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A Review on Optimization of Nonlinear RF Circuits
Abstract
High data rate mobile communication requires wideband high-frequency circuits and systems. A high power amplifier (HPA) is an essential part of the communication systems and operates at saturation to achieve high efficiency. On the other hand, the design of an HPA is a challenging issue due to the nonlinear behavior of the high frequency and high power transistors in the saturation region. Inevitable tradeoffs between the efficiency, output power, gain, linearity, and bandwidth performances requires additional effort and time to optimize the design. Therefore automatized optimization is able to provide high performance designs with nonlinear components. This study discusses and reviews several studies that are reported for designing and optimizing of nonlinear microwave circuits. In addition, an automatized optimization methodology is presented with a GaN MMIC at X-Band.
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This paper investigates the impact of open second-harmonic impedance on the static AMAM characteristic of Class F-1 power amplifiers. The phenomenon of double gain compression in harmonically tuned PAs is analyzed for the first time. The study starts with presenting a theoretical postulation explaining the occurrence of inflection points in AM/AM profiles of Class F-1 PAs. Thereafter, a simplified mathematical model providing closed-form expressions for predicting the locations of output power levels of the first and second AM/AM inflection points is presented. For practical validation, an active load pull system is used to synthesize Class F-1 load conditions using a 0.7 mm NXP GaN die at a frequency of 2.6 GHz. The gain curve profile varies quite significantly as a function of second harmonic load reactance and it has been measured to capture the AM/AM inflection phenomenon at low power levels. Under Class F-1 harmonic loading conditions at 28 and 48 V, the active load pull measurements demonstrate the phenomenon of AM/AM inflection and agree well with the mathematical analysis presented in the paper.
Class-F type amplification stands on proper termination of harmonics such as short for even harmonics and open for odd harmonics. Moreover, termination of only the first few harmonics is practical for high-frequency circuits, while obtaining satisfactory short and open terminations at high frequencies is a challenging design issue. In the present study, a topology of harmonics termination for Class-F load network with 2nd and 3rd harmonics and its relative analytical analysis are presented. The proposed output termination structure for Class-F type amplification provides an improved short termination of 2nd harmonic; therefore, the efficiency of the power amplifier increases. In addition, the topology implemented with the microstrip lines has a tunable structure, and it is suitable for very high-frequency applications due to its straightforward architecture. An X-Band high power amplifier for a small satellite transmitter is designed and fabricated with the proposed method. A 0.25 μm GaN on SiC HEMT having a total gate width of 1.25 mm is used. The PA achieves the peak PAE of 55% at 8.1 GHz while the output power is 36 dBm at the 3 - dB compression point. The Class-F PA has higher than 50% PAE and 35.5 dBm output power in the band of 7:9 - 8:2 GHz . The measured linear power gain is 16.2 dB.
This paper presents the design and characterization of a load modulated balanced amplifier for telecom base station applications adopting a novel mode of operation. The theory of operation is described explaining the main differences compared to Doherty amplifiers, in particular the RF bandwidth advantages and, on the other hand, the intrinsic nonlinear behavior. The specific design strategy that adopts prematching for back-off broadband matching is explained in detail. A prototype, based on 25-W GaN packaged devices, has been fabricated and measured with single tone CW and modulated signal stimulus. For CW conditions, on the 1.7-2.5-GHz band, the peak output power is between 63 and 78 W, with power added efficiency higher than 48%, 43%, and 39% at saturation, 6- and 8-dB output power back-off, respectively. With a modulated signal for Long Term Evolution the amplifier provides an average output power of around 10 W, with efficiency higher than 40%, and can be linearized by adopting a low complexity predistorter. If compared to previously published power amplifiers targeting similar power and bandwidth, the measurement shows very good performance, demonstrating the potential of this novel technique in the field of efficiency enhanced transmitters.
