Harmonic Balance Simulation Research Articles

Generalized Class-E Power Amplifier With Shunt Capacitance and Shunt Filter

This paper presents a generalized analysis of the Class-E power amplifier (PA) with a shunt capacitance and a shunt filter, leading to a revelation of a unique design flexibility that can be exploited either to extend the maximum operating frequency of the PA or to allow the use of larger active devices with higher power handling capability. The proposed PA fulfills zero voltage switching (ZVS) and zero voltage derivative switching (ZVDS) conditions, resulting in a theoretical dc-to-RF efficiency of 100%. Explicit design equations for the load-network parameters are derived, and the analytical results are confirmed by harmonic-balance simulations. Two PA prototypes were constructed with one designed at low frequency and the other at high frequency. The first PA, which employs a MOSFET and a lumped-element load-network, delivered a peak drain efficiency $(\!D\!E)$ of 93.3% and a peak output power of 37 dBm at 1 MHz. The second PA, which employs a GaN HEMT and a transmission-line (TL) load-network to provide the drain of the transistor with the required load impedances at the fundamental frequency as well as even and odd harmonic frequencies, delivered a peak $D\!E$ of 90.2% and a peak output power of 39.8 dBm at 1.37 GHz.

An Optimized-Load-Impedance Calculation and Mining Method Based on <italic>I–V</italic> Curves: Using Broadband Class-E Power Amplifier as Example

Broadband power amplifier (PA) designs are usually based on optimized-load-impedance ( $Z_{\text{opt}}$ ) searching, which can be guided by PA theories or a load–pull method. After practical devices’ internal dc I–V curves and package parasitics are known, a direct impedance calculation method is realized in this paper to offer an alternate option. This method can offer relatively accurate $Z_{\text{opt}}$ solutions, which can satisfy both operation-mode and performance requirements for a practical device. Following this large-signal impedance calculation method, waveform regulations can be added to the design steps to realize a specific PA operation-mode through a robust and fast small-signal S-parameter simulator. A waveform-guided perturbation and selection process is introduced to expand the impedance space further. A broadband quasi-class-E PA example, across 1.4 to 2.4 GHz, is given to demonstrate this method. Its output matching network is designed accordingly to fit into the expanded impedance space, and its theoretical performances are verified by harmonic balance simulations and measurements. Finally, this PA is fabricated and measured to offer 39.0 to 40.9 dBm power, 11.6 to 14.0 dB gain, and 73% to 83% drain efficiency in its frequency band.

Piecewise Semi-Analytical Formulation for the Analysis of Coupled-Oscillator Systems

A new simulation technique to obtain the synchronized steady-state solutions existing in coupled oscillator systems is presented. The technique departs from a semi-analytical formulation presented in previous works. It extends the model of the admittance function describing each individual oscillator to a piecewise linear one. This provides a global formulation of the coupled system, considering the whole characteristic of each voltage-controlled oscillator (VCO) in the array. In comparison with the previous local formulation, the new formulation significantly improves the accuracy in the prediction of the system synchronization ranges. The technique has been tested by comparison with computationally demanding circuit-level harmonic balance simulations in an array of Van der Pol-type oscillators and then applied to a coupled system of FET-based oscillators at 5 GHz, with a very good agreement with measurements.

Fast and Accurate Rectenna Design Method

A fast and accurate method is proposed for the design of rectennas. It combines the use of a full-wave electromagnetic simulator, for the analysis of the antenna, and a harmonic balance simulator, for the analysis of the rectifying circuit, with a single-point experimental calibration procedure. The proposed method can be used to speed up the development of general rectenna configurations. Two case studies are presented to illustrate the proposed concept. The first one is general, whereas the second is efficiency-optimization-oriented. A modified planar printed dipole without a matching network is used in both cases. An efficiency of 48% is achieved for a power density of 57 μW/cm² around 5 GHz. A good agreement is obtained between simulation and experimental results.

