Harmonic Balance Simulation Research Articles

Signal growth in a pure time-modulated transmission line and the loss effect.

We present the first comprehensive study for signal growth in transmission lines (TL) with purely time-modulated characteristic impedance (infinite superluminality). This study pioneers the investigation into the effects of varying the cell's electrical length and the impact of loss on momentum bandgaps and amplification levels. It also thoroughly examines how time-modulated transmission line truncation by a static load influences the sensitivity of amplification gain to the relative phase between the incoming signal and modulation, comparing these findings with the case of parametric amplification. Varying is accomplished by loading TLs with a sinusoidally time-modulated capacitor (TMC). The study starts with a simple lumped model cell to facilitate understanding of the phenomena. Following this, transmission lines are introduced, and the effects of incorporating loss are examined. To accomplish this, three models are investigated: a lossless L-C TL lumped model loaded with a shunt lossless TMC and a TL loaded with a shunt lossless and lossy TMC. Dispersion diagrams are plotted and momentum bandgaps are identified at a modulation frequency double the signal frequency. Within the momentum bandgap, only imaginary frequencies are found and correlated to momentum bandgap width and signal growth level. Signal growth is confirmed using harmonic balance and transient simulations, and the results are consistent with the dispersion diagram outcomes.

Conjointly active and passive modelings with deep neural networks as fully automated optimizations for upper-mid band 6G communications

Today wireless systems include the fifth and sixth generations (5G and 6G) technologies and are growing day by day that result in exponentially increasing data traffic. For providing a reliable and high performance radio frequency (RF) designs especially for 6G networks, amplifiers and antenna as active and passive components play important roles. In the 5G/6G communication systems, the propagation loss is considerably large and its compensation requires high output power generated from the amplifiers for guaranteeing the satisfied quality of transmitted signal. From another point of view, the installed antennas must be able to optimally manage the radiated signals and handle/compensate nonlinear performances of the RF circuitry. Hence, advanced modeling and multi-objective optimization algorithms are required for designing and optimizing high performance amplifiers and antennas in terms of output power, gain, efficiency, linearity, and bandwidth. Concurrently optimizing active and passive components is not straightforward and typically it requires additional efforts by the RF designers. To tackle this drawback, a two-step methodology is proposed: (1) configuring the initial structure of active and passive devices, and (2) sizing the configured devices. In this work, various methods are introduced for structuring the topology of circuits and then artificial intelligence, including machine learning and neural networks, is preferred among other surrogate modelling for sizing the designs. These neural networks are satisfied due to the accurate modeling responses and are able to provide an automated optimization process leads to employ multi-objective optimization methods. In this work, an automated optimization process for comprehensive design of high-performance amplifiers with antennas through bottom-up optimization (BUO) method and long short-term memory (LSTM)-based deep neural networks (DNNs) is proposed. At the output layer of DNNs, the multi-objective multi-verse optimizer (MOMVO) method is employed for optimizing various specifications of active device (i.e., amplifier), and passive device (i.e., antenna), concurrently. In the presented method, all the electromagnetic (EM) design rules are implemented which results in reducing simulation time in the harmonic balance simulation environment that also provides ready to fabricate layouts. The novelty consists of the all-inclusive style that (1) reduces the manual breaks, aka time-to-market, and (2) delivers ready-to-fabricate layouts of the device that exhibits global optimum performances, automatically. The validation of the proposed method is verified by designing and optimizing high power amplifier (HPA) with antenna in the frequency band from 9.0 GHz to 9.6 GHz, suitable for upper-mid band 6G communications.

