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Rational approximation of frequency-domain responses by Vector Fitting
Abstract
The paper describes a general methodology for the fitting of
measured or calculated frequency domain responses with rational function
approximations. This is achieved by replacing a set of starting poles
with an improved set of poles via a scaling procedure. A previous paper
(Gustavsen et al., 1997) described the application of the method to
smooth functions using real starting poles. This paper extends the
method to functions with a high number of resonance peaks by allowing
complex starting poles. Fundamental properties of the method are
discussed and details of its practical implementation are described. The
method is demonstrated to be very suitable for fitting network
equivalents and transformer responses. The computer code is in the
public domain, available from the first author
... The advantages of the new approach are demonstrated for frequencydependent network equivalent (FDNE) modeling and wideband modeling of transformers, focusing on accuracy and computational efficiency. [14] II. INPUT -OUTPUT TRANSFER FUNCTION The modeling starts from a given port-admittance matrix Y(s), which defines the relation between port voltage v and current i. ...
... It is assumed that a rational model for Y has been identified which satisfies (2). This can be easily achieved by fitting the elements of Y using the pole relocating algorithm known as vector fitting (VF) [14]. This leads to a pole-residue model (3) which can be expanded into a model in standard state-space form (4). ...
... However, in our case, we simply try to calculate a Q which is as close to T as possible. This problem is addressed by the orthogonal procrustes problem (14) Which seeks to calculate the orthogonal matrix Q which rotates B as close as possible into A. [15] With the obtained transformation matrix Q, we calculate the matrix using (7) which is fitted by the rational model (3) using vector fitting (VF). ...
Employing an admittance matrix in the frequency domain by rational functions for power transformers is a well-known method which improves calculation efficiency. This model must be passive in order to avoid unstable time domain simulations. All of the methods have made efforts to overcome the problem of preserving the passivity of the final model. In this paper a similarity transformation matrix which better reveals the eigenvalues of the admittance matrix is presented. The chosen transformation preserves the passivity and symmetry of the original data.
... The FRVF, first documented in its current and complete form in [27], originated as an improved version of the standard Vector Fitting (VF) algorithm, proposed by Gustavsen and Semlyen in 1999 for the modelling of large multiport electrical circuits [28]. With respect to the VF algorithm, the FRVF adds a relaxed non-trivial constraint in the pole identification step [29,30] and exploits the matrix form of the linear problem and the QR decomposition, such that A = QR, for fast computation. ...
... where H ∞ governs the behaviour at high frequencies and can be decomposed into two components: the real quantity d and the term se [28]. An unknown function σ(s) is defined as: ...
... The poles of σ(s) f (s) are the same rational approximation of σ(s). By forcing σ(s) to approach unity at very high frequencies (that is to say, for H ∞ = 1), the ambiguity is removed from the solution for σ(s) [28]. This leads to: ...
Predicting flutter remains a key challenge in aeroelastic research, with certain models relying on modal parameters, such as natural frequencies and damping ratios. These models are particularly useful in early design stages or for the development of small UAVs (maximum take-off mass below 7 kg). This study evaluates two frequency-domain system identification methods, Fast Relaxed Vector Fitting (FRVF) and the Loewner Framework (LF), for predicting the flutter onset speed of a flexible wing model. Both methods are applied to extract modal parameters from Ground Vibration Testing data, which are subsequently used to develop a reduced-order model with two degrees of freedom. Results indicate that FRVF and LFinformed models provide reliable flutter speed, with predictions deviating by no more than 3% (FRVF) and 5% (LF) from the N4SID-informed benchmark. The findings highlight the sensitivity of flutter speed predictions to damping ratio identification accuracy and demonstrate the potential of these methods as computationally efficient alternatives for preliminary aeroelastic assessments.
... In the last decades, a large number of algorithms have been proposed to tackle the problem, e.g. via interpolation [3,21,31,42,46] or optimization [10,12,36]. More recently, the Adaptive Antoulas-Anderson (AAA) algorithm [43] has been intensively and successfully applied to solve a vast collection of rational approximation problems [5,8,23,25,26,29,57], including MOR of LTI systems [18,28]. ...
... On the other hand, treating the problem as is, employing for example global optimization techniques, would be practically feasible only in case very few optimization variables are involved. In the following, we show that problem (36) can be relaxed to a convex problem that preserves the exactness of the model stability constraints, at the price of a slight modification of the target cost function. We start by noticing that the non-linear matrix inequality involved in the problem can be turned into a linear one by applying a congruence transformation. ...
... We remark that solving iteratively problem (52) can be done efficiently, following e.g., the approach described in [13,Sec.V], since the matrix L (30) and performing the same subsequent derivations. Notice that the size of the semidefinite program (36) does not depend on the number of input or output of the system. Hence, the choice of model structure (47) guarantees that stable MIMO ROMs can be built in a particularly efficient manner, as will be experimentally shown in Sec. 6. ...
In recent years, the Adaptive Antoulas-Anderson (AAA) algorithm has been applied extensively for generating Reduced Order Models (ROMs) of large scale Linear Time-Invariant systems starting from measurements of their transfer functions. When used for Model Order Reduction (MOR) of an asymptotically stable system, the ROMs generated applying AAA are not guaranteed to preserve this fundamental property, thus rendering them impractical in many relevant applications. To overcome this issue, we propose a novel algebraic characterization for the stability of ROMs represented under the AAA barycentric structure. We then translate such characterization into a set of convex semidefinite constraints that can be embedded into the AAA optimization routine to explicitly maximize the model accuracy while ensuring its stability. Finally, we generalize the resulting modeling framework to allow for efficient stable MOR of Multi-Input-Multi-Output systems. An extensive set of numerical experiments provides practical evidence for the effectiveness of the proposed approach and compares its performance with those of available state-of-the-art methods.