Linearity improvement in E‐mode ferroelectric GaN MOS‐HEMT using dual gate technology

In this work, an enhancement mode dual gate ferroelectric gallium nitride metal oxide semiconductor-high electron mobility transistor (GaN MOS-HEMT) is proposed with enhanced linearity characteristics. The different DC characteristics of the device are analysed and compared with available experimental data of single gate un-recessed ferroelectric GaN MOS-HEMT. In order to analyse the linearity performance of the devices, a look up table-based large signal model is developed directly from technology computer-aided device simulation results built by feeding different small signal parameters. The different linearity characteristics such as input third-order intercept point (IIP3), the input gain compression point (P1dB), third-order intermodulation (IM3) and the carrier to intermodulation power ratio of both the devices are compared by harmonic balance simulation of the developed large signal models. The interlink between IIP3 and IM3 with transconductance indicates that the broader the transconductance distribution with respect to different gate voltage generates higher IIP3 and lower IM3, which results in an improved linearity performance. The dual gate device shows improved linearity performance resulting in applicability in radiofrequency front end receiver.

High-Efficient Broadband CPW RF Rectifier for Wireless Energy Harvesting

In this paper, the implementation of a compact, wideband, single layer, and simple RF rectifier design is discussed. The proposed rectifier configuration is based on the coplanar waveguide transmission line. The rectifier is constructed using the voltage doubler circuit in conjunction with a broadband matching network. To obtain a small circuit size, the matching circuit is constituted with a series dual-inductive lumped element. The overall rectifier dimensions are very compact $22.5\times 31\,\,\text {mm}^{2}$ . In order to accurately characterize the rectifier performance, the electromagnetic and harmonic balance simulations are conducted using the Agilent, ADS software. The comparison shows good agreement between the simulation and measurement results. For instance, the peak measured efficiency is 74.8% at 10-dBm RF input power and the corresponding simulated value is 75% with a terminal load of 1 $\text{k}\Omega $ . The efficient frequency range is extending from 0.1 to 2.5 GHz, with an efficiency of more than 45% at input power 10 dBm.

Enhanced Broadband RF Differential Rectifier Integrated with Archimedean Spiral Antenna for Wireless Energy Harvesting Applications

This work addresses the design and implementation of a broadband differential rectifier (DR) combined with an Archimedean spiral dipole antenna (ASDA) for wireless power harvesting at low incident power densities below 200 W/cm. The proposed design exhibits an improved RF-DC conversion efficiency over a wide frequency range from 1.2 to 5 GHz. This frequency band is associated with several wireless communication services, for instance, ISM, WLAN, 5G, LTE, and GPS applications. The receiving planar ASDA exhibits circular polarization and has an average measured gain of 4.5 dBi from 1.2 to 5 GHz. To enable a wide operating bandwidth, the rectifier circuit is constituted by two architectures, designated A and B. Each scheme is designed to harvest power efficiently across a specific bandwidth. The optimal performance of both rectifiers are obtained using the nonlinear harmonic-balance simulations. The antenna–rectifier integration yields a compact rectenna with a high-efficiency performance over the intended bandwidth from 1.2 to 5 GHz for an input power of 9 dBm and terminal load resistance of 1 k. The total measured RF-DC conversion efficiency is maintained above 30% across the entire frequency range with a peak value of 61% achieved at 1.2 GHz. In comparison with similar architectures, the proposed rectenna maintains a stable output efficiency despite the wide fluctuations in the input frequency and also has a minimum footprint size (58 × 55 mm).