Rigorous design and experimental validation of non-reciprocal bandpass filter with ultrawide high reverse isolation bandwidth

This work presents a rigorous design and experimental validation of a non-reciprocal bandpass filter (NBPF) employing time-modulated resonators with progressive phase shift sinusoidal modulation signal to achieve low forward transmission insertion loss (IL) and ultrawide high reverse isolation bandwidth. Analytical design equations have been derived for numerical design of NBPF asl well as providing insights of nonreciprocal frequency response. Additionally, these proposed analytical spectral S-parameters offer the flexibility to design NBPFs with arbitrary termination port impedances. This work also demonstrates a systematic approach to find modulation parameters without relying on optimization methods. It has established empirical relationships between the modulation parameters and the filters specifications to achieve low forward transmission IL and reverse isolation exceeding 15 dB at entire frequency range. The accuracy of the proposed NBPF design has been confirmed by comparing with harmonic balance simulation results of microstrip line NBPF. For experimental validation, three prototypes of NBPF with different termination port impedances are designed, implemented, and manufactured. In experimental results, the frequency of NBPFs can be continously tuned from 1.62 GHz to 1.94 GHz (17.98 %) with maximum forward transmission IL (|S21|) of 5.40 dB and input/output return loss higher than 12 dB. For each frequency tuning state, the reverse isolation (|S12|) exceeds higher than 20 dB across the entire frequency range.

Analyzing Unsteady Turbomachinery Flow Simulations With Mixing Entropy

Abstract The prediction of unsteady aerodynamic loads is a central problem during the design of turbomachinery. Over the last 20 years, harmonic balance methods have been shown to be highly efficient for this task. A CPU-cost optimal setup of a harmonic balance simulation, however, requires knowledge of relevant harmonics. In the case of a single blade row with a periodic disturbance this question amounts to the classical problem of harmonic convergence, a problem which is solely due to the nonlinearity of the unsteady flow physics. In contrast, for multi-stage configurations, the choice of harmonics is further complicated by the fact that the interactions of disturbances with blade rows may give rise to a vast spectrum of harmonics that possibly have important modal content, e.g., Tyler–Sofrin modes. The aim of this paper is to show that the mixing entropy attributed to circumferential modes of a given harmonic can serve as a disturbance metric on the basis of which a criterion could be derived whether a certain harmonic should be included or not. The idea is based on the observation that the entropy due to the temporal and circumferential mixing of the flow at a blade row interface may be decomposed, up to third-order terms, into independent contributions from different frequencies and mode orders. For a given harmonic balance (and steady) flow result, the mixing entropy attributed to modes that are simply mixed out, rather than resolved in the neighboring row, is shown to be a natural indicator of a potential inaccuracy. We present important features of the mixing entropy for unsteady disturbances, in particular a close relationship to sound power for acoustic modes. The problem of mode selection in a 1.5-stage compressor configuration serves as a practical example to illustrate our findings.

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.

Large-scale harmonic balance simulations with Krylov subspace and preconditioner recycling

Large-scale harmonic balance simulations with Krylov subspace and preconditioner recycling

Harmonic balance simulation of the influence of component uniformity and reliability on the performance of a Josephson traveling wave parametric amplifier

A Josephson traveling wave parametric amplifier (JTWPA), which is a quantum-limited amplifier with high gain and large bandwidth, is the core device of large-scale measurement and control systems for quantum computing. A typical JTWPA consists of thousands of Josephson junctions connected in series to form a transmission line and hundreds of shunt LC resonators periodically loaded along the line for phase matching. Because the variation of these capacitors and inductors can be detrimental to their high-frequency characteristics, the fabrication of a JTWPA typically necessitates precise processing equipment. To guide the fabrication process and further improve the design for manufacturability, it is necessary to understand how each electronic component affects the amplifier. In this paper, we use the harmonic balance method to conduct a comprehensive study on the impact of nonuniformity and fabrication yield of the electronic components on the performance of a JTWPA. The results provide insightful and scientific guidance for device design and fabrication processes.