... A numerically robust and theoretically elegant representation is the so-called barycentric form of rational approximants [13]. For univariate functions f : C → C many effective rational approximation frameworks have been developed using the barycentric form; see, e.g., [2,6,13,20,23,29,36] and the references therein. Recently, several efforts have been made to extend these barycentric form based methods to the multivariate setting where we aim to approximate a function f : C d → C by a multivariate rational function r : C d → C; see, e.g., [5,19,22,32]. ...
... , d, consider the sampling points Z (j) in (24) and the interpolation nodes λ (j) in (26). Let D be the corresponding sample tensor in (25) and L d be the associated higher-order Loewner matrix in (29). If rank(D) = 1 and there exists α = 0 such that L d vec(α) = 0, then we can choose α such that rank(α) = 1. ...
... , d. Let L d be the corresponding higher-order Loewner matrix in (29) and α ∈ C n 1 ×···×n d an arbitrary tensor. Letλ (j) = λ (j) 1 , . . . ...
Approximations based on rational functions are widely used in various applications across computational science and engineering. For univariate functions, the adaptive Antoulas-Anderson algorithm (AAA), which uses the barycentric form of a rational approximant, has established itself as a powerful tool for efficiently computing such approximations. The p-AAA algorithm, an extension of the AAA algorithm specifically designed to address multivariate approximation problems, has been recently introduced. A common challenge in multivariate approximation methods is that multivariate problems with a large number of variables often pose significant memory and computational demands. To tackle this hurdle in the setting of p-AAA, we first introduce barycentric forms that are represented in the terms of separable functions. This then leads to the low-rank p-AAA algorithm which leverages low-rank tensor decompositions in the setting of barycentric rational approximations. We discuss various theoretical and practical aspects of the proposed computational framework and showcase its effectiveness on four numerical examples. We focus specifically on applications in parametric reduced-order modeling for which higher-dimensional data sets can be tackled effectively with our novel procedure.
... When the available data are acquired or generated in the complex frequency plane, the calculation of the poles and residues (or equivalently the poles and zeros) is quite straightforward through contour integrals [25,26,22,27]. However, in practice, many spectra are acquired over a real spectral range [ω A , ω B ] ⊂ R. In the latter case, much fewer methods exist but this longstanding problem has been attracting more and more attention [28,29,30,31,32]. Existing methods mainly approximate the response functions as meromorphic functions, the parameters of which are either the poles and zeros (Cauchy method [31] and AAA algorithm [32] indirectly), or the poles and residues (matrix pencil method [28], harmonic inversion [29], vector fitting [30]). ...
... However, in practice, many spectra are acquired over a real spectral range [ω A , ω B ] ⊂ R. In the latter case, much fewer methods exist but this longstanding problem has been attracting more and more attention [28,29,30,31,32]. Existing methods mainly approximate the response functions as meromorphic functions, the parameters of which are either the poles and zeros (Cauchy method [31] and AAA algorithm [32] indirectly), or the poles and residues (matrix pencil method [28], harmonic inversion [29], vector fitting [30]). They rely on matrix algebra, but can be quite sensitive to an initial guess of some parameters, as in the case of the vector fitting method. ...
The Singularity Expansion Method Parameter Optimizer - SEMPO - is a toolbox to extract the complex poles, zeros and residues of an arbitrary response function acquired along the real frequency axis. SEMPO allows to determine this full set of complex parameters of linear physical systems from their spectral responses only, without prior information about the system. The method leverages on the Singularity Expansion Method of the physical signal. This analytical expansion of the meromorphic function in the complex frequency plane motivates the use of the Cauchy method and auto-differentiation-based optimization approach to retrieve the complex poles, zeros and residues from the knowledge of the spectrum over a finite and real spectral range. Both approaches can be sequentially associated to provide highly accurate reconstructions of physical signals in large spectral windows. The performances of SEMPO are assessed and analysed in several configurations that include the dielectric permittivity of materials and the optical response spectra of various optical metasurfaces.
... Darwin Blanco is with Standards & Technology Ericsson AB, Stockholm, Sweden. and vector fitting [10], [11]. The Loewner framework and vector fitting are categorized as non-intrusive (or data-driven) MOR methods [12]. ...
... Vector fitting is well described in [10], [13]. We repeat the main steps in the scalar case for completeness. ...
Frequency domain sweeps of array antennas are well-known to be time-intensive, and different surrogate models have been used to improve the performance. Data-driven model order reduction algorithms, such as the Loewner framework and vector fitting, can be integrated with adaptive frequency sampling algorithms, in an iterative scheme, to reduce the number of full-wave simulations required to accurately capture the requested frequency behavior of multiport array antennas. In this work, we propose two novel adaptive methods exploiting a block matrix function which is a key part of the Loewner framework generating system approach. The first algorithm leverages an inherent matrix parameter freedom in the block matrix function to identify frequency points with large errors, whereas the second utilizes the condition number of the block matrix function. The first method effectively provide a frequency domain error estimator, which is essential for improved performance. Numerical experiments on multiport array antenna S-parameters demonstrate the effectiveness of our proposed algorithms within the Loewner framework, where the proposed algorithms reach the smallest errors for the smallest number of frequency points chosen.
... HVDC cables are meticulously modeled using cascaded proportional-integral (PI) sections with parallel RL branches [15], [16]. The parameters of the parallel RL branches are estimated via the vector-fitting approach [17]. The study shows that long HVDC cables introduce oscillations and potential instability into the interconnected system, which is analyzed using eigenvalue analysis techniques [18]. ...
