Vector Fitting Algorithm Research Articles

Enhancing rational approximation of wideband resonant MIMO systems with frequency-domain data

Vector Fitting (VF) is a successful methodology for both time and frequency domain estimation of dynamic systems. These algorithms have gained popularity as a system identification tool due to their ability to provide a practical procedure for accurately approximating the measured frequency response behavior of complex and highly resonant dynamic systems using rational models. Instrumental Variable (IV) techniques can also be added to the VF algorithm to enhance the optimality of the solutions in the presence of measurement noise. In this context, this paper presents a multiple-input multiple-output (MIMO) generalization of instrumental variable (IV-)VF. The proposed algorithm can be applied to accurately estimating models formed by either continuous- or discrete-time rational basis functions (RBFs). We show that such a generalization preserves the main implementation features observed in Standard VF, but with improved optimality property for its solutions. Additionally, in order to alleviate convergence problems that may arise from the fixed high-frequency normalization, we also present in this paper how to incorporate in the proposed IV-VF method the relaxed formulation found in standard VF for both the continuous and discrete-time cases. The resulting MIMO IV-VF algorithm can be easily understood by VF users, and its applicability is discussed by using an estimation case study with two power system pieces of equipment (an 88 kV Transformer and a 550 kV Instrument Transformer) for which actual frequency domain measurements were collected in the field. It is shown that the proposed algorithm significantly improves the modeling accuracy when compared to standard VF algorithms.

Rapid Multi-Objective Optimization Strategy of TSV-Based 3D Inductor Using Vector Fitting and Evolutionary Algorithm

Rapid Multi-Objective Optimization Strategy of TSV-Based 3D Inductor Using Vector Fitting and Evolutionary Algorithm

Wideband parametric baseband macromodeling of linear and passive photonic circuits via complex vector fitting

A novel wideband parametric baseband macromodeling technique for passive photonic devices and circuits is presented. It allows to efficiently estimate the baseband scattering representations of a linear, passive photonic system as a function of a set of design variables, such as geometrical layout or substrate features. The proposed technique relies on the interpolation of macromodels computed via a complex vector fitting (CVF) algorithm, by adopting a methodology based on amplitude and frequency scaling that preserves, by construction, the physical properties of the system, such as causality, stability and passivity. For a specified combination of the design parameters, a rational CVF model is derived that can be simulated by a wide range of ordinary differential equation (ODE) solvers or circuit simulators. Additionally, time-domain simulations using the computed model can be performed at arbitrary optical carrier frequencies, thus allowing for the simulation of multi-wavelength systems. Two application examples are presented to demonstrate the flexibility and advantages of the proposed method.

Enhancing CBFM With Adaptive Frequency Sampling for Wideband Scattering From Objects Buried in Layered Media

An adaptive frequency sampling (AFS) strategy is proposed in conjunction with characteristic basis function method (CBFM) to investigate the problem of wide-band scattering from objects buried in layered media. Conventionally, the CBFM is implemented to reduce the solution time at a single frequency. However, wide-band analysis of the above problem is still time consuming when a large number of frequency samples (FSs) are employed with uniform sampling. To mitigate this issue and improve efficiency, the AFS algorithm is proposed to select FSs with a reduced number via an iterative process based on error-estimators. The vector fitting (VF) algorithm is used to derive the rational model, which allows to generate the scattered fields in the band efficiently. The key feature of the proposed AFS is it minimizes the number of the FSs and guarantees the convergence of the solution simultaneously. Numerical results demonstrate that the number of required samples is significantly reduced without compromising the accuracy of the solution.

