity. The gain of the HI-PA is lower by about 1 dB as compared with the fundamental PA in the linear region, but remains higher in saturation. Harmonic balance simulations using a nonlinear model provided by the manufacturer show the same trends for gain as the measured data. Measured results show that higher efficiency can be achieved for a constant output power with HI-PA by changing the operating bias point. For instance, the drain efficiency of the PA improves from 58% with no injection to 75% with injection for an output power of 40 dBm by changing the drain bias from 28 to 22 V. A two-tone linearity test is performed at 22 V 1.8 V. The two tones are kept 5 MHz apart and with 2.45 GHz and third-order intermodulation (IMD3) products generated at 2.455 GHz 2.46 GHz. Simultaneously, , , or and is injected at the output, each requiring a different The HI-PA using a TriQuint 6-W GaN discrete high-electron mobility transistor (HEMT) in a class-AB PA achieves a high total drain efficiency of 89% with external second-harmonic injection at the output at a bias voltage of 22 V. This efficiency is very close to the theoretical efficiency of 89.9% from Fig. 3, though one would expect it to be lower. The reason being that the theory is derived for an ideal device with ideal – curves which do not take into account knee voltage of the transistor and does not generate any odd-order harmonics. In practice, the PA always generates some harmonic content even at lower input power levels. The gain of the amplifier reduces by 1 dB as compared with the amplifier without any harmonic injection. Fig. 12 shows a comparison of the measured performance for the HI-PA and PA without harmonic injection. These measurements are optimized for high efficiency and hence the amplifier is nonlinear at . It is seen that a better performance is achieved with the discrete device as compared with the results presented in Fig. 10 for the packaged device, as expected. As seen in the theoretical analysis (30), harmonic injection implies a shift in the bias voltage in order to get the optimum performance from the amplifier. Fig. 13 shows the performance of the HI-PA at different drain bias voltages for a fundamental input drive of 16.2 dBm. The HI-PA is then optimized in order to get high efficiency along with linearity. As explained in Section II-C, for a CW amplifier, the values of can give an estimate of the linearity. It is seen 24 V, the drain efficiency of the from Fig. 13 that, at HI-PA is improved by over 20% as compared with the class-AB DANI et al.: PA EFFICIENCY AND LINEARITY ENHANCEMENT USING EXTERNAL HARMONIC INJECTION Fig. 14. Contour plots of (a) measured fundamental output power . dBm and (b) drain efficiency Fig. 12. Comparison of measured (a) , (b) , and (c) gain for discrete die prototype of HI-PA optimized for maximum efficiency. Fig. 13. Drain efficiency with the ratio , 0.1 and for different bias voltages 16.2 dBm. PA with no injection, and the output power at the third harmonic is lowered by 30 dB for an input drive level of 16.2 dBm. At this bias point, conditions for high linearity 4103 in and high drain efficiency are obtained with a nominal fundareduction of 0.26 dB over the mental output power PA without any injection. Note that, when harmonic injection is performed, it results in higher fundamental output power due to reduction in other harmonic content. In order to keep the output power constant and reduce the dc power dissipation, the drain supply voltage can be reduced to a certain extent, as shown in Fig. 13. The supply voltage reduction is only advantageous up to a device-dependent lower value when the output power starts decreasing. In the case of packaged prototype, 22 V was found to be optimal, and, in the case of discrete prototype, this value was 24 V. Note that the PA with no injection is not designed for high efficiency, since it is biased in class-AB, and the gate bias is kept the same for the PA with and without injection. All of the measurements presented in the remainder of the paper will be at 24 V. As discussed in Section II-A, the injected second-harmonic signal needs to be at a particular phase and amplitude in order to shape the voltage and current waveforms at the virtual drain of the amplifier. If the input drive is kept constant, various parameters affecting the performance of the HI-PA such as , drain efficiency , drain current , and power at the vary with the amplitude and harmonics phase of injected second harmonic power as shown in Figs. 14 and 15. Figs. 14 and 15 show that, for 9 dBc and a phase shift of 80 , high drain efficiency of 79% is achieved using (32) along with extremely low values of . Note that this efficiency takes into account the power of the injected 4104 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 60, NO. 12, DECEMBER 2012 Fig. 16. Comparison of drain efficiency and gain for HI and PA with no injec. tion as a function of Fig. 15. (a) Drain current (amperes) and (b) third-harmonic output power in dBm for 16.2 dBm depicting the variation in the values of these parameters as a function of phase and power level (dBc w.r.t. ) of the injected second harmonic . Analogous plots can be . obtained for the second-harmonic output power signal. However, the efficiency of the injector circuit is not included in this proof-of-concept experiment in which the HI-PA is not fully integrated. The value of obtained at this point is approximately 37 dBm, which is only 0.2 dB lower than the fundamental output power obtained with no injection (Fig. 8). The measurements show that, if is not at the optimum phase and amplitude, the performance of the amplifier can severely degrade. When the