... To enhance model accuracy, incorporating multiple cascaded PI sections is advantageous, particularly for long HVDC cables, although this comes at the expense of increasing the order of both the HVDC cable and the overall system. The parallel RL values are determined using the vector fitting approach [17] while generation G and capacitance C can be derived from the cable data. The cable model can be succinctly represented in the standard state-space form provided below with the states x cable be the voltages and the inductor currents of the capacitors. ...
Innovative dynamic models for the DC modular multilevel converter (DC-MMC) in rotating frame are presented in this paper, which are specifically designed to enhance converter design and stability analysis. Open-loop and closed-loop models are developed using three frames, providing a detailed examination of the impact of 2ndand 3rd harmonic components on the model accuracy. A novel contribution of this paper is the integration of a 2ndharmonic current suppression controller (SHCSC) within the closed-loop model, offering new insights into its effects on system stability. The DC-MMC model is further extended by coupling it with high-voltage direct current (HVDC) cables on each side, forming an interconnected system model that accurately represents a more authentic scenario for future DC grids. The proposed model is rigorously validated against PSCAD benchmark model, confirming their precision and reliability. The interconnected system model is then utilized to analyze the influence of cable length on system stability, demonstrating practical applications. The closed-loop model is subsequently employed for stability assessment of the inter-connected system, showcasing its applicability in real-world scenarios. Additionally, a damping controller is designed using participation factor and residue approaches, offering a refined approach to oscillation damping and stability optimization. The effectiveness of the controller is evaluated through eigenvalue analysis, supported by simulation results, underscoring its potential for enhancing system stability.
... The above results allow us to conclude that, although ϕ is bounded over any closed interval K ⊂ R \ {0}, if the strip of holomorphy of C(k) is too narrow, cf. (13), then almost-resonances will appear, in the form of "spikes" in ϕ, cf. (16) and (18). ...
... Indeed, rational approximation sometimes relies on unstable poles to improve approximation accuracy. It is commonly suggested to either keep such non-physical poles, to "flip" them [13], i.e., change the sign of their imaginary parts, or to remove them [18]. Four more spurious poles can also be observed in Fig. 2 (top), along a band with imaginary part between −1 and −0.2. ...
This paper describes novel algorithms for the identification of (almost-)resonant behavior in scattering problems. Our methods, relying on rational approximation, aim at building surrogate models of what we call "field amplification", defined as the norm of the solution operator of the scattering problem, which we express through boundary-integral equations. To provide our techniques with theoretical foundations, we first derive results linking the field amplification to the spectral properties of the operator that defines the scattering problem. Such results are then used to justify the use of rational approximation in the surrogate-modeling task. Some of our proposed methods apply rational approximation in a "standard" way, building a rational approximant for either the solution operator directly or, in the interest of computational efficiency, for a randomly "sketched" version of it. Our other "hybrid" approaches are more innovative, combining rational-approximation-assisted root-finding with approximation using radial basis functions. Three key features of our methods are that (i) they are agnostic of the strategy used to discretize the scattering problem, (ii) they do not require any computations involving non-real wavenumbers, and (iii) they can adjust to different settings through the use of adaptive sampling strategies. We carry out some numerical experiments involving 2D scatterers to compare our approaches. In our tests, two of our approaches (one standard, one hybrid) emerge as the best performers, with one or the other being preferable, depending on whether emphasis is placed on accuracy or efficiency.
... However, to improve the model accuracy for frequencies above 1 MHz, additional ladder circuit levels may be needed. The ladder circuit parameters are found using the process shown in Figure 8 by iteratively approximating the impedance and admittance of the system using vector fitting, a pole relocation method proposed by Gustavsen, whose work covers the process in great detail [33]. Vector fitting approximates an s-domain function, F(s), with the following form: ...
... Energies 2025, 18, x FOR PEER REVIEW 9 of 29 process in great detail [33]. Vector fitting approximates an s-domain function, F(s), with the following form: ...
This paper comprehensively presents an approach for modeling form wound coils of a motor driven by an inverter, with focus on the electric stresses on the coil insulation. A 10 kV SiC testbed for medium voltage form wound coils was developed to support and validate the modeling techniques discussed. A finite element analysis (FEA) model of the motor coil is presented using COMSOL 6.1. The FEA model was used to determine parameters for an electrical model based on the multi-conductor transmission line theory. The linking of these models allows for a rapid analysis of the electrical stresses the insulation can be exposed to. An experimental method for model validation using the empirical transfer function estimation (ETFE) approach to find the impedance response of the testbed for comparison to the proposed electrical model is presented and employed. The paper also uses the model to analyze the impact of insulation delamination and voids and to demonstrate the implementation of a metric called insulation state of health monitoring for both healthy and damaged coils.
... Modal delays must thus be identified and extracted from matrix for a proper rational fit. Vector Fitting (VF) is the adopted methodology for rational fittings in EMT analysis [2]. To enable the application of VF in the rational fit of , Gustavsen and Semlyen introduced in 1998 a method to estimate modal delays [3]. ...
... Since rational approximations with VF produces only real or complex conjugated pairs of poles and residues, the corresponding zeros too will be either real or complex conjugate pairs [2]; therefore, the compensation of any zero in the right-side of the s-plane implies the synthesis of either a first or a second-order all-pass function. Hence, to introduce the delay equalization strategy into the estimation process, the all-pass function is now represented as follows ...