Analysis and Design of Injection-Locked Oscillators Coupled to an External Resonator

This work investigates the nonlinear dynamics of an injection-locked power oscillator inductively coupled to an external resonator. This allows a high-efficiency power transfer while ensuring a constant oscillation frequency versus the coupling factor, unlike free-running implementations. The investigation focuses on the impact of the external-resonator elements on the locking range, output power, efficiency, and phase noise. The aim is to derive a strategy for an optimum selection of these elements. Initially, the effect of the coupled resonator is theoretically studied using a simple oscillator model, based on a cubic nonlinearity. For practical oscillators, two kinds of analysis methods, compatible with the use of commercial harmonic-balance (HB) simulators, are presented. The first one is semianalytical and is based on the extraction of a phase-dependent nonlinear admittance function from HB simulations. The system response is predicted in a flexible and computationally efficient manner, but coupling effects are considered at the fundamental frequency only. The second set of methods is fully based on HB and relies on the combination of a nonlinear immittance function and a Thevenin/Norton equivalent. The impact of the external resonator on the stability properties is analyzed through bifurcation detection. The phase-noise spectrum is predicted with a semianalytical formulation that demonstrates the benefit of the injection-locked operation. For validation, the methods have been applied to a Class-E oscillator at 13.56 MHz.

Analysis of Device Mismatches Effect on the Performance of UWB Receiver Front-End in Wireless Body Area Network Sensor Nodes

Today it is important to manufacture high quality integrated circuits which are insensitive to device mismatches. This paper presents an analysis of MOSFET transistors mismatches effect on the performance of UWB receiver front-end which constitute the most important part of Wireless Body Area Network sensor node. The receiver is based on Balun LNA with 25% fully differential double-balanced passive mixer. A PMOS and NMOS transistors mismatch models were proposed to determine LNA output offset voltage and mixer offset current respectively. The analysis result suggests that, to minimize NMOS current mismatch, and thus reducing second-order inter modulation distortion, the overdrive voltage must be maximized. A Monte Carlo and harmonic balance simulations were performed using 0.18µm CMOS process to evaluate the impact of mismatch as well as Vth mismatch on the receiver gain and IIP2. Simulation results show that IIP2 of the receiver is less sensitive to mixer NMOS mismatch but receiver gain is more sensitive. The receiver IIP2 confidence interval in case of NMOS mismatch is [24.674, 24.77]dBm and in case of NMOS Vth mismatch is [24.659, 24.857]dBm. This show the robustness of the proposed UWB receiver front end. Therefore the proposed circuit meets the requirement of UWB system perfectly which make it suitable for WBAN applications.

An Extra Low-Mass Harmonic Radar Transponder for Insect Tracking Applications

The design, construction, and performance of a harmonic radar transponder with a total mass of less than 500 μg is presented. The transponder is intended for insect tracking applications and consists of very fine wire and a small Schottky diode. It is designed for fundamental and harmonic frequencies of 10 GHz and 20 GHz, respectively. Compared to existing harmonic radar transponders, this transponder is easy to construct because the loop inductor can be implemented with a simple bend in the dipole conductor without degrading performance. Through careful design optimization, the conversion loss of the transponder is not impacted by the measures taken to minimize its mass. The expected harmonic power versus the transmitted power is estimated based on the link analysis between the transmitter and receiver of the radar, with the link analysis itself being performed via calculation, harmonic balance simulation, and full-wave simulation. The link analysis simulation predicted a received power of -66.4 dBm for a transmitted power of +22 dBm and a range of 2.4 m. The measured received power level at the harmonic frequency, obtained from the broadside of the transponder in an anechoic test chamber, is approximately -70 dBm, which agrees well with the link analysis. Simulated and measured transponder radiation patterns are also compared and show good agreement. Low-mass transponders such as this enable tracking of smaller insects without reducing their lifespan or compromising their ability to fly at natural altitudes and ranges.