Two-Scale Envelope-Domain Analysis of Injected Chirped Oscillators

The response of chirped oscillators under the injection of independent signals, for spectrum sensing in cognitive radio, and under self-injection, for radio frequency identification, is analyzed in detail. The investigation is performed by means of a semianalytical formulation, based on a realistic modeling of the free-running oscillator, extracted from harmonic-balance simulations or from experimental measurements, through a new characterization technique. In the new formulation, the oscillator is linearized about a free-running solution that varies with the control voltage. This enables its application to oscillators having a frequency characteristic that deviates from the linear one. In the case of injection by independent signals, the two-scale envelope-domain formulation will enable an efficient handling of the difference between the slow chirp frequency and the beat frequency. The input carriers can be detected from their dynamic synchronization intervals or, at lower input-power levels, from the dynamics of the beat frequency. Noise perturbations are introduced into the formulation, which enables an estimation of the minimum detectable signal. In the case of a self-injected oscillator for radio frequency identification, an insightful formulation is derived to predict the propagation and tag-resonance effects on the instantaneous oscillation frequency. The tag-resonance signature gives rise to a distinct modulation of the oscillation frequency during the chirp period, which can be detected from the variation of the oscillator bias current. The analysis methods are illustrated through their application to a chirped oscillator, operating in the band 2–3 GHz.

Advanced Extrapolation Technique for Neural-Based Microwave Modeling and Design

In this paper, an advanced multidimensional extrapolation technique for neural-based microwave modeling and design is proposed to address the modeling challenges in microwave simulation and optimization, such as electromagnetic (EM) optimization and large-signal harmonic balance simulation. A standard neural model is accurate only within the particular range where it is trained by training data, and is unreliable if it is used outside this range. Our proposed method aims to address this issue by extrapolation to provide information and guide the neural network outside the training range. The given training data can be randomly distributed in the input space, and the boundaries of the training region can be arbitrary. The unknown target values of the model outside the training range are formulated as optimization variables and are determined by optimization, such that the first-order continuity of model outputs versus inputs is preserved and the second-order derivatives are minimized everywhere. Formulas for the first-order continuity and the second-order derivatives are derived through the cubic polynomial functions. In this way, the formulation of the proposed method guarantees a good model accuracy inside the training region and makes the model maximally smooth across all directions everywhere outside the training region. Compared with existing extrapolation methods for neural networks, the proposed extrapolation technique makes neural models more robust, resulting in faster convergence in microwave simulation and optimization involving neural model inputs as iterative variables. The validity of the proposed technique is demonstrated using both EM optimization example and nonlinear microwave simulation examples.

Motion in frequency domain for harmonic balance simulation of electrical machines

PurposeThe high computational effort of steady-state simulations limits the optimization of electrical machines. Stationary solvers calculate a fast but less accurate approximation without eddy-currents and hysteresis losses. The harmonic balance approach is known for efficient and accurate simulations of magnetic devices in the frequency domain. But it lacks an efficient method for the motion of the geometry.Design/methodology/approachThe high computational effort of steady-state simulations limits the optimization of electrical machines. Stationary solvers calculate a fast but less accurate approximation without eddy-currents and hysteresis losses. The harmonic balance approach is known for efficient and accurate simulations of magnetic devices in the frequency domain. But it lacks an efficient method for the motion of the geometry.FindingsThe three-phase symmetry reduces the simulated geometry to the sixth part of one pole. The motion transforms to a frequency offset in the angular Fourier series decomposition. The calculation overhead of the Fourier integrals is negligible. The air impedance approximation increases the accuracy and yields a convergence speed of three iterations per decade.Research limitations/implicationsOnly linear materials and two-dimensional geometries are shown for clearness. Researchers are encouraged to adopt recent harmonic balance findings and to evaluate the performance and accuracy of both formulations for larger applications.Practical implicationsThis method offers fast-frequency domain simulations in the optimization process of rotating machines and so an efficient way to treat time-dependent effects such as eddy-currents or voltage-driven coils.Originality/valueThis paper proposes a new, efficient and accurate method to simulate a rotating machine in the frequency domain.

