Nonlinear Microwave Circuits Research Articles

Guest Editorial Mini-Special Issue on the 2023 International Workshop on Integrated Nonlinear Microwave and Millimetre-Wave Circuits

Guest Editorial Mini-Special Issue on the 2023 International Workshop on Integrated Nonlinear Microwave and Millimetre-Wave Circuits

Non-Linear Simulation by Harmonic Balance Techniques of Load Modulated Power Amplifier Driven by Random Modulated Signals

The simulation of the steady state and the non-linear stability of a load modulated power amplifier (LMPA) driven by a random modulated generator, fully performed in the frequency domain by harmonic balance (HB) techniques, is presented. The non-linear microwave circuit and the driving pseudo-random modulated (PRM) generator are integrally defined in the frequency domain. The simulation is implemented and performed using commercially available circuit simulation software. The demodulation of the output signal of the LMPA is implemented with optimally matched filters as software-defined demodulation. The simulated dynamic results of a Quasi-MMIC GaN Doherty power amplifier (DPA) are shown and compared to the measured results with a 16-QAM driving signal at 10 MS/s. The time-domain measurement allows the validation of the new simulation technique through the comparison of both the measured and the simulated error vector magnitude (EVM), the left and right adjacent channel power ratios (ACPRs) versus the average output power. This new simulation is then called pseudo-random modulated harmonic balance (PRM-HB) simulation. The full PRM-HB simulation of an LMPA driven by a random modulated signal, performed in the frequency domain at the design circuit level, results in an advanced simulation tool in the frame of the design of RF circuits and subsystems for telecommunication applications.

Signal-Flow-Graph Analysis of Weakly Nonlinear Microwave Circuits Around a Large-Signal Operating Point

A signal flow graph (SFG) representation of small-signal responses of nonlinear microwave circuits around a large-signal operating point is developed using the X-parameters. It is shown that, unlike the SFGs for linear circuits, negative-frequency nodes need to be included explicitly. The development elucidates the circuit-operational meaning of the elusive T-type small-signal X-parameters, which represent the interaction between positive-and negative-frequency components. As an example, such an SFG is used to derive a closed-form expression of the output power of an amplifier as a function of the load reflection coefficient. It is then used to plot approximate load-pull power contours. The result is consistent with the expressions of the optimum load reflection coefficient derived by Root et al. (EuMIC 2017) and power contours derived by Peláez-Pérez et al. (TMTT 2013). SFGs provide an alternative systematic means to derive closed-form expressions in terms of X-parameters and gain illuminating insights into the workings of weakly nonlinear circuits.

A MIXED TIME-FREQUENCY DOMAIN APPROACH FOR THE QUALITATIVE ANALYSIS OF AN HYSTERETIC OSCILLATOR

Frequency domain techniques, like harmonic balance and describing function, are classical methods for studying and designing electronic oscillators and nonlinear microwave circuits. In most applications spectral techniques have been used for determining the steady-state behavior of nonlinear circuits that exhibit a single periodic attractor. On the other hand, the global dynamics of nonlinear networks and systems is usually investigated through time-domain techniques, that require to introduce rather complex and sophisticated concepts. Recently some HB based techniques have been proposed for investigating bifurcation processes in nonlinear circuits that present several attractors (the authors have considered systems that admits of a Lur’e representation). Their approach presents the advantages of providing a simple and qualitative description of the system dynamics, that can be effectively exploited for design purposes. In this manuscript we will examine a third order hysteretic oscillator, that cannot be decribed in the classical Lur’e form and we will show that its dynamics can be investigated through the joint application of the describing function technique and of a suitable time-domain method for estimating Floquet’s multipliers.

Simulation and Automated Modeling of Microwave Circuits: State-of-the-Art and Emerging Trends

Microwave modeling and simulation are essential to designing microwave circuits and systems. Although fundamental concepts and approaches for modeling and simulation are mature, the drive to higher frequencies, tighter design margins, and more functionality/complexity of circuits continue to defy the capabilities of existing modeling and simulation methods. Newer algorithms are being developed to address the speed, accuracy and robustness of design algorithms. Coupled with the advent of more powerful computers and algorithms, microwave design automations are solving much more complex problems in much shorter time than previously imaginable. This paper describes the advances and state-of-the-art in automated modeling and simulation. Automated data-driven modeling approaches covering data sampling/generation, model structure adaptation, and model training/validation are described. Simulation of nonlinear microwave circuits is described covering formulations of simulation equations and advanced solution algorithms addressing problem size, convergence speed and solution accuracy. The descriptions highlight fundamental concepts, advanced methodologies, and future trends of development.