Oscillators Based on Step-Impedance and Slow Wave Transmission Lines for Sensing Applications

This work investigates the capabilities of oscillators based on step-impedance and slow wave structures to sense dielectric constants. The material under test (MUT) is placed over the structure and the objective is to achieve a high sensitivity of the oscillation frequency with the advantage of low phase noise, enabled by the high quality factor of the structure. With the aid of simplified analytical models, we will initially study the variation of the resonance frequency of a istep-impedance transmission line with the dielectric constant of the MUT, paying attention to the influence of the number of line sections. The study includes the derivation of analytical expressions for the sensitivity of the resonance frequency. Next, the structure will be connected to the oscillator active core, which will be modeled with a numerical nonlinear admittance function extracted from harmonic-balance (HB) simulations. The resulting semianalytical formulation will provide insight into the variation of the oscillation frequency and amplitude with the dielectric constant of the MUT, as well as the variation of the phase-noise spectral density. It will also enable a versatile test and optimization of the various structures to achieve high sensitivity with low phase noise. The methods have been successfully applied to an field-effect transistor (FET)-based oscillator at about 2 GHz.

A core drain current model for β-Ga2O3 power MOSFETs based on surface potential

For the first time, a core drain current model based on surface potential without any implicit functions is developed for beta-phase gallium oxide (β-Ga2O3) power metal-oxide-semiconductor field-effect transistors (MOSFETs). The surface potential solution is analytically deduced by solving the Poisson equation with appropriate simplification assumptions in accumulation, partial-depletion, and full-depletion modes. Then, the drain current expression is analytically derived from the Pao–Sah integral as a function of the mobile charge density obtained from the surface potential at the source and drain terminals. In addition, nonlinear resistors in the source/drain access region are considered. It continuously predicts the characteristics of β-Ga2O3 power MOSFETs in all operation modes, including accumulation mode, partial-depletion mode, and full-depletion mode. Furthermore, the validity of the model is verified by comparing the results of the model with the numerical simulations carried out with the technology computer-aided design (TCAD) tool ATLAS Device Simulator from Silvaco. Good agreement between the proposed model and TCAD simulations is shown for β-Ga2O3 power MOSFETs with different intrinsic channel lengths, channel doping concentrations, and channel thicknesses. Ultimately, the Gummel symmetry test and the harmonic balance simulation test are performed to validate the model’s robustness and convergence.

Simulating the behaviour of travelling wave superconducting parametric amplifiers using a commercial circuit simulator

Kinetic inductance travelling wave parametric amplifiers (KI-TWPAs) have been simulated using Keysight’s Advanced Design System. A lumped element representation of a superconducting transmission line has been developed using nonlinear inductor elements to represent the current dependency of the intrinsic kinetic inductance. This provides a fast, simple and easily modifiable method for analysing the parametric processes that occur in nonlinear kinetic inductance devices, such as parametric oscillators or amplifiers while also allowing the accompanying RF network to be optimised. This methodology is used to model the behaviour of a select number of KI-TWPA designs operating in the 3-wave mixing (3WM) and 4-wave mixing (4WM) regimes. S-parameters and dispersion properties are extracted from large signal S-parameter simulations, and the gain curve of each amplifier design is generated from harmonic balance simulations. The results are compared to the gain expression from the analytic solutions of the coupled mode equations while making reference to the gain curves of the equivalent KI-TWPAs from literature.

A parametric frequency domain approach to analysis and design of critical design parameters of nonlinear energy harvesting systems: Parametric output spectrum and power generation functions

Nonlinear energy harvesting systems have been extensively studied which usually involve quite a few critical design parameters to determine. However, a straightforward parametric design method for such a purpose is still far from developed. Traditional methods with harmonic balance or numerical simulation usually cannot reveal a straightforward relationship between the energy harvesting performance and system parameters, and in most cases face relatively large computational costs during parametric studies. To this aim, a novel Volterra series-based frequency domain method is established in this study with respect to a typical nonlinear harvesting system. A concise parametric characteristic approach, namely the linear and nonlinear characteristic output spectrum (lnCOS) method, is thus proposed to achieve parametric characteristic expression for the nonlinear output frequency response and power generation function (PGF) of the system, referred to as the lnCOS function and the Parametric-PGF respectively. Compared with the harmonic balance method, the vibration responses/power generation can be obtained in a more straightforward way without solving complex algebraic equations, and the lnCOS method directly links the power outputs to critical system parameters in a concise parametric way. It is for the first time to provide explicit expressions for vibration responses and power generations with respect to structural parameters of a nonlinear energy harvesting system, which cannot be achieved in a traditional way. Based on the obtained explicit expressions of the lnCOS function/Parametric-PGF, the sensitivity analysis of vibration response and power generation with respect to structural parameters can be directly performed. Our work provides an efficient way to characterize both the linear and nonlinear effects on a nonlinear energy harvesting system and demonstrates an efficient approach to the analysis and design of nonlinear energy harvesting systems.