Traveling-wave line models, such as the ULM, are widely used in time-domain EMT simulations for power systems. These models require the rational approximation of both, the characteristic admittance matrix Y_c and the propagation matrix H. The rational fitting of H is challenging due to the inclusion of a mix of modal delays in all its elements. These delays must thus be identified and extracted before proceeding to calculating its rational approximation. This paper proposes a new iterative method to estimate time delays employing all-pass filters and delay equalizations. As opposed to other currently used methods, the one proposed here ensures causality and minimum-phase features in the synthesized H model. Three test cases are included: 1) a synthetic transfer function, 2) a system of underground cables, and 3) the EMT response of an aerial line. The obtained results show that the proposed method maintains causality and attains similar accuracies with fewer iterations as a state-of the art method based on rms-error minimization.
... In this direction, an approach based on Vector Fitting [51], [52] that performs the frequency domain identification of a vector of transfer functions using a common denominator has been recently proposed in [53] to analyze the stability of multistage amplifiers. An advantageous aspect of using a Vector Fitting based algorithm for stability analysis of microwave circuits is that the identified transfer functions H n (s) are described using a partial fraction representation (H n is the n th component of a vector of transfer functions): (1) where p k are the system poles, common to all the H n , r n,k is the residue corresponding to pole p k and transfer function H n , and D is the direct gain. ...
... It is often necessary to divide the original problem into sub-bands when a ratio of two polynomials is used for the identification [23]. However, with a partial fraction representation as in (1), large numbers are avoided and the frequency response identification is numerically better conditioned [51]. This has an important consequence: large bandwidth responses can be identified without dividing the frequency response into narrower sub-bands and algorithms for automatic order selection of the transfer function are simplified and can be more effective. ...
Pole-zero identification refers to the obtaining of the poles and zeros of a linear (or linearized) system described by its frequency response. This is usually done using optimization techniques (such as least squares, maximum likelihood estimation, or vector fitting) that fit a given frequency response of the linear system to a transfer function defined as the ratio of two polynomials. This kind of linear system identification in the frequency domain has numerous applications in a wide variety of engineering fields (such as mechanical systems, power systems and Electromagnetic Compatibility). In the microwave domain, rational approximation is increasingly used to obtain black-box models of complex passive structures for model order reduction and efficient transient simulation. In this paper we will focus on a different application of pole-zero identification. We will review the different ways in which pole-zero identification can be applied to nonlinear circuit design (for power amplifier stability analysis and beyond). We will give a comprehensive view on recent approaches through illustrative application examples. Other uses of rational approximation techniques are beyond the scope of this paper.
... Next, the frequency-domain identification technique can be used to obtain a vector of transfer functions (14) from the vector of frequency responses in (13). Vector Fitting (VF) [25]- [27] will be used here as the identification algorithm. It has the benefit of allowing the identification of a vector of frequency responses with a same set of poles, which makes it especially well-suited for the modeling of MIMO linear systems. ...
... If the frequency band of interest is very wide and in the presence of transmission lines, the number of stable poles required to obtain an accurate rational model can be very high. For this reason, the use of Vector Fitting [25]- [27] is key to obtaining an accurate model of the HTF because the algorithm is particularly well suited to handle high-order transfer functions at high frequencies. ...
This paper proposes an efficient method for the calculation of the stabilization parameters in RF power amplifiers operating in periodic large-signal regimes. Stabilization is achieved by applying the principles of linear control theory for Periodic Linear Time-Varying (PLTV) systems. A numerical method is proposed to obtain the Harmonic Transfer Function that represents the system linearized around the large-signal steady state. Then, a feedback analysis is performed to calculate the closed-loop poles of the PLTV system. The proposed approach is demonstrated with two examples. Firstly, a three-stage amplifier that exhibits a low-frequency oscillation for increasing values of input power is correctly stabilized. Next, the stabilization of an unstable design that exhibits an odd-mode parametric oscillation is presented. The results of the proposed technique are compared to those obtained with the conventional parametric stability simulation. These examples serve to illustrate the capability and efficiency of the proposed approach.
... In practice, D-type bifurcations analyzed with pole-zero identification are commonly detected through the crossing of a pair of complex conjugate poles crossing at ±j2πfin (n = 1), instead of the crossing of a real pole (n = 0) [23], [24]. This is because common identification algorithms are more accurate in the detection of resonant peaks in the frequency response by means of pairs of complex conjugate poles than fitting the non-resonant magnitude and phase changes associated with real poles [25]. ...
... The commercial automatic pole-zero identification algorithm in [17], [30] starts from an even number of initial complex conjugate poles that is related to the number of resonances estimated in the frequency response. Then, these poles are reallocated via different frequency domain identification algorithms (least squares or Vector Fitting [25], [31], [32] for instance). A maximum value for the phase error along the analyzed frequency band is established as fitting goal condition. ...
Understanding the nature of potential instabilities is indispensable for the stabilization of power amplifiers. Pole-zero identification is one of the techniques that can be used to determine the stability of a design in large-signal operation. In this work, the possible presence of poles at the fundamental frequency linked to the long-term memory parameters of the transistor's model (self-heating and traps) is presented and discussed. The paper shows how their effect on the identified frequency responses around the fundamental frequency may compromise the stability analysis results and the assessment of stability margins. The low observability of the poles at the fundamental frequency highlights the importance of an accurate identification of real poles in low-frequency bands. A specific algorithm for the automatic frequency domain identification of non-resonant frequency responses and a procedure for detecting and reducing overfitting of real poles is proposed in this article. The benefits of the proposed methodology to correctly detect and analyze real poles at low frequencies is demonstrated through Monte-Carlo sensitivity analyses of two different amplifier designs.
... Alternatively, to overcome said assumptions while being black-box compatible, an analytical model can be obtained from the frequency response data via vector fitting algorithms at the expense of higher computational burden and model order [11], which might hinder their application to largescale systems. In addition, the required eigenvalue information to perform FD eigenvalue sensitivity analysis may not be available due to the computational burden for large-scale systems that comes from finding roots of its determinant from the entire system matrix. ...