Integrated Cooperative Ambient Power Harvester Collecting Ubiquitous Radio Frequency and Kinetic Energy

An integrated cooperative harvester capable of collecting both electromagnetic [(principally radio frequency (RF)] and kinetic energy simultaneously is proposed and studied in this paper. One single substrate board is demonstrated to accommodate and integrate an electromagnetic kinetic generator, an F-shaped antenna, and a rectifying circuit, having a compact credit-card size. Cooperative harvesting mode enables a significant enhancement of the diode’s power conversion efficiency (PCE) and also greatly strengthening the resilience of harvester at the same time. Measured results suggest that SMS7630 and HSMS-2850-based cooperative harvesters can increase dc output power by 50% to around 100% when both injecting power sources exhibit below −37 dBm with reference to the total added amount of dc power harvested directly from separate RF and kinetic excitation modes. A closed-form method is proposed and used to predict the enhanced efficiency performance of the cooperative harvester within a low power range of interest (from −50 to −30 dBm). Our analysis highlights the roles of the diode’s nonlinear junction resistance and capacitance. Good agreements are observed when comparing the analysis results with harmonic balance simulations by ADS. Furthermore, this method is carefully verified by measured results of two different diode-based harvesters. Measured dc output power rectified from cooperative harvesting mode is reported to be 1.16 and 1.97 times larger with SMS7630 and HSMS-2850, respectively, when both power sources are −40 dBm, compared to the conventional single RF harvesting mode. The proposed cooperative harvester with its strong resilience is highly suitable for ambient power harvesting in an ever-changing environment.

Fan tonal noise from aircraft aeroengines with short intake: A study at approach

This work assesses the risks of increased fan noise for high bypass ratio aeroengines with short intakes. The close proximity between the fan and inlet contributes to the increase in radiation of the rotor-alone tones and reinforces the interaction of the inflow distortion with the fan. Thus, the closer the fan is to the inlet, the higher the risk for noise generation. This article discusses the results of Harmonic Balance simulations performed on a conceptual turbofan operated at the approach condition. The inflow distortion created by the nacelle incidence is dominated by the circumferential component [Formula: see text] = 1. Its presence is visible throughout the nacelle. A thorough analysis of the unsteady pressure and velocity fields shows that the new acoustic source created by the periodic unsteady loading of the rotor cutting the inflow distortion is negligible compared to the rotor–stator interaction. But the amplitude of the rotor–stator interaction tones is affected by the unsteadiness of the rotor wake shape, particularly in the tip region where a pronounced flow separation on the rotor blade is created at a certain range of azimuthal position. The variations of the flow incidence at the rotor leading edge, due to the axial and tangential components of the mean velocity, cannot explain that flow separation. Instead, the origin is attributed to the azimuthal variations of the radial component of the mean flow velocity near the casing which slightly points inward to the spinner, i.e. in the opposite direction of the casing contour line. The flow separation induces a pronounced scattering of the wake azimuthal components mw = hB into [Formula: see text] in the tip region, whereas the same effect is rather limited on the rest of the blade height. This leads to a moderate increase of the tonal sound power level compared to the case with clean inflow. The azimuthal scattering due to the propagation of the sound waves through the distortion is found to be weak in the bypass duct. However, this effect is very important in the inlet lip vicinity, where the strong asymmetry of the flow modifies the path of the sound waves up to the far field.

Multitone Excitation Analysis in RF Energy Harvesters—Considerations and Limitations

The effect of multitone excitation on the dc response of a voltage-doubler radio frequency energy harvester is analyzed. Theoretical analysis as well as frequency and time domain (TD) simulations were conducted to clarify the findings. Measurements were also carried out to validate the results. The measured, simulations and theoretical results are in good agreement. This paper focuses on evaluating the performance of a voltage doubler rectifier under multitone excitation (input power is the same in the single-tone and multitone case). Based on TD and harmonic balance simulations, theoretical and measurement analyses, it is evident that the application of multiple tones simultaneously within the matched frequency band and with the same average available power results in a lower average output dc power when compared with the single-tone case with the same input power. This trend is evident over a broad low input power range of −50 to −10 dBm (0.01–100 $\boldsymbol {\mu }\text{W}$ ).