ТЕОРИЯ ГАРМОНИЧЕСКОГО БАЛАНСА ДЛЯ СХЕМОТЕХНИЧЕСКОГО ПРОЕКТИРОВАНИЯ

The analysis and generalization of the main publications on the methods of synthesis and analysis of non-linear active microwave circuits based on the use of the harmonic balance method are presented. As a result of some classification of mathematical approaches and techniques used in the context of this method, a selection and review of basic algorithms was made, the sequential application of which makes it possible to obtain the final result for a scheme of any complexity. The principles of drawing up the initial system of differential equations for electronic circuits and reducing it to a system of linear algebraic equations are considered. A detailed and, at the same time, simplified interpretation of the approaches involving the use of projection methods and Krylov subspaces is given in order to make them easier to understand. Both the complete and the restart generalized method of minimal residuals are considered, in which the desired solution is obtained in the course of an iterative process, at each stage of which subspaces of lower dimension are constructed. The possibilities of simulators and application packages intended for circuit design of electronic circuits are considered. The problem of matching a power amplifier in large signal mode using the APLAC simulator, which is NI AWR technology for designing high-frequency circuits, is discussed.

Effective time‐domain approach for the assessment of the stability characteristics and other non‐linear effects of RF and microwave circuits

This study describes a systematic approach for the stability analysis of RF and microwave non-linear circuits in the time-domain and that can be useful also for the verification of other non-linearities, like intermodulation. The time-domain analysis is the most reliable approach for the evaluation of complex non-linear phenomena but, in general, the transient behaviour of non-linear circuits is difficult to verify at high frequencies, where distributed elements are common. The solution here addressed overcomes this limitation and it may be applied, without restrictions, also to monolithic microwave integrated circuits and EM-based designs. Examples of application to hybrid prototypes are provided, and the comparison between simulations and measurements illustrates the accuracy and reliability of the proposed approach.

Guest Editorial

This Transactions’ Mini-Special Issue contains six expanded papers from the 2018 International Workshop on Integrated Nonlinear Microwave and Millimetre-wave Circuits (INMMiC), Brive-La-Gaillarde, France, July 5–6, 2018. The INMMiC is a forum for promoting, discussing, and spreading ideas in the field of nonlinear microwave and millimeter-wave circuits and systems. The INMMiC Workshop was originally launched in Duisburg, Germany, in 1990, and achieved its present form in Rome, Italy, in 2004. Subsequently, the INMMiC has been held at different venues around Europe, namely, Aveiro, Portugal, in 2006; Málaga, Spain, in 2008; Göteborg, Sweden, in 2010; Vienna, Austria, in 2011; Dublin, Ireland, in 2012; Leuven, Belgium, in 2014; Taormina, Italy, in 2015; and Graz, Austria in 2017. The INMMiC 2018 was held in Brive-La-Gaillarde, France, on July 5–6, 2018.

Experimental Investigation of Bifurcation Behavior in Nonlinear Microwave Circuits

We present an experimental technique to study bifurcations in periodically forced nonlinear microwave circuits, including even physically unstable (periodic) steady states. The designer specifies a key node in the circuit being studied (often associated with an active device) and the method synthesizes a voltage waveform to match the waveform at the selected node so that no current flows across the interface. This null condition is maintained while a parameter, such as bias voltage, is varied over a specified range. The addition of the external nulling source is able to stabilize a steady state that would be unstable in the original circuit. Various applications are presented.

Guest Editorial

Welcome to this Transactions’ Mini-Special Issue containing expanded papers from the 2015 IEEE International Workshop on Integrated Nonlinear Microwave and Millimetre-wave Circuits (INMMiC 2015), Taormina, Italy, October 1–2, 2015. INMMiC is a forum for promoting, discussing, and spreading ideas in the field of nonlinear microwave and millimeter-wave circuits and systems. INMMiC was originally launched in Duisburg, Germany, in 1990, and it achieved its present form in Rome, Italy, in 2004. Subsequently INMMiC has been held at different venues around Europe: 2006 in Aveiro, Portugal; 2008 in Málaga, Spain; 2010 in Göteborg, Sweden; 2011 in Vienna, Austria; 2012 in Dublin, Ireland; 2014 in Leuven, Belgium; and 2015 in Taormina, Italy. INMMiC 2017 will be held in Graz, Austria, April 20–21, 2017.