A Gray-Box Stability Analysis Method of Grid-Connected Inverter Considering Synchronization Dynamics

The Grid-Connected Inverter (GCI) can be considered a gray box when circuit and controller parameters are missing due to intellectual property rights or parameter variations caused by aging, which poses an impediment to assessing the stability of the system. This paper presents a gray-box stability analysis method based on impedance identification of GCI considering the synchronization dynamics. The impedance frequency responses of GCI are measured by the frequency scanning method on the dq-frame. Meanwhile, the influence of synchronization dynamics and background harmonics is theoretically investigated. A vector fitting (VF) algorithm, co-designed with impedance identification, is then applied to generate polynomial transfer functions. Based on the obtained transfer functions, the stability of the GCI can be judged by the distance relationship between the prohibited area boundary and the center of the gershgorin-circle through the distance formula. Finally, the experiments of both RT-LAB and experimental prototypes are conducted to verify the feasibility of the proposed method.

Wideband Simulations of Periodic Structures by the Hybrid Spectral FDTD/TD-VFz Method

The spectral finite-difference time-domain method is combined with the discrete-time time-domain vector fitting (TD-VFz) algorithm to analyze three-dimensional (3-D) periodic structures excited by oblique incident plane waves over a wide frequency band. The time signal reconstruction with TD-VFz produces an untruncated response over an infinite time window allowing an accurate spectral result, especially near the cutoff frequency separating the spectrum propagating wave region of the evanescent wave region.

A New Approach on Modeling of Pulsed-Power Loads

Pulsed power sources provide high-power waves with high frequency (HF) content for a wide range of applications. Conventionally, a simple resistance is employed for the most prevalent pulsed power loads. However, the load model can highly affect the efficiency of source design. This paper contributes to fill this gap by considering the HF behavior of the loads. Accordingly, a new approach has been proposed to present a general HF lumped-circuit model for pulsed power loads. The frequency response of the loads is identified using the Finite Integration Technique (FIT). Subsequently, an equivalent circuit model is established based on the vector fitting (VF) algorithm. The applicability of the approach has been verified by comparison with two experiments, i.e. Vircator and coaxial reactor as the loads. It is indicated that the HF-based model can significantly improve the simulation results to be in good agreement with experiments. The results present that while a pure-resistive load model could lead to about 30% error in the estimation of the characteristics of output voltage, it is about less than 5% in the case of the proposed HF load model.

Stabilized Sanathanan-Koerner Iteration for Rational Transfer Function Approximation of Scattering Parameters

Abstract The vector-fitting algorithm has been the predominant method of macromodeling of high-frequency circuits for the past 20 years. Vector fitting is based on a partial-fractions basis to avoid the ill-conditioning of the Vandermonde matrices in direct fitting of the polynomial coefficients of a rational function. In a recent formulation of the Sanathanan- Koerner iteration, an orthogonal basis is obtained using the Arnoldi iteration significantly improving the conditioning and accuracy of rational approximation. This Stabilized Sanathanan-Koerner (SSK) iteration has been mostly applied on closed-form functions or data with no noise. We will utilize this SSK formulation for high-frequency package interconnect macromodeling. The method will be tested using empirical scattering parameters on a noisy transmission line system. The resultant curve fit will then be compared to several other polynomial and rational linear approximation methods, including the standard polynomial approximation. We show that the SSK method compares favorably to these different methods and provide a curve fit that can provide a reasonable approximation over different frequency ranges.

Leveraging a data-driven approach to simulate and experimentally validate a MIMO multiphysics vibroacoustic system

The objective of the present work is to evaluate the performance of a data-driven approach to simulate a multiphysics vibroacoustic field. Of particular interest is the complex propagation of anechoic waves in a finite 1D-structure submerged in water. Experimental data from an underwater beam actuated by two macro-fiber composites (MFCs) serves as the experimental framework for the data-driven model. Next, the Vector-Fitting (VF) algorithm estimates a state-space model of the multiphysics dynamics with voltage signals to the two MFCs as inputs, and the structural and the acoustic responses as the outputs. The multi-input–multi-output data-driven model is then used to determine the parameters that result in the optimal anechoic wave, a process that was carried out previously by experimental iterations. The optimal time-domain simulations are validated with experimental results.