In this letter, the design of a broadband asymmetric Doherty power amplifier (ADPA) with an 800 MHz (41% fractional) bandwidth is presented. The post-matching structure and low-order impedance transformation networks (ITN) are employed to achieve the broadband performance. Meanwhile, different drain biases on the main and peaking devices are used to run the asymmetric operation. The proposed ADPA shows high-efficiency performance at 8–9 dB output power back-off (OBO) throughout the whole 800 MHz band. The ADPA has been designed and implemented using commercial GaN HEMTs to validate the OBO and broadband characteristics. Maximum output power ranges from 43.7 to 45.2 dBm, 50.4%-56.2% efficiencies at 8–9 dB OBO are measured from 1.55 to 2.35 GHz.
We review the space-mapping (SM) technique and the SM-based surrogate (modeling) concept and their applications in engineering design optimization. For the first time, we present a mathematical motivation and place SM into the context of classical optimization. The aim of SM is to achieve a satisfactory solution with a minimal number of computationally expensive "fine" model evaluations. SM procedures iteratively update and optimize surrogates based on a fast physically based "coarse" model. Proposed approaches to SM-based optimization include the original algorithm, the Broyden-based aggressive SM algorithm, various trust-region approaches, neural SM, and implicit SM. Parameter extraction is an essential SM subproblem. It is used to align the surrogate (enhanced coarse model) with the fine model. Different approaches to enhance uniqueness are suggested, including the recent gradient parameter-extraction approach. Novel physical illustrations are presented, including the cheese-cutting and wedge-cutting problems. Significant practical applications are reviewed.
The effects of the nonlinear drain-to-source capacitance (CDS), or the varactor effect, on the operation of continuous-mode class-B/J power amplifiers (PAs) are described in terms of the mathematical analysis as well as experimental validation. The nonlinear CDS generates harmonic current when the class-B/J drain voltage is applied across it, modifying the theoretical load trajectories on the Smith chart. This is shown to significantly deviate from the continuous-mode assertion of constant power and efficiency. A novel measurement approach called voltage-pull is used to investigate the effects of the second harmonic generation by the varactor in a class-B/J PA mode. Theoretical and experimental results showed the nonlinear CDS degrades the drain efficiency compared to class-B on one half of the class-B/J design space while improving the other half due to the energy upconversion and downconversion, respectively. This paper provides an approach for die nonlinear characteristic optimizations for transistor manufacturers.
Design and optimization of wideband nonlinear
RF circuits are time-consuming and challenging design issue.
Both electronic design automation (EDA) and numerical analysis
tools can be used to design microwave circuits. However, they
have different capabilities regarding calculation and optimization
processes. In this study, EDA and numerical analysis tools are
operated collaboratively and sequentially to create an automated
co-simulation based iterative design and optimization methodology
for wideband power amplifier design. Co-simulation based
optimization provides uninterrupted and un-intervened design
and optimization process. Consequently, the required effort and
time to achieve a high-performance circuit are reduced dramatically.
Keysight ADS is used nonlinear analysis of the circuits and
MATLAB used to control process and for optimization. Initial
circuit parameters were generated by simplified real frequency
technique (SRFT). A Class-AB X-Band high power amplifier
including a GaN HEMT was designed and optimized with the
proposed methodology.
Since Nathan Sokal's invention of the class-E power amplifier (PA), the vast majority of class-E results have been reported at kilohertz and millihertz frequencies, but the concept is increasingly applied in the ultrahigh-frequency (UHF) [1]-[13], microwave [14]-[20], and even millimeter-wave range [21]. The goal of this article is to briefly review some interesting concepts concerning high-frequency class-E PAs and related circuits. (The article on page 26 of this issue, »A History of Switching-Mode Class-E Techniques» by Andrei Grebennikov and Frederick H. Raab, provides a historical overview of class-E amplifier development.