Check the Stability: Stability Analysis Methods for Microwave Circuits

The instability problems often faced by the designers of nonlinear microwave circuits are the cause of significant qualitative discrepancies between simulations and measurements, even when using powerful simulation tools based on harmonic-balance (HB) analysis and numerical optimization algorithms. Critical anomalies resulting from instability phenomena most often invalidate the prototype and demand intense investigation and resolution efforts, which may substantially increase production cycles and the final cost. Understanding instability requires awareness of two facts: 1) two or more steady-state solutions can coexist for the same values of the circuit elements and 2) stable solutions must be able to recover from the small perturbations that are always present in real life. To realize the complexity of the problem, one must take into account the fact that circuits containing nonlinear components, such as transistors and diodes, are governed by a set of nonlinear differential algebraic equations [1]-[4]. Time differentiation comes from the existence of reactive elements, involving this operation in their constitutive relationships, and nonlinearity comes from the presence of semiconductor devices, containing nonlinear functions in their intrinsic models. Nonlinear differential equation systems provide four main types of steady-state solutions: dc, periodic, quasi-periodic (having two or more fundamental frequencies with nonrational relationships), and chaotic (nonperiodic) [1], [5]-[7]. Unexpected solutions are often observed in nonlinear circuits since, in addition to the frequencies delivered by the input sources, there may be frequency components coming from the circuit self-oscillation. For instance, under a periodic excitation at ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</sub> , the solution measured may not be periodic at ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</sub> . Instead, it may be quasi-periodic at ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</sub> and an oscillation frequency ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> , it may exhibit a subharmonic oscillation at ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</sub> /2 or exhibit a continuous spectrum (chaos) [7], [8].

An Adaptive Graph Sparsification Approach to Scalable Harmonic Balance Analysis of Strongly Nonlinear Post-Layout RF Circuits

In the past decades, harmonic balance (HB) has been widely used for computing steady-state solutions of nonlinear radio-frequency (RF) and microwave circuits. However, using HB for simulating strongly nonlinear post-layout RF circuits still remains a very challenging task. Although direct solution methods can be adopted to handle moderate to strong nonlinearities in HB analysis, such methods do not scale efficiently with large-scale problems due to excessively long simulation time and prohibitively large memory consumption. In this paper, we present a novel graph sparsification approach for automatically generating preconditioners that can be efficiently applied for simulating strongly nonlinear post-layout RF circuits. Our approach allows to sparsify time-domain circuit modified nodal analysis matrices that can be subsequently leveraged for sparsifying the entire HB Jacobian matrix. We show that the resultant sparsified Jacobian matrix can be used as a robust yet efficient preconditioner in HB analysis. Our experimental results show that when compared with the prior state-of-the-art direct solution method, the proposed solver can more efficiently handle moderate to strong nonlinearities during the HB analysis of RF circuits, achieving up to $20{\boldsymbol \times }$ speedups and $6{\boldsymbol \times }$ memory reductions.

Future Device Modeling Trends

Good transistor models are essential for efficient computer-aided-design (CAD) of nonlinear microwave and RF circuits, monolithic microwave integrated circuits (MMICs), power amplifiers (PAs), and nonlinear RF systems. Increasingly complicated demands of the various semiconductor technologies (e.g., GaAs pHEMTs, InP double heterojunction bipolar transistors (DHBTs), silicon on insulator (SOI), LDMOS, GaN HFETs, etc.), and their applications in terms of power and frequency of operation and complexity of applied signals (e.g., modern communication signals with high peak-toaverage ratios) have placed commensurate requirements on the accuracy and generality of the device models used for design. New semiconductor material systems (e.g., GaN) have been developing at such a fast rate that conventional compact modeling approaches may not be able to keep up. These and other challenges have spawned much research into the advanced nonlinear device modeling techniques that are the focus of this article.

Large-Signal Oscillator Design Procedure Utilizing Analytical $X$-Parameters Closed-Form Expressions

New analytical behavioral model formulations based on the polyharmonic distortion (PHD) model have been successfully used to describe the nonlinear behavior of transistors and circuits. In this paper, the PHD model and its associated analytical <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> -parameters formulation will be utilized to provide an analytical design procedure for use in nonlinear microwave circuit design. For RF oscillator design, the negative-resistance method based on the analytical manipulation of scattering parameters is very popular due to its high rate of success in oscillation frequency prediction. However, it cannot be used to accurately predict the oscillator performance because it is based on linear parameters. To overcome this limitation, new analytical expressions based on large-signal <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> -parameters have been developed for use in transistor-based oscillator circuit design. The robustness of this new approach has been validated by designing and manufacturing a 5-GHz microwave oscillator.