A State-Space-Based Method to Model Vccin Feedthrough Noise in Microprocessors With Fully Integrated Voltage Regulators

Modern high-performance server microprocessors need to integrate multiple intellectual property (IP) blocks in a single chip to achieve higher performance. Due to the diverse supply voltage requirements of these IP blocks, their supply voltages are generated locally on the chip using fully integrated voltage regulators (FIVRs). FIVR is designed using on-chip power bridges, bridge drivers, control circuits, on-chip metal–insulator–metal (MIM) capacitors, package inductors, and package power planes. The input power supply (Vccin) of all the FIVRs is shared to minimize the platform cost and is generated using a voltage regulator module (VRM) on the motherboard. The noise coupling between FIVRs due to the common input supply is referred to as the Vccin feedthrough noise. The existing method of modeling the Vccin feedthrough noise is based on circuit simulation. Circuit models are highly complex as different components such as the FIVR power bridges, bridge drivers, MIM capacitors, package inductors, package power planes, and the Vccin network need to be modeled. Convergence issues are very frequent in full chip circuit simulations and it typically takes many iterations. Hence design optimization is difficult to carry out using circuit simulations. In this article, a state-space-based time-domain method is proposed to model the Vccin feedthrough noise based on G-parameters and S-parameters of the FIVR and the Vccin network. The state-space models are generated using the vector-fitting algorithm. The proposed method simplifies the construction of the Vccin feedthrough models and improves the convergence time.

Estimating experimental dispersion curves from steady-state frequency response measurements

Dispersion curves characterize the frequency dependence of the phase and the group velocities of propagating elastic waves. Many analytical and numerical techniques produce dispersion curves from physics-based models. However, it is often challenging to accurately model engineering structures with intricate geometric features and inhomogeneous material properties. For such cases, this paper proposes a novel method to estimate group velocities from experimental data-driven models. Experimental frequency response functions (FRFs) are used to develop data-driven models, which are then used to estimate dispersion curves. The advantages of this approach over other traditionally used transient techniques stem from the need to conduct only steady-state experiments. In comparison, transient experiments often need a higher-sampling rate for wave-propagation applications and are more susceptible to noise.The vector-fitting (VF) algorithm is adopted to develop data-driven models from experimental in-plane and out-of-plane FRFs of a one-dimensional structure. The quality of the corresponding data-driven estimates is evaluated using an analytical Timoshenko beam as a baseline. The data-driven model (using the out-of-plane FRFs) estimates the anti-symmetric (A0) group velocity with a maximum error of 4% over a 40 kHz frequency band. In contrast, group velocities estimated from transient experiments resulted in a maximum error of 6% over the same frequency band.

Handling Initial Conditions in Vector Fitting for Real Time Modeling of Power System Dynamics

This paper develops a predictive modeling algorithm, denoted as Real-Time Vector Fitting (RTVF), which is capable of approximating the real-time linearized dynamics of multi-input multi-output (MIMO) dynamical systems via rational transfer function matrices. Based on a generalization of the well-known Time-Domain Vector Fitting (TDVF) algorithm, RTVF is suitable for online modeling of dynamical systems which experience both initial-state decay contributions in the measured output signals and concurrently active input signals. These adaptations were specifically contrived to meet the needs currently present in the electrical power systems community, where real-time modeling of low frequency power system dynamics is becoming an increasingly coveted tool by power system operators. After introducing and validating the RTVF scheme on synthetic test cases, this paper presents a series of numerical tests on high-order closed-loop generator systems in the IEEE 39-bus test system.