A novel design space of load modulated (LM) continuous Class-B/J power amplifiers (PAs) is proposed, and a detailed mathematical analysis is presented. The combination of LM technique and continuous mode is introduced to enhance the wideband performance of LM PAs. An inversely proportional relationship between Z
L
and C
out
is derived, which indicates that the nonlinear C
out
can affect the optimal load impedances at output power backoff (OPBO) and the design parameter β should be appropriately adjusted. The measured results of a PA prototype show that the PAE is 39%-45% under LM when OPBO = 5 dB across 1.6-2.4 GHz. Compared with the same PA with fixed Vvar, the proposed PA with optimum V
var
at 2 GHz achieves a PAE improvement with 11% when OPBO = 5 dB.
Signals designed for high-capacity communications result in high peak-to-average ratio (PAR) waveforms that the transmitter power amplifier (PA) must amplify with low distortion. With emerging fifth-generation (5G) wireless systems, carrier frequencies and signal bandwidths are expected to increase significantly from the current S- and C-band allocations. All these factors are contributing to the increasingly challenging goals of maintaining transmitter PA efficiency for the following reasons: efficiency is not constant over output power; Highly efficient PAs are usually nonlinear; RF load variations can adversely impact efficiency. This article briefly reviews supply modulation to enhance the efficiency of several PA types, followed by a discussion of the main challenges and some predictions about where this technology might be headed in the future.
In this paper, the class-J mode of operation is investigated when sinusoidal, half-sinusoidal (HS), triangle, pulse, and reduced conduction angle voltage waveforms are shaped at the gate node of the transistor. Output power, maximum power-added efficiency (PAE), large signal gain (LSG), and load-pull contours are presented and compared for each input signal. It is shown that PAE of a class-J power amplifier (PA) is improved when an HS voltage is realized at the gate node of the transistor. This enhancement can also be observed for a pulse input with 20% duty cycle, however, at the expense of reduced output power and LSG. A proof-of-concept, two-stage class-J PA is designed and fabricated in a 0.25-μm AlGaAs-InGaAs pHEMT technology. The prototype PA is based on an HS gate voltage for the second-stage transistor, which operates in class-J mode. The driver stage also works in class-J mode to provide an HS voltage at its drain node, and the interstage matching network is designed to convey this waveform to gate of the second-stage transistor. Chip dimensions are 1.99 × 1.24 mm
2
, and 26.6-27.2-dBm output power with ≥50% PAE is achieved over the 0.92-1.4-GHz bandwidth.
This paper proposes a multiobjective Bayesian optimization strategy to deal with the complex matching problem of designing a broadband 20-W gallium nitride Doherty power amplifier (DPA). Applying the proposed method, the DPA's matching networks are optimized to align the desired and the realized impedance trajectories both at saturation and power backoff from 1.5 to 2.4 GHz. By combining a programming environment with the commercial electronic design automation (EDA) software, the optimization process is implemented automatically. Comparison designs reveal that the proposed method outperforms the built-in optimizer of the commercial EDA software for broadband DPA designs. When tested with a single-carrier 20 MHz long-term evolution signal, the measured results show that the DPA achieves an average efficiency of 45.3%-53.6% at 7 dB backoff across the band, with the adjacent channel power ratio levels better than -45.6 dBc after using digital predistortion.
This brief introduces a broadband double-matching (DM) solver called fast simplified real frequency technique (FSRFT). FSRFT is essentially a greatly accelerated variant of the well-known classical simplified real frequency technique (SRFT). The basic idea that turns the classical SRFT into a "fast" SRFT relies on two main approaches: the selective target data approach (STDA) and the constraint optimization approach (COA). STDA constructs an optimization target data set formed of only critically selected target data whose element number is equal to or slightly greater than the order of the system unknowns n plus 1, n + 1. In order to exhibit speed performance comparison between SRFT and FSRFT, an example design is considered. An exemplary DM problem, dealing with an n = 6th order low-pass Chebyshev-type equalizer design to match the given generator and load impedances, has been solved by SRFT within 29 s using 90 target data in a typical computer-e.g., Intel 2.20-GHz i7 CPU with 8-GB RAM. On the other hand, the same problem has been solved by the newly proposed FSRFT within only 0.6 s using only n + 1 = 7 critically selected target data in the same computer. FSRFT introduced herein works in any domain, i.e., lumped, distributed, and mixed.