Distributed Nonlinear Model and Fast Analysis for In-Band IMD3 Prediction of Surface Acoustic Wave Resonators.

The motivation of this work is to analyze the in-band intermodulation distortion (IMD) occurring in surface acoustic wave (SAW) devices, using a recently developed fast method based on the input-output equivalent sources (IOES). The method calculates the equivalent current sources of a given harmonic (H) or IMD, which when applied at the boundaries of any uniform nonlinear region produce the same nonlinearities as the full distributed circuit. The accuracy of the method is validated with a very simplified SAW resonator with ten digits, which is modeled by a discretized Mason-based circuit. The IOES method provides equal results to the ones obtained through harmonic balance (HB) simulations, performed by means of commercial software, being the first 1000 times faster. Once the accuracy of the method is guaranteed, it is used to analyze the measured in-band IMD3 of several lithium tantalite 42° cut leaky SAW (LSAW) resonators with different pitches and duty factors at the B66 long term evolution (LTE) frequency band. Those resonators are comprised of 100 and 20 electrode pairs for the active region and each of the reflectors, respectively, which implies the analysis of a very large distributed nonlinear problem with thousands of nonlinear local sources. The IOES method takes 35.4 s in simulating 51 frequency points, whereas this simulation is not possible using a commercial HB simulator on a general-purpose computer.

Acoustic Feedback and Its Impact on Fan Flutter in Short Aeroengine Intakes

The authors have investigated pressure waves being emitted by the structural motion of a rotor blade and their impact on fan flutter sensitivity. This impact results from an acoustic feedback loop caused by reflection phenomena at the intake lip and may lead to a specific form of aeroelastic instability. Harmonic balance simulations of the fan/intake system over a wide operating range and blade eigenfrequencies were carried out. A significant impact on the rotor aerodynamic damping was observed as compared with the isolated rotor analysis due to the presence of the intake. This influence was either stabilizing or destabilizing, depending mainly on the phase relation between the emitted and reflected unsteady pressure fields. Computational fluid dynamics (CFD) data were analyzed accordingly, taking advantage of two-dimensional wave split as well as three-dimensional in-duct modal decomposition techniques. By deriving a simple analytical model based on eigenvalue analysis, a landscape showing regions of aerodynamic damping and excitation could be drawn over the investigated frequency–mass flow range. This analytical model and its verification with CFD data can help in identifying destabilizing areas at an early design stage, and hence support design decisions regarding flutter stability or give indications where analysis with higher fidelity is required.

A 2.4-GHz 10-W Class Bridge Rectifier and Its Efficiency Analysis With the Behavioral Model

This article presents a 2.4-GHz band 10-W class bridge rectifier that achieves a high-measured efficiency of 90.2%. To clarify the mechanism for obtaining high efficiency, a theoretical analysis with the novel behavioral model of bridge rectifiers is demonstrated. The behavior model includes three functional blocks, namely; 1) impedance transform, which defines the RF impedance for the bridge diode; 2) RF impedance matching; and 3) rectification with the ideal current waveform of the bridge diode. Closed-form formulas for the rectification efficiency and the maximum input power are derived using the behavioral model. The RF impedance can be optimized to obtain high efficiency with the derived formulas, similar to source-pull operations in harmonic balance simulations or experiments. The fundamental limitation on the rectification efficiency is obtained for a bridge rectifier with the optimum RF impedance. In the experimental investigation of the 2.4-GHz 10-W class bridge rectifier, the measured efficiency of 90.2% agreed well with the theoretical value of 91.2% with represented formulas. Furthermore, the derived formulas are applied to the 5.8-GHz 1-W rectifier, and its effectiveness can be confirmed. It is evident that the analyzed efficiencies are close to the fundamental limitations.