A Robust Circuit and Controller Parameters’ Identification Method of Grid-Connected Voltage-Source Converters Using Vector Fitting Algorithm

This article presents a vector fitting (VF) algorithm-based robust circuit and controller parameters’ identification method for grid-connected voltage-source converters (VSCs). The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$dq$ </tex-math></inline-formula> -domain impedance frequency responses (IFRs) of the VSCs are first measured using the frequency scanning method, based on which the corresponding measured phasor-domain IFRs are calculated. Then, polynomial transfer functions are generated by applying the VF algorithm on the measured phasor-domain IFRs, from which the circuit and controller parameters, i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$LCL$ </tex-math></inline-formula> filter parameters, digital sampling time, current controller parameters, and phase-locked loop parameters, are identified. The influence of measurement noise on parameters’ identification accuracy and corresponding countermeasure to mitigate the adverse influence are also theoretically investigated. The proposed method is able to identify the circuit and controller parameters when detailed parameters are missing due to industrial secrecy or parameters variation caused by operating condition change, temperature fluctuation, or aging. The effectiveness of the proposed circuit and controller parameters’ identification method is validated by theoretical demonstration, OPAL-RT-based real-time simulation, and experimental validation.

Two VHDL-AMS-based models of multi-conductor power cables for EMI simulations

Due to the fast switching of the commutation cell in a motor drive system, high-frequency electromagnetic disturbances propagate through cables to reach the loads. Mismatched impedances lead to an overvoltage at the loads terminals. In order to study this EMI issue, accurate circuit models of power cables are required. In this context, this paper proposes two multi-conductor cable models. Our approach takes into consideration the frequency dependence of the cable per-unit-length parameters, the coupling between conductors and propagated induced phenomena. The power cables used in motor drive systems are characterized by their rotational symmetry. Thus, the complexity of the model is reduced by means of both a set of mathematical functions approximating the frequency behavior of the cable and using a modal domain-based transfer matrix. So, the VHDL-AMS frequency model can be easily deduced. Once the model is established, the simulation of any cable length becomes an easy task. The model is implemented in ANSYS Simplorer software. After that, the time domain model is built up based on the vector-fitting algorithm. In order to validate our proposal, a comparison of the common mode and the differential mode frequency responses of the reduced cable model with those of the well-known cascaded cell model and measurements found in the literature is performed. The results demonstrate the accuracy and the speed of our approach in the predefined frequency range. Also, a time-domain comparison between the frequency model and the time model is performed. The results obtained validate the two models.

GVF: GPU-Based Vector Fitting for Modeling of Multiport Tabulated Data Networks

Modeling of multiport data characterizing high-speed modules, such as packages, vias, and complex multiconductor interconnects is becoming increasingly important in signal and power integrity applications. Vector fitting (VF) algorithm has been widely used by designers for macromodeling and system identification from such multiport tabulated data. Since VF and strategies based on it require many iterations to arrive at an optimal number of converged poles, it is highly desired to reduce the computational cost of each VF iteration. This article advances the applicability of VF to exploit the emerging massively parallel graphical processing units (GPUs) by developing necessary parallelization strategies and investigates their performance while using different GPU libraries. For large problem sizes (an increasing number of poles and ports), numerical results demonstrate that the proposed method while using MAGMA libraries provides significant speedup compared with existing multi-CPU-based parallel VF techniques.

A Gray-Box Hierarchical Oscillatory Instability Source Identification Method of Multiple-Inverter-Fed Power Systems

This article presents a gray-box hierarchical instability source identification method of multiple-inverter-fed power systems, which enables stability analysis at system, component, and parameter levels sequentially. Impedance frequency responses of all components are first obtained using frequency scanning method. System impedance network model is then established by connecting these individual components based on system topology, which is further lumped into a loop impedance model (LIM). The vector fitting (VF) algorithm is then used to generate system state–space model from the impedance frequency responses of the LIM for eigenvalue-based stability analysis. If the system is assessed to be unstable, problematic components are further identified by performing impedance-based stability criterion at terminals of all components, where the numbers of right-half-plane poles are obtained by the VF algorithm. Finally, circuit and controller parameters of the identified problematic components are further identified using the VF algorithm, which are retuned to improve system stability. The proposed hierarchical instability source identification method is implemented in a multiple-paralleled grid-connected inverter system. Simulation results obtained in MATLAB/Simulink platform and real-time simulation verification results obtained in OPAL-RT platform are given to validate the correctness of the theoretical analysis results.