Oscillation Modes in Symmetrical Wireless-Locked Systems

Time synchronization of multiple elements of a wireless network can be achieved through the wireless coupling of their oscillator circuits. Most previous works on wireless locking of oscillators analyze the system in an idealized manner, representing the oscillator elements with phase models and describing the propagation effects with constant scalar coefficients and time delays. Here, a realistic analysis of the wireless system is presented, which relies on the extraction of the oscillator models from harmonic-balance (HB) simulations and takes into account the antenna gains and propagation effects. The most usual network configurations, corresponding to ring, fully connected, and star topologies, are investigated in detail. In symmetric conditions, the oscillation modes are detected through an eigenvalue/eigenvector calculation of an equivalent coupling matrix. For each particular mode, the system is analyzed in the following manners: by means of an analytical formulation, able to provide all the coexistent solutions, and through a circuit-level HB simulation of an equivalent system with a reduced number of oscillator elements. The stability properties are determined by means of a perturbation system of general application to any coupled structure. A specific formulation is also derived to predict the impact of discrepancies between the oscillator elements. All the results have been validated with independent circuit-level simulations and measurements.

Effects of Noisy and Modulated Interferers on the Free-Running Oscillator Spectrum

A new methodology for the prediction of oscillator phase dynamics under the effect of an interference signal is presented. It is based on a semianalytical formulation in the presence of a noisy or modulated interferer, using a realistic oscillator model extracted from harmonic-balance simulations. The theoretical analysis of the phase process enables the derivation of key mathematical properties, used for an efficient calculation of the interfered-oscillator spectrum. The resulting quasi-periodic spectrum is predicted, as well as the impact of the interferer phase noise and modulation over each spectral component, in particular over the one at the fundamental frequency. It is demonstrated that under some conditions, the phase noise at this component is pulled to that of the interference signal. Resonance effects at multiples of the beat frequency are also predicted. In addition, the effects of interferer phase and amplitude modulation on the oscillator phase dynamics have been studied and compared. For that analysis, efficient simulation techniques have been developed. The analyses have been validated with experimental measurements in an FET-based oscillator at 2.5 GHz, obtaining excellent agreement.

A comprehensive study of cryogenic cooled millimeter-wave frequency multipliers based on GaAs Schottky-barrier varactors

AbstractThe benefit of cryogenic cooling on the performance of millimeter-wave GaAs Schottky-barrier varactor-based frequency multipliers has been studied. For this purpose, a dedicated compact model of a GaAs Schottky-barrier varactor using a triple-anode diode stack has been developed for use with a commercial RF and microwave CAD tool. The model implements critical physical phenomena such as thermionic-field emission current transport at cryogenic temperatures, temperature dependent mobility, reverse breakdown, self-heating, and high-field velocity saturation effects. A parallel conduction model is employed in order to include the effect of barrier inhomogeneities which is known to cause deviation from the expected I--V characteristics at cryogenic temperatures. The developed model is shown to accurately fit the I--V --T dataset from 25 to 295 K measured on the varactor diode stack. Harmonic balance simulations using the model are used to predict the efficiency of a millimeter-wave balanced doubler from room to cryogenic temperatures. The estimation is verified experimentally using a 188 GHz balanced doubler cooled down to 77 K. The model has been further verified down to 14 K using a 78 GHz balanced doubler.