Approach for Extreme Learning Machine-Based Microwave Power Device Modeling

The nonlinear representation of active devices plays an important role in microwave circuit design. Whereas, it takes a long time to extract a large amount of large signal data, and the problem of memory resource and CPU occupancy becomes significant. In order to address the problems in traditional large-signal modeling methods, in this paper an X-parameter modeling method for microwave power devices based on extreme learning machine (ELM) is proposed. To demonstrate the effectiveness of this method, a double layer back propagation (BP) neural network model is established. Then, harmonic balance simulations are used to verify the accuracy of these two models. After comparisons, it is proved that the three harmonic errors of double layer BP neural network model are 9.525dBm, 1.309dBm and 14.593dBm, respectively, and the three harmonic errors of ELM model are 0.673 dBm, 0.314 dBm, 3.09 dBm, respectively. Furthermore, the three harmonic modulus errors of double layer BP neural network model are 0.031, 0.002, 7.665e-4, respectively, and the errors of ELM model are 0.005, 0.001, 8.38e-5, respectively. Finally, in order to verify the accuracy of the predicted model in circuit design, the predicted X-parameter is used in the design of power amplifier. Moreover, the errors of the double layer BP neural network prediction model at 2.5 GHz, 5 GHz and 7.5 GHz are 1.142 dBm, 1.436 dBm and 2.294 dBm, respectively. The output power error of the ELM model at 2.5 GHz, 5 GHz and 7.5 GHz are 0.089 dBm, 0.311 dBm and 0.309 dBm, respectively. These experimental results demonstrate that the established ELM model is an efficient and valid approach for modeling GaN high electron mobility transistor types of nonlinear microwave devices.

A Time-Domain Multi-Tone Distortion Model for Effective Design of High Power Amplifiers

This paper proposes a new time-domain multi-tone distortion (TD-MTD) model suitable for accurately predicting the non-linear behavior of packaged high power radio frequency (RF) transistors over a range of discrete non-uniformly distributed frequencies. This proposed TD-MTD model uses a single expression rather than multiple distinct frequency specific behavioral models to describe the underlying behavior of the high power RF transistor at multiple fundamental frequencies. Furthermore its extraction is carried out using a time-domain representation of the travelling waves that can be acquired using a generic vector load-pull characterization system and without imposing additional requirements. The proposed model is extracted as an artificial neural network (ANN) and is implemented as a Netlist to serve in a harmonic balance simulator based power amplifier design process. The proposed model is validated in two phases. First, its ability to reproduce the large-signal behavior of a high power RF LDMOS transistor was demonstrated in simulation. Then, the TD-MTD model was used to validate the design of a high power two-way asymmetric Doherty power amplifier and the simulated output-power-dependent power efficiency, AM/AM, AM/PM and input return loss characteristics were compared to those obtained in measurement. The excellent agreement between the simulation and measurement results confirms the usefulness of the proposed model despite the simplicity of its extraction routine and measurement data.

The Development of Frequency Tripler Based on Six-Anode Schottky Varactors.

The development of a millimeter-wave unbalanced frequency tripler based on the nonlinear characteristics of planar Schottky varactors is presented. The entire module is designed by hybrid integration. A frequency multiplier circuit model was established to reflect the influence of diode parameters and the impedance matching on the multiplier in different frequency bands. The effect of junction imbalance on the output power of the frequency multiplier was investigated and the multiplier was improved based on the basic design. The addition of a cut microstrip stub in the improved diode unit reduced the impact of a power imbalance on frequency multiplier performance. The characteristics of the multiplier circuit were analyzed by the full-wave electromagnetic simulation of the three-dimensional structure and the harmonic balance simulation of the circuit. Test results showed that the peak output power of the improved frequency tripler was 12.6 mW at 277 GHz with an input power of 200 mW, an effective 12% improvement over the basic design.