This paper proposes a generalized passivity sensitivity analysis for power system stability studies. The method uncovers the most effective instability mitigation actions for both device-level and system-level investigations. The particular structure of the admittance and nodal models is exploited in the detailed derivation of the passivity sensitivity expressions. These proposed sensitivities are validated for different parameters at device-level and at system-level. Compared to previous stability and sensitivity methods, it does not require detailed system information, such as exact system eigenvalues, while it provides valuable information for a less conservative stable system design. In addition, we demonstrate how to utilize the proposed method through case studies with different converter controls and system-wide insights showing its general applicability.
... This paper presents a comprehensive approach to measure the impedance of passive components, primarily chokes in this paper, across a wide frequency range and to model the measured data using the vector fitting algorithm. Vector fitting is a numerical technique that efficiently generates rational function models, which accurately represent the complex impedance behavior over the measured frequency spectrum [1]- [3]. The outputs of this fitting algorithm are state-space models that can be translated into SPICE syntax in order to simulate them in AC-as well as in transient simulations. ...
This paper presents an enhanced method to model impedance data of inductive components for accurate simulation in transient SPICE simulations by utilizing vector fitting algorithms. These models are extended by a core loss model based on the improved generalized Steinmetz equation (iGSE). Through practical measurements, we demonstrate that the resulting models closely replicate the behavior of real inductors. These results present a more precise approach to simulating inductive components, particularly in scenarios where losses play a crucial role.
... The estimation of additive model parametrizations has previously been considered in a vector fitting procedure [16], which considers fitting first-order poles and residues to the given data, as well as in the estimation of nonlinear finite impulse responses and generalized Hammerstein models [17], [18]. Recent results in additive system identification include the direct continuous-time identification method introduced in [19], which is based on the simplified refined instrumental variable method (SRIVC) [20], as well as a block coordinate descent approach with variants for both offline and online parameter estimation [21], [22]. ...
Multivariable parametric models are critical for designing, controlling, and optimizing the performance of engineered systems. The main objective of this paper is to develop a parametric identification strategy that delivers accurate and physically relevant models of multivariable systems using time-domain data. The introduced approach adopts an additive model structure, offering a parsimonious and interpretable representation of many physical systems, and employs a refined instrumental variable-based estimation algorithm. The developed identification method enables the estimation of parametric continuous-time additive models and is applicable to both open and closed-loop controlled systems. The performance of the estimator is demonstrated through numerical simulations and experimentally validated on a flexible beam system.
... Apart from the Cauchy methods, other types of approaches, such as vector-fitting methods (see e.g., [17,27,29,60]) and optimization-based approaches (e.g., [5,6,30,37]) are proposed in the literature (see [52] for more details and references). As an efficient variant of the vector-fitting method, [8,9] develop RKFIT based on the rational Krylov subspace, for both scalar-valued and matrix-valued functions. ...
In the duplexer filtering of the three-port network microwave application, three coupled rational functions need to be reconstructed by fitting sampled responses measured at a set of frequencies efficiently and effectively. Traditionally, the Cauchy method and its modifications are common approaches that find the underlying rational functions by solving the linearized system. However, with increasing degrees of polynomials, Cauchy methods encounter difficulties when applied to modern duplexer hardware, particularly due to the ill-conditioned Vandermonde matrices involved and a large variation of order of magnitude (dB) in the noise-contaminated and lossy responses. In this paper, by relying on recent achievements from the rational approximation literature using barycentric representations, namely the “adaptive Antoulas-Anderson” (AAA) algorithm and Lawson’s iteration for scalar-valued as well as matrix-valued functions, we propose a vector-valued AAA-type approach (v-AAA-Lawson) for addressing the duplexer filtering problem, which is also applicable for the general matrix-valued rational approximations. Numerical experiments demonstrate that the new method is able to improve significantly the performance of the Cauchy methods both in accuracy and robustness.
... No Passo 2 utilizou-se a conhecida ferramenta Vector Fitting (GUSTAVSEN; SEMLYEN, 1999;GUSTAVSEN, 2006;DESCHRIJVER et al., 2008) que possibilita a estimação de sistemas através de funções racionais. Assim, o VF foi empregado para se estimar uma função de transferência que representasse a impedância harmônica do aterramento, possibilitando os cálculos a serem realizados no próximo passo. ...