A Parametric Conducted Emission Modeling Method of a Switching Model Power Supply (SMPS) Chip by a Developed Vector Fitting Algorithm

This paper proposes a modeling method to establish a parametric-conducted emission model of a switching model power supply (SMPS) chip through a developed vector fitting algorithm. A common SMPS chip LTM8025 was taken as an example to explain the modeling process. According to the integrated circuit (IC) electromagnetic modeling (ICEM) standard, the parametric conducted emission model is divided into two parts: IC internal activity (ICIA) and IC passive distribution network (ICPDN). The parameters of ICIA are identified by measured data and correlated with key components; an improved vector-fitting algorithm is proposed to solve the fitting problem of ICPDN without phase information. This parametric model can be used with commercial simulation software together to achieve predictions of conducted emissions from power modules. The experiment results show that the maximum and 90% confidence interval of the forecast errors are 9.677 dB and (−4.56 dB, 6.52 dB) respectively, which achieve the international standard requirements and have sufficient accuracy and effectiveness.

Baseband Macromodeling of Linear Photonic Circuits for Time-Domain Simulations

In this paper, we propose a novel approach to build real-valued baseband models of linear, time-invariant, passive photonic devices, circuits, and systems, which allows modeling of photonic wavelength filter circuits with their full dispersion effects in an accurate way. The proposed technique starts from the scattering parameters of the photonic systems under study and leverages on the modeling power of the vector-fitting algorithm, thereby leading to both complex- and real-valued baseband state-space models. The modeling procedure is robust and applicable to general linear passive photonic devices and circuits, and the physical properties of the resulting models for the time-domain simulation, such as stability and passivity, can be properly guaranteed. The built models are systems of first-order ordinary differential equations (ODEs), which can be efficiently simulated in a variety of ODE solvers at baseband frequencies rather than in the optical frequency range. We demonstrate the applicability and accuracy of the proposed method on two examples of photonic filter circuits.

Frequency-Dependent Pi Model of a Three-Core Submarine Cable for Time and Frequency Domain Analysis

In this paper, a Frequency-Dependent Pi Model (FDPi) of a three-core submarine cable is presented. The model is intended to be used for the representation of submarine cables in an Offshore Wind Power Plant (OWPP) scenario for both time and frequency domain analysis. The frequency-dependent variation of each conductive layer is modeled by a Foster equivalent network whose parameters are tuned by means of Vector Fitting (VF) algorithm. The complete formulation for the parameterization of the model is presented in detail, which allows an easy reproduction of the presented model. The validation of the model is performed via a comparison with a well-established reference model, the Universal Line Model (ULM) from PSCAD/EMTDC software. Two cable system case studies are presented. The first case study shows the response of the FDPi Model for a three-core submarine cable. On the other hand, the second case study depicts the response of three single-core underground cables laying in trefoil formation. This last case shows the applicability of the FDPi Model to other types of cable systems and indirectly validates the response of the aforementioned model with experimental results. Additionally, potential applications of the FDPi model are presented.

Performance of Vector Fitting Algorithm Applied to Bandpass and Baseband Systems

This article presents the performance evaluation of Vector Fitting Algorithm (VFA) from a system identification perspective. In this paper, VFA has been first applied to known baseband and bandpass systems such as Butterworth lowpass and bandpass filters to analyze the algorithm’s pole-residue extraction ability for band-limited noisy data. The poles identified by the algorithm for different bandwidths and noise powers are compared with the actual system poles of the baseband and bandpass systems. It is concluded that the algorithm is capable of identifying the actual system poles even if the capture bandwidth is less than the 3 dB bandwidth, which is a significant observation of this paper. It is also seen that the system identification performance with noisy data is better for baseband systems when compared to bandpass systems. Further, a practical investigation has been done to evaluate VFA performance for modeling a microstrip coupled line filter in the presence of noise.