Performing RF amplification in a way that is simultaneously linear and efficient has long been a challenge in power amplifier (PA) design. In short, single-transistor PAs can either be operated in a linear (but inefficient) current-source mode of operation or as efficient (but nonlinear) switches, but these two modes are mutually exclusive. The outphasing concept has been explored to address this tradeoff by using efficient but nonlinear elements in an architecture providing linear output power control.
William H. Doherty was an American electrical engineer, best known for his invention of the Doherty amplifier. Born in Cambridge, Massachusetts, in 1907, Doherty attended Harvard University, where he received his bachelor's degree in communication engineering (1927) and his master's degree in engineering (1928). He then joined the American Telegraph and Telephone Company Long Lines Department in Boston. He remained there for a few months before joining the National Bureau of Standards, where he researched radio phenomena. Doherty joined Bell Telephone Laboratories (now Bell Labs) in 1929 and, while there, worked on the development of high power radio transmitters that were used for transoceanic radio telephones and broadcasting.
This paper proposes a novel, optimization-oriented strategy for the design of broadband, high-efficiency power amplifiers (PAs) using Bayesian optimization (BO). The optimization algorithm optimizes the drain waveforms by maximizing the fundamental output power while minimizing the harmonic and dissipated components. The optimization process is automated using simulation software. Circuit-based BO and electromagnetic-based (EM-based) BO are applied to design 10 and 30 W PAs. The 10 W PA designed using circuit-based BO achieves a drain efficiency higher than 60% with output power greater than 39.8 dBm from 1.5 to 2.5 GHz, while the 30 W PA designed using EM-based BO offers a drain efficiency higher than 57% with output power greater than 43.8 dBm across the band. Upon comparison of results, it is revealed that the proposed strategy outperforms a commercial electronic design automation (EDA) software's built-in optimizer, thus demonstrating that the EM-based BO is well-suited to the challenge of high power designs.
In this paper, a novel Doherty power amplifier (DPA) configuration with an auxiliary transformer is proposed for broadband operation. Theoretical analysis reveals that the proposed design can offer enhanced power utilization factor (PUF) and wideband Doherty behavior. The optimal design of the auxiliary transformer is identified with an aim to provide good impedance matching and minimal group delay. Based on the proposed theory, a 1.6–2.4-GHz 20-W DPA with improved PUF is designed, simulated, and measured. Under continuous wave excitation, measurement results indicate that the proposed DPA can achieve a PUF of 0.9, good Doherty behavior over the entire frequency band with a 6-dB back-off efficiency of 55%–64%, and saturated efficiency of 68%–76%. In addition, by the use of single-carrier WCDMA signal (centered at 2 GHz) with peak-to-average power ratio of 6.6 dB and at an average output power of 37 dBm, an average drain efficiency of 56% is obtained with adjacent channel leakage power ratio of better than 37 dBc.
A general matching method is presented in this paper for broadband power amplifier (PA) design. A novel cost function is proposed for the real frequency technique (RFT), which could straightforwardly describe PA optimal impedance along with frequency change. The new function is also developed to design a broadband transformer for the PA output matching network (MN). Based on Richard transformation, a commensurate transmission line is deployed so that the PA matching could be convenient expressed by a real positive function. More important is that the function could be directly implemented with a distributed MN through synthesis theory. Therefore, this method is practical for computer-aided design and has less calculation amount with new function for the RFT. To verify the method, a step-by-step design of a broadband PA is given. For large signals, power gain is 14.2-16.8 dB across 0.9-2.8 GHz, while output power is around 39.5 dBm. The maximum power-added efficiency is from 52.2% to 85.1%.