O correto dimensionamento do sistema de aterramento garante confiabilidade e segurança no funcionamento das linhas de transmissão sob condições de descargas atmosféricas. Quanto a isso, ao longo do tempo, esforços têm sido realizados buscando-se novas técnicas para aprimoramento da resposta transitória dos aterramentos, principalmente para solos de alta resistividade. Nesse sentido, uma das técnicas há muito conhecida é a utilização de materiais de baixa resistividade ao redor dos eletrodos de aterramentos, tais como a bentonita e o concreto. No entanto, apesar do tempo decorrido, poucos trabalhos na literatura realizam um estudo aprofundado sobre a caracterização da resposta transitória
de aterramentos encapsulados por concreto, o que teria especial importância na avaliação dos mesmos para uso em aterramentos de torres de linhas de transmissão. Diante do exposto, e visando contribuir com o estado da arte, neste trabalho, a resposta transitória de eletrodos de aterramento encapsulados por concreto é investigada. Para isso, realizaram-se simulações através do método dos elementos finitos, considerando uma rigorosa modelagem eletromagnética. Foram analisados aterramentos encapsulados por concreto do tipo eletrodo vertical (haste) com 3 e 30 m, assim como de eletrodo horizontal (contrapeso) com 15, 30, 45 e 60 m. Consideraram-se excitações de correntes impulsivas de primeira descarga e descarga subsequente. O efeito da dependência da frequência dos parâmetros do solo foi considerado. Além disso, solos com resistividades baixa, média e alta (300 Ω.m a 4000 Ω.m) foram avaliados. Diferentes comprimentos de revestimento de concreto foram levados em consideração, a fim de verificar situações de encapsulamento parcial ou total. Para todos esses casos, a impedância harmônica (de 60 Hz a 4 MHz) foi simulada, sendo calculadas a respectiva elevação de potencial no aterramento e a impedância impulsiva. Ao todo, foram avaliados 150 aterramentos diferentes, sendo cada um submetido a duas descargas, totalizando 300 respostas impulsivas. Os resultados indicam que a concretagem do eletrodo,
em solos de média e alta resistividade produziu reduções médias na impedância impulsiva de até 40%, para a haste vertical com 3 m, considerando primeira descarga e subsequente. Já para os eletrodos horizontais, a redução na impedância impulsiva, em solos de média e alta resistividade, foi de até 22%, para primeira descarga, enquanto que para descarga subsequente os resultados variaram consideravelmente entre 10% a 26%, a depender do comprimento do eletrodo e da resistividade do solo. Além disso, constatou-se que o uso de concreto no aterramento reduz o comprimento efetivo do eletrodo.
... In contrast, the estimation of additive model parametrizations has primarily been explored in single-input singleoutput (SISO) approaches. One such method is vector fitting (Semlyen, 1999), which considers fitting first-order pole models. Recent advancements in additive system identification include the direct continuous-time identification method introduced in González et al. (2024), which is based on the simplified refined instrumental variable method (SRIVC) Young and Jakeman (1980), as well as a block coordinate descent approach with variants for both offline and online parameter estimation González et al. (2023); Classens et al. (2024). ...
Multivariable parametric models are essential for optimizing the performance of high-tech systems. The main objective of this paper is to develop an identification strategy that provides accurate parametric models for complex multivariable systems. To achieve this, an additive model structure is adopted, offering advantages over traditional black-box model structures when considering physical systems. The introduced method minimizes a weighted least-squares criterion and uses an iterative linear regression algorithm to solve the estimation problem, achieving local optimality upon convergence. Experimental validation is conducted on a prototype wafer-stage system, featuring a large number of spatially distributed actuators and sensors and exhibiting complex flexible dynamic behavior, to evaluate performance and demonstrate the effectiveness of the proposed method.
... Note that applying a direct inverse fast Fourier transform may result in non-causal discrete-time models especially when dealing with band-limited scattering parameters [4]. A similar issue arose in microwave engineering around 20 years ago, but it was largely addressed by the introduction of the vector fitting (VF) modeling method [5], along with other advancements that enhanced its performance [6,7]. ...
For accurate circuit-level simulations, passive photonic integrated circuits (PICs) must be modeled in the time domain, using their frequency responses. The complex vector fitting (CVF) method has been established as a reliable technique for constructing time-domain state-space models of passive PICs. This paper introduces a more advanced method, called the baseband Loewner matrix approach, which offers enhanced efficiency compared to CVF. The resulting models are significantly more compact while maintaining accuracy. The advantages of the proposed method are demonstrated through three passive PIC case studies.
... The frequency responses, shown in Figures 8 and 9, are fitted using the method presented in [23]. Fitting the frequency response simplifies the simulation of transmission line transients. ...
Insulation coordination studies are of great importance in power grid reliability. In this paper, a new method is proposed for modeling trapped charge sources (TCS) in switching transient studies. The TCS is used to take into account the voltage stored in line capacitors during reclosing operation after fault occurrence. The proposed model is designed based on the active filter concept, thus overcoming the limitations of conventional TCS for simulating transient states in EMTP/ATPDraw. Given the natural frequencies of a transmission line to which the proposed TCS (PTCS) is connected, it injects the appropriate frequencies and eliminates voltage oscillations which limit the use of TCS. To verify the efficiency of the PTCS, it is implemented in a real system with a shunt reactor, and the results are then compared with field measurements. A comparison of the results shows that the PTCS eliminates the voltage oscillations in the simulation before closing and provides a smooth voltage with the desired amplitude. Using the proposed model, the maximum line switching overvoltage is correctly calculated; this, in turn, results in a more accurate transmission line insulation design, which is technically and economically beneficial.
... The measured worst case RL and peak insertion loss in the passband are 18 and 2 dB, respectively, for Filter I and 11 and 2.1 dB for Filter II. The unloaded Q-factor measured for both filters is approximately 550 extracted using vector fitting method [43]. ...
This article introduces a novel frequency-variant coupling structure with high adjustability for waveguide filters. Although this structure is proposed for rectangular cavities, the same principle can be applied to other cavity geometries. The proposed structure is compatible with standard micromachining technologies. Using this coupling structure, two inline N th-order waveguide filters with N -1 and N +1 transmission zeros (TZs) are developed. A fourth-order bandpass filter (BPF) with three TZs at 245, 285, and 288 GHz is designed, operating at a center frequency of 270 GHz with a 2% fractional bandwidth. In addition, a second-order BPF with a center frequency of 280 GHz and a 0.71% fractional bandwidth, utilizing three TZs at 230, 300, and 308 GHz, is also developed. Prototypes of both filters are fabricated using silicon micromachining. The measured insertion loss and return loss (RL) in the passband are 2 dB and better than 18 dB, respectively, for the fourth-order filter and 2.1 dB and better than 11 dB for the second-order filter, respectively.