Simulation Method for Complex Multivalued Curves in Injection-Locked Oscillators

A new methodology is presented for the efficient harmonic-balance simulation of injection-locked oscillators with complex multivalued and disconnected curves. It is illustrated through its application to high-order subharmonically injection-locked oscillators. A graphical technique is applied to analyze the oscillator-phase sensitivity with respect to the input signal, required for the injection-locked operation. The intricate synchronized-solution curves are obtained with the new method, which enables a global exploration of all the coexistent periodic solutions. These solutions can belong to different curve sections, in a multivalued response, or to disconnected synchronization curves. The method is based on the calculation of a series of phase-dependent outer-tier admittance functions, which provide the oscillator response to the injection signal. Coexistent solutions are simultaneously obtained through a contour-plot intersection, without the need for continuation techniques. The method is illustrated through application to an oscillator synchronized to low-frequency sinusoidal signal by means of a nonlinear-transmission line. The analysis and design techniques have been successfully validated through comparison with independent simulations and measurements.

Optimal Definition of Class F for Realistic Transistor Models

An optimal three-harmonic definition of class F at the intrinsic level is presented for realistic transistor models exhibiting IV characteristics with a nonzero knee voltage. This updated class-F definition is needed for use with the recently reported embedding device model , which predicts in a single harmonic balance simulation the voltage and current waveforms required at the package reference planes to sustain an intrinsic mode of operation. Optimal class-F operation is obtained by setting to infinite the third-harmonic output impedance of the transistor IV characteristics instead of using an open load for the third-harmonic termination. This is achieved by fine tuning the class-F quasi-rectangular drain voltage waveform. The required third-harmonic component of the drain voltage in the optimal class F is then found to be generated by the lossless inductive termination of the third-harmonic component of the drain displacement current arising from the nonlinear drain-to-source capacitance. The proposed class-F definition is verified for a gallium nitride (GaN) high electron mobility transistor using third-harmonic load–pull simulations with a realistic GaN transistor model. The optimal third-harmonic load termination predicted using the class-F definition is found to be in full agreement with the one obtained from the drain efficiency contour plots. A close agreement is also obtained for the predicted and measured optimal third-harmonic load termination, bringing experimental support for the proposed class-F definition.

Physics-Based Multi-Bias RF Large-Signal GaN HEMT Modeling and Parameter Extraction Flow

In this paper, a consistent DC to RF modeling solution for Al gallium nitride (GaN)/GaN high electron mobility transistors is demonstrated that is constructed around a surface-potential-based core. Expressions for drain current and intrinsic terminal charges in the form of surface-potential are used to simultaneously model the DC characteristics and the intrinsic capacitances of a commercial GaN device. Self-heating and trapping effects are incorporated to account for the non-linear nature of the device. We discuss the parameter extraction flow for some of the key model parameters that are instrumental in fitting the DC characteristics, which simultaneously determines the bias-dependent intrinsic capacitances and conductances that significantly eases the RF parameter extraction. Parasitic capacitances, gate finger resistance, and extrinsic bus-inductances are extracted, from a single set of measured non-cold-FET S-parameters, using the model process design kit. The extraction procedure is validated through overlays of broadband (0.5–50 GHz) S-parameters, load-pull and harmonic-balance (10 GHz) simulations against measured data, under multiple bias conditions to successfully demonstrate the model performance at large-signal RF excitations.

Monte Carlo modelling of Schottky diode for rectenna simulation

Before designing a detector circuit, the electrical parameters extraction of the Schottky diode is a critical step. This article is based on a Monte-Carlo (MC) solver of the Boltzmann Transport Equation (BTE) including different transport mechanisms at the metal-semiconductor contact such as image force effect or tunneling. The weight of tunneling and thermionic current is quantified according to different degrees of tunneling modelling. The I-V characteristic highlights the dependence of the ideality factor and the current saturation with bias. Harmonic Balance (HB) simulation on a rectifier circuit within Advanced Design System (ADS) software shows that considering non-linear ideality factor and saturation current for the electrical model of the Schottky diode does not seem essential. Indeed, bias independent values extracted in forward regime on I-V curve are sufficient. However, the non-linear series resistance extracted from a small signal analysis (SSA) strongly influences the conversion efficiency at low input powers.