Although many generations of microwave transistors have continuously fed a never ending effort of both physical and equivalent circuit device modeling, many microwave engineers have recently turned their attention to behavioral modeling. Behavioral models are measurement-based mathematical representations of the observable inputoutput characteristics of devices, regardless of their technology or of their internal composition. Therefore, not only they protect the intellectual property of the device's manufacturer, as they are only limited by the available laboratory measurement based capabilities. Unfortunately, they also require a great deal of ingenuity in the model formulation and their predictive capability is generally worse than the one of physics-based models. The aim of this paper is to present an overview of microwave transistor behavioral models addressing their mathematical formulation and their laboratory parameter extraction procedures. Both time-domain and frequency-domain behavioral models are addressed, giving a particular emphasis on the implementations that have seen a wider acceptance.
A new, nonlinear, frequency-domain behavioral model, based on Padé approximation theory, is presented in this paper. The new Padé model is similar in concept to the X-parameter method, but it uses a rational function instead of the X-parameter model's polynomial expression. The basic theory of the proposed Padé model and the model extraction methodology are provided. The proposed Padé approach increases the available modeling space beyond that available to the X-parameter method, and the performance of a selection of this family of models under various loading conditions is described in this paper. Along with the examples displaying the ability of the new model to predict fundamental frequency behavior, a second-harmonic example is also provided, with simulation results presented in both the frequency domain and the time domain. Based on the simulation studies and experimentally measured results, the new Padé model shows good accuracy and a high level of stability. All of these characteristics show that the proposed Padé model has high potential in RF design.
This paper derives the theory and presents measurements of a new power amplifier based on the Doherty power amplifier topology. It is theoretically shown that the proposed amplifier can simultaneously provide high efficiency at both full output power and at output power back-off, over a much improved bandwidth compared to the conventional Doherty power amplifier. It is also shown that the proposed amplifier allows reconfiguration of the efficiency in power back-off without the need of tunable elements.
The author introduces a formulation for two-dimensional Pade approximant theory, and proves a theorem on convergence under the stringent assumption of uniform boundedness. Finally he discusses some general problems connected with rational functions in several variable theory.
A shape-preserving response prediction methodology for microwave design optimization is introduced. The presented technique allows us to estimate the response of the microwave structure being optimized (fine model) using a computationally cheap representation of the structure (coarse model). The change of the coarse model response is described by the translation vectors corresponding to certain (finite) number of characteristic points of the response. These translation vectors are subsequently used to predict the response change of the fine model. The presented method has very good generalization capability and it is not based on any extractable parameters, which makes it easy to implement. Applications for microwave design optimization are discussed. The robustness of the proposed approach is demonstrated by extensive comparison with space mapping, which is one of the most efficient optimization approaches in microwave engineering so far.
We present a tutorial on Bayesian optimization, a method of finding the maximum of expensive cost functions. Bayesian optimization employs the Bayesian technique of setting a prior over the objective function and combining it with evidence to get a posterior function. This permits a utility-based selection of the next observation to make on the objective function, which must take into account both exploration (sampling from areas of high uncertainty) and exploitation (sampling areas likely to offer improvement over the current best observation). We also present two detailed extensions of Bayesian optimization, with experiments---active user modelling with preferences, and hierarchical reinforcement learning---and a discussion of the pros and cons of Bayesian optimization based on our experiences.
Print reproduction from microfilm. Thesis (Ph. D.)--Cornell University.
A review of the developments in high-efficiency class F RF/microwave power amplifiers (PA) is presented. The basic topology of a PA includes an active device, dc feed, output-matching network, and input-matching network, that is controlled by its drive and dc bias to act as a current source. A Class-F PA boosts both efficiency and output power using harmonic resonators in the output network to shape the drain waveforms such that the load appears to be a short at even harmonics and open at odd harmonics. In modern PAs, a high efficiency performance, coupled with the requirement for a high level of output power, gain, bandwidth, and linearity are planned to be designed.
Simplified Real Frequency Technique
- B S Yarman
B. S. Yarman, Simplified Real Frequency Technique. Dordrecht:
Springer Netherlands, 2008, pp. 183-224. [Online]. Available: https:
//doi.org/10.1007/978-1-4020-8418-8 9