... for which extensive discussions have been made traditionally in e.g., [2,11,16,22,23,40,53,55,56,60,65,66,67] and recently in e.g., [4,15,17,21,30,43,44,69,71]. ...
In this paper, we propose a novel dual-based Lawson's method, termed b-d-Lawson, designed for addressing the rational minimax approximation under specific interpolation conditions. The b-d-Lawson approach incorporates two pivotal components that have been recently gained prominence in the realm of the rational approximations: the barycentric representation of the rational function and the dual framework for tackling minimax approximation challenges. The employment of barycentric formulae enables a streamlined parameterization of the rational function, ensuring natural satisfaction of interpolation conditions while mitigating numerical instability typically associated with Vandermonde basis matrices when monomial bases are utilized. This enhances both the accuracy and computational stability of the method. To address the bi-level min-max structure, the dual framework effectively transforms the challenge into a max-min dual problem, thereby facilitating the efficient application of Lawson's iteration. The integration of this dual perspective is crucial for optimizing the approximation process. We will discuss several applications of interpolation-constrained rational minimax approximation and illustrate numerical results to evaluate the performance of the b-d-Lawson method.
... Now, taking the frequency characteristics obtained from the frequency scan/admittance prediction method, vector fitting may be used to create an approximate model that exhibits the input-output frequency response very close to the impedance or admittance characteristics obtained from the admittance prediction method and frequency scans. Here, a vector fitting package described in [14], [15] and [16] is used -different implementations of this package are available for MATLAB and Python languages. For this project, the MATLAB package is used. ...
With the large growth in the share of inverter-based resources (IBRs) in the Australian power system, one aspect important for ensuring stable operation of the Australian power system is evaluating the system stability and stability margin in the small signal domain, considering the IBRs connected to the system. In the research roadmap submitted to CSIRO, ‘stability margin evaluation’ and ‘small signal stability screening methods’ were identified as two critical research topics; this stage of the project aims to continue to address these two research topics.
The full report is available at: https://www.csiro.au/en/research/technology-space/energy/g-pst-research-roadmap/stage3-reports
... will correspond to the FTF. In order to obtain an analytic transfer function, we apply the Vector Fitting algorithm [51,52]. The experimental FTF and the transfer function of the state-space model are shown in Fig. 8. ...
We create and describe an inhomogeneous Helmholtz equation solver, helmholtz-x, written in an open-source framework. The mesh is generated with Gmsh and the solver uses DOLFINx and UFL from FEniCSx. The performance, validity, stability and extensibility of the solver are demonstrated through several examples of thermoacoustic instability, from the one-dimensional Rijke tube to the three-dimensional MICCA combustor. The implementation of Bloch-type boundary conditions is explained and tested. The adjoint capability of the solver is also shown, and used to obtain derivatives of the eigenvalue with respect to shape parameters. This is exploited to find shape changes that reduce the thermoacoustic growth rate.
... After model reduction, the vector fitting and passivity enforcement algorithms [32]- [36] are applied to the admittance matrix. In this way, the obtained model can be implemented in an EMT software environment. ...
This paper presents a comprehensive model for power transformers, by considering eddy current losses in both the core and conductors. This is achieved through a meticulous analytical approach that ensures high fidelity in representing the transformer's electromagnetic properties. The consideration of magnetic flux effects on inductance and resistance values significantly enhances the model's accuracy and validity. Traditional analytical methods often resort to simplified approaches due to the complexity of these calculations. The paper addresses these limitations by evaluating the eddy current losses in the core and conductors, and by providing a detailed understanding of each component's impact on transformer behavior. Furthermore, by considering the core and conductor effects on the magnetic field distribution, the model handles a wide range of frequencies, making it suitable for conducting comprehensive transient analysis. To validate the model, comparisons with the finite element method and empirical measurements are conducted. Additionally, a reduced-order transformer model is developed using admittance matrix reduction. This approach focuses on the nodes of interest, effectively eliminating not-observed nodes and reducing computational complexity without compromising accuracy. In this way, voltages at specific points of interest are computed efficiently, maintaining the accuracy of the original model.
... This approach is limited as, in general, the transformation matrix is frequency dependent (Noda et al., 1996;Gustavsen et al., 1995). Gustavsen and Semlyen (1999) proposed the pole relocation algorithm known as Vector Fitting (VF) allowing an efficient and accurate representation of frequencydependent functions regardless of their dimensions. The usage of VF led to the development of frequencydependent line models in both modal domain and phase coordinates (Gustavsen and Semlyen, 1998a,c,b). ...
This paper compares two different methods to obtain rational approximations for the Universal Line Model (ULM) formulation. These approximations allow the representation of frequency-dependent transmission lines in the time-domain. The two fitting methods evaluated in this work are the pole relocation algorithm, known as Vector Fitting (VF), and the Frequency Partitioned Fitting (FpF). Both methods have their fit result of the admittance matrix and the propagation matrix in frequency-domain compared. The time-domain results for both approximations are obtained by simulating a field test in an actual 500 kV double-circuit transmission line in Japan. The simulations were ran on PSCAD and XTAP software packages which use, respectively, VF and FpF. The fit results in the frequency-domain shows that the two approximations are similar and present small relative errors, all of them in an order of magnitude of 0.1 %. Similar results were obtained for time-domain results, where both methods obtain similar representations and the simulation results are very close. Simulation results are also compared with measurement data of the field test. Although some differences are observed between simulations and the measured data, their general behavior are close.
In this paper, we introduce a novel wavelet‐based algorithm for reconstructing time‐domain impulse responses from band‐limited scattering parameters (frequency‐domain data) with a particular focus on ship hull applications. We establish the algorithm and demonstrate its convergence, as well as its efficiency for a class of functions that can be expanded as exponential functions. We provide simulation results to validate our numerical results.