Two frequency domain black-box models of microwave nonlinear circuit based on measurement

Two frequency domain black-box models of microwave nonlinear circuit based on measurement

Dynamic Behavioral Modeling of Nonlinear Microwave Devices Using Real-Time Recurrent Neural Network

A novel real-time recurrent neural network (RTRNN) approach is presented for dynamic behavioral macromodeling of nonlinear microwave devices. A modified real-time recurrent learning algorithm is developed to train the neural network model. This proposed RTRNN model can directly be developed from input-output waveform data without having to rely on the internal details of the devices. Once trained, this model provides fast and accurate prediction on the analog behaviors of the nonlinear microwave devices under modeling, which can readily be incorporated into high-level circuit simulation and optimization. This RTRNN approach enhances the neural modeling speed and accuracy. Moreover, it also provides additional flexibility in handing diverse needs of nonlinear microwave circuit designs in the time domain, such as single-tone and multiple-tone simulations and large-signal simulations by comparison to the previously published neural models. The validity of this proposed approach is illustrated through behavioral macromodeling of two typical microwave devices: power amplifiers and pHEMTs.

Automated time domain modeling of linear and nonlinear microwave circuits using recurrent neural networks

In this article, a recurrent neural network (RNN) method is employed for dynamic time-domain modeling of both linear and nonlinear microwave circuits. An automated RNN modeling technique is proposed to efficiently determine the training waveform distribution and internal RNN structure during the offline training process. This technique extends a recent automatic model generation (AMG) algorithm from frequency-domain model generation to dynamic time-domain model generation. Two types of applications of the algorithm are presented, transient electromagnetic (EM) behavior modeling of microwave structures, and time-domain envelope modeling of power amplifiers (PA). For transient EM modeling, we consider EM structures with varying material and geometrical parameters. AMG automatically varies the EM structural parameters during training and drives time-domain EM simulators to generate necessary amount of data for RNN to learn. AMG aims to model the transient behavior with minimum RNN order while satisfying accuracy requirements. In modeling PA behavior, an envelope formulation is used to specifically learn the AM/AM and AM/PM distortions due to third-generation (3G) digital modulation input. The RNN PA model is able to model these time domain distortions after training and can accurately model the amplifier behavior in both time (AM/AM, AM/PM) and frequency (spectral re-growth). © 2008 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2008.

On the Application of the Describing Function Technique to the Bifurcation Analysis of Nonlinear Systems

Spectral techniques, like harmonic balance, are classical numerical tools for designing nonlinear oscillators and microwave circuits. Recently these techniques have been exploited for investigating complex dynamics in nonlinear systems. In this manuscript we firstly show that limit cycle Floquet's multipliers and the related bifurcation phenomena can be estimated through a spectral approach, entirely based on the describing function technique. Then we consider some significant case studies, and we show that our method yields more accurate results than the other describing function-based approaches proposed in the literature

Analysis of Nonlinear Microwave Circuits by Particle Swarm Algorithm

In this paper, the particle swarm algorithm is applied to the analysis of nonlinear microwave circuits. It should be emphasized that the particle swarm algorithm of this study is utilized for analyzing nonlinear microwave circuits, but not for optimization. The problem of analyzing a nonlinear microwave circuit was successfully transformed into the minimization of a scalar fitness function. This scalar fitness function is then minimized by particle swarm algorithm. Numerical simulation shows that the results calculated by this study are consistent with those calculated by the harmonic balance technique. Our analysis is very efficient and straightforward. In addition, our analysis imposes no limitation on the nonlinearity of microwave circuits. Therefore, this study can be applied to the analysis of many other nonlinear problems in electromagnetics.

State-space dynamic neural network technique for high-speed IC applications: modeling and stability analysis

We present a state-space dynamic neural network (SSDNN) method for modeling the transient behaviors of high-speed nonlinear circuits. The SSDNN technique extends the existing dynamic neural network (DNN) approaches into a more generalized and robust formulation. For the first time, stability analysis methods are presented for neural modeling of nonlinear microwave circuits. We derive the stability criteria for both the local stability and global stability of SSDNN models. Stability test matrices are formulated from SSDNN internal weight parameters. The proposed criteria can be conveniently applied to the stability verification of a trained SSDNN model using the eigenvalues of the test matrices. In addition, a new constrained training algorithm is introduced by formulating the proposed stability criteria as training constraints such that the resulting SSDNN models satisfy both the accuracy and stability requirements. The validity of the proposed technique is demonstrated through the transient modeling of high-speed interconnect driver and receiver circuits and the stability verifications of the obtained SSDNN models