second-harmonic voltage is out of phase relative to the optimal value, the amplitude of increases, making the amplifier extremely nonlinear. The efficiency reduces from 80% to 40% while the output power drops more than 3 dB. If is higher than the optimum value (in this case, 9 dBc), then, even at the optimum phase of the injected harmonic, the HI-PA is highly nonlinear. This is due to undesired additional second-harmonic content in the output voltage and current waveforms generated by the device under hard drive. A sweep is performed at in order to achieve the optimal performance of the HI-PA at various input drive levels. Since an amplifier undergoes AM–AM and AM–PM distortion, the optimal phase and amplitude of the injected second-harmonic changes for different input drive levels. Fig. 16 shows a comparison of the gain and drain efficiency obtained for HI-PA and PA without harmonic injection as a function of . The efficiency obtained at each input drive level is for an optimal value of amplitude and phase which are also dependent on Fig. 17. Comparison of and as a function of for HI and PA with no injection. The graph also shows the amplitude of as a function of in order to achieve high efficiency and linearity performance for the HI-PA. . The overall gain of the HI-PA is reduced by 1 dB, and the 1-dB compression point of the HI-PA is shifted to a higher of 15.7 dBm, implying improved linearity. The drain efficiency improvement ranges from 8% to 20% as the input drive level increases. The comparison of and for the HI-PA and as PA with no injection is shown in Fig. 17 along with a function of . As derived in [17], the transconductance at the third harmonic is inversely proportional to the amplitude of the second harmonic. The value for gain of an amplifier under nonlinear operation can be given from (31) as (33) where and are values for transconductance at fundamental and third harmonic. Therefore, it is seen that the value of required to lower the value of for the HI-PA is exactly equal to . The nominal class-A operation of the PA without harmonic injection with a 50% drain efficiency is achieved at 13 dBm. Fig. 17 shows that, at this input drive level, the amplitude of required in order to achieve an optimum performance in terms of efficiency and linearity for the HI-PA is 10 dBc. This result matches with the theoretical analysis presented in Fig. 4 where, for a 100% injector efficiency, the ratio of to is 0.1 for a class-A bias point. DANI et al.: PA EFFICIENCY AND LINEARITY ENHANCEMENT USING EXTERNAL HARMONIC INJECTION Fig. 18. Comparison of power at , for HI-PA and PA without harmonic injection as a function of input drive level. to The graph also shows the power injected at the second harmonic tone . achieve lowest Two-tone measurements with optimization for the amplitude and phase of the injected second harmonic in order to achieve lowest IMD3 products in both lower and upper sidebands are performed. This measurement is similar to the one presented in Section IV-A for the packaged device prototype HI-PA where either or are injected at the output of the HI-PA. Here, the injection of affects the perat and at due to formance of active impedance synthesis at the injection port. It is important to note that the reduction in results from mixing of and distortion products caused due to second-order nonlinearities. It is seen that the reduction in using external second-harmonic injection is greater than 15 dB for different input drive levels, whereas at and at remain unaffected. Fig. 18 shows the reduction in power levels for and achieved for different fundamental input drive levels along with the amount of injected power. For practical communication signals, the harmonic injection path needs to be modified in order to inject an exactly doubled spectrum of the signal. As seen in the two-tone measurements, injection at one harmonic tone only affects the distortion products which are a function of that harmonic tone frequency. Since, a modulated signal in general is a multitone signal, it will require a injected signal with twice the modulation bandwidth and RF carrier. This can be accomplished by baseband signal up-conversion. V. DISCUSSION The results above show that a PA with harmonic injection in the output can be both efficient and linear. In the demonstrated results above, we start with a class-AB PA, which is not perfectly linear. In fact, the theory shown in Section II assumes that some second-harmonic content is generated by the active device. If the transistor fails to generate second-harmonic power and presents an impedance other than that of the fundamental frequency output termination, the necessary negative impedance cannot be synthesized using harmonic injection. In this case, harmonic injection at both the input and output of the transistor would be required. 4105 Fig. 19. Minimum and measured at virtual drain of the 16.2 dBm. The minimum for is obtained with HI-PA for 17.8 dBc w.r.t. , whereas minimum for is 8.9 dBc. obtained for It is of interest to discuss some limitations on linearity and efficiency that are practically achievable. We have shown that the third harmonic, which directly affects IMD performance, is minimized for a specific phase and amplitude of the injected second harmonic. However, the injected signal also affects the nonlinear content in the waveform produced by the transistor, which can be evaluated by measuring the level of the second harmonic at the output. The amount of injected second-harmonic power that results in a minimum of harmonic content in the output is shown in Fig. 19. Note that the second and third harmonic have minima for different injected power levels of the second harmonic. The amplitude of needed to lower is approximately 10 dB less than that needed to lower . Also, the phase shift for injection differs by 50 . As seen from Fig. 14, the drain efficiency drops by approximately 10% between these two points in amplitude and phase. For a modulated input signal, the third-order nonlinearities have to be minimized since they create in-band distortion which is extremely difficult to filter. The third-order distortion products are a function of the amplitude of second harmonic produced by the amplifier itself. Hence, a point in the amplitude and phase of the injected signal needs to be chosen to optimize linearity by a tradeoff between the second- and third-harmonic content. The main practical limitation on efficiency is the implementation of an efficient injector circuit. Fig. 5 shows that an injector efficiency of 40% or higher, with a power level 10 dB below the output power, is required to obtain an overall efficiency above 80%. This might present a challenge for very high-power PAs, but is otherwise not a difficult constraint. In the prototype characterization presented in this paper, a passive doubler was used to produce the harmonic. This is not only inefficient but also impractical for amplifying a real signal, in which case significant distortion would be introduced. The linearity tests (Fig. 11) show that a clean doubled frequency spectrum needs to be injected. Therefore, for signal amplification, a different approach is needed than was done for the CW tests in this paper. A topic of current research is integration of an up-converter in the injection circuit with a synchronized second baseband input. For very broadband signals, the phase control that achieves linearity might prove to be challenging. An interesting extension of the concept to a broadband HI-PA 4106 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 60, NO. 12, DECEMBER 2012 will require a three-port injection network design with good harmonic isolation and low fundamental frequency loss over a broad frequency range. In summary, this paper presents a theoretical Fourier analysis of PAs with harmonic injection at the output, showing limits on total circuit efficiency. The theory is experimentally verified on class-AB 2.45-GHz PAs using both packaged and discrete GaN HEMTs, resulting in 75% and 80% efficiencies, respectively, at the 1-dB compression points. The linearity of the PA is shown to be significantly improved for specific phase and amplitude of injected second harmonic, with simultaneous improvement in efficiency. To the best of our knowledge, these are the first reported efficient linear PAs of this type using solid-state devices. REFERENCES [1] F. 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Theory Tech., vol. 49, no. 6, pp. 1148–1154, Jun. 2001. Asmita Dani (S’011) received the B.Tech degree in electronics engineering from the University of Mumbai, Mumbai, India, in 2008, and the M.S.E.E. degree from the University of Colorado at Boulder in 2010, where she is currently working toward the Ph.D. degree in microwave circuits and system design. Her research interests include linearization and efficiency enhancement of power amplifiers using external harmonics, monolithic microwave integrated circuit design and analysis of communication system transmitters and receivers. Michael Roberg (S’09) received the B.S.E.E degree from Bucknell University, Lewisburg, PA, in 2003, the M.S.E.E. degree from the University of Pennsylvania, Philadelphia, in 2006, and the Ph.D. degree from the University of Colorado at Boulder in 2012. From 2003 to 2009, he was an Engineer with Lockheed MartinMS2, Moorestown, NJ, where he was involved with advanced phased-array radar systems. His current research interests include high-efficiency microwave PA theory and design, microwave power rectifiers, monolithic microwave integrated circuit (MMIC) design, and high-efficiency radar and communication system transmitters. He is currently with TriQuint Semiconductor—Defense Products and Foundry Services, Richardson, TX, where he is involved with wideband high-efficiency GaN MMIC power amplifier design. Zoya Popović (S’86–M’90–SM’99–F’02) received the Dipl.Ing. degree from the University of Belgrade, Belgrade, Serbia, Yugoslavia, in 1985, and the Ph.D. degree from the California Institute of Technology, Pasadena, in 1990. Since 1990, she has been with the University of Colorado at Boulder, where she is currently a Distinguished Professor and holds the Hudson Moore Jr. Chair with the Department of Electrical, Computer and Energy Engineering. In 2001, she was a Visiting Professor with the Technical University of Munich, Munich, Germany. Since 1991, she has graduated 44 Ph.D. students. Her research interests include high-efficiency, low-noise, and broadband microwave and millimeter-wave circuits, quasi-optical millimeter-wave techniques, active antenna arrays, and wireless powering for batteryless sensors. Prof. Popović was the recipient of the 1993 and 2006 Microwave Prizes presented by the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) for the best journal papers and the 1996 URSI Issac Koga Gold Medal. In 1997, Eta Kappa Nu students chose her as a Professor of the Year. She was the recipient of a 2000 Humboldt Research Award for Senior U.S. Scientists of the German Alexander von Humboldt Stiftung. She was elected a Foreign Member of the Serbian Academy of Sciences and Arts in 2006. She was also the recipient of the 2001 Hewlett-Packard (HP)/American Society for Engineering Education (ASEE) Terman Medal for combined teaching and research excellence.

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