This paper presents a comprehensive assessment of the accuracy of high-frequency (HF) earth meters in measuring the tower-footing ground resistance of transmission line structures, combining simulation and experimental results. The findings demonstrate that HF earth meters reliably estimate the harmonic grounding impedance (R25kHz) at their operating frequency, typically 25 kHz, for a wide range of soil resistivities and typical span lengths. For the analyzed tower geometries, the simulations indicate that accurate measurements are obtained for adjacent span lengths of approximately 300 m and 400 m, corresponding to configurations with one and two shield wires, respectively. Acceptable errors below 10% are observed for span lengths exceeding 200 m and 300 m under the same conditions. While the measured R25kHz does not directly represent the resistance at the industrial frequency, it provides a meaningful measure of the grounding system’s impedance, enabling condition monitoring and the evaluation of seasonal or event-related impacts, such as damage after outages. Furthermore, the industrial frequency resistance can be estimated through an inversion process using an electromagnetic model and knowing the geometry of the grounding electrodes. Overall, the results suggest that HF earth meters, when correctly applied with the fall-of-potential method, offer a reliable means to assess the grounding response of high-voltage transmission line structures in most practical scenarios.
In this article, we discuss the behavioral modeling of the power amplifier (PA) for system-level simulations through its most advanced approach, named TPM model, based on a simplification of the Volterra series following the method of separation of the low and high frequency memory effects present in PA. The model, relying on frequency domain CW characterization of the PA, shows a limitation when applied to high-power radar applications, for which this article investigates an alternate solution based on time-domain pulsed RF characterization.
Stability of multi-vendor power electronicsdominated grids requires proper specifications for converters interoperability, e.g. tuning boundaries for the converters’ control loops to be respected by all vendors. The confidentiality and variety of implementation of the converters’ control system by different vendors and the nonlinear dynamics of power converters make this task arduous. This paper formulates this research question as a robust stability problem, embedding frequency-dependent uncertainties in the converter control systems and analytically quantifying the amount of uncertainty able to warranty the power grid stability. Grid following converters are considered and modeled with a nonlinear grey-box approach, in contrast with conventional small-signal black-box models which suffer from inaccuracy when their operating point deviates from the nominal one. This paper also demonstrates both analytically and with Hardware-In-the-Loop (HIL) experiments that different converters operating points (e.g. active power injections, ac bus voltages) radically affect their stability, empathizing the necessity of nonlinear converters models to realize accurate power grid stability analyses.
We present a novel modular implementation of the discontinuous Galerkin time-domain (DGTD) method to effectively address electromagnetic problems involving general dielectric dispersive media modeled through vector fitting. This approach includes an extended dispersive perfectly matched layer to directly truncate dispersive materials, allowing for the modeling of open domains. The proposed modular and concise DGTD implementation, based on the complex conjugated pole residue model, offers flexibility and simplifies the handling of complex medium problems. We apply the formulation to both two-dimensional and three-dimensional canonical scattering problems, demonstrating good agreement with their respective analytical solutions.
In this study, we present a purely data-driven method that uses the Loewner framework along with nonlinear optimization techniques to infer quadratic dynamical systems with affine control that admit Volterra series (VS) representations from input-output time-domain measurements. The proposed method extensively employs optimization tools for interpolating the symmetric generalized frequency response functions (GFRFs) derived in the VS framework. The GFRF estimations are obtained from the Fourier spectrum (phase and amplitude) of the quasi-steady-state system response under harmonic excitation. Appropriate treatment of these measurements under the developed framework allows the identification of low-order in-state dimension quadratic state-space models with nontrivial stable equilibria, such as in the Lorenz ’63 forced system. We thus can achieve low-order global model identification for certain classes of systems that can bifurcate to multiple equilibria after collecting measurements solely from a local stable operational regime. The advanced framework rigorously tests the identification and reduction features across various examples with increasing dimensions and varying complexities. It effectively addresses the inference of a reduced model from measurements of a viscous flow, assuming adherence to the Burgers’ model with Robin boundary conditions. These results provide new prospects for developing low-order predictive models based on input-output data suitable for control.
Engineered subsurface structures inspired by phononic materials have shown favorable capabilities in flow control. Current efforts rely on tuning a structural resonance to the frequency of an unstable fluid mode, causing the deformations of the interfacing solid to destructively interfere with a wave-like flow instability. Although promising, the technique is most effective at targeting a single frequency mode with high-Q resonance. Additionally, several studies have shown the reduction in the perturbation energy to be spatially localized to the flow region above the subsurface strip, with a severely worsening effect downstream. Motivated by a desire to overcome these limitations, we present a different approach to methodically extract an undesirable pressure disturbance from a fluid column in a manner that capitalizes on the broad frequency range of a phononic bandgap, with the goal of permanently confining the perturbation energy within the subsurface structure. An acoustic diode (AD), i.e., a unidirectional transmitter of mechanical energy, comprised of a bi-layered phononic crystal and an auxiliary medium, interacts with a fluid cavity and provides a terminal energy sink. Two distinct ADs are presented that demonstrate active (time-variant) and passive (strain-dependent) paths to concept realization. The AD’s performance is described in terms of the time-transient energy distribution in the fluid and the interacting structure, as well as spatial wave profiles at critical time instants. The results show the system’s ability to achieve robust extraction of undesirable fluid oscillations with minimal residual energy. The concept is then tested in a fully developed plane channel flow with a superimposed perturbation, demonstrating the sustained nature of the subsurface AD’s energy trapping mechanism in addition to its ability to induce downstream attenuation.
Neuro-transfer function (neuro-TF) approaches are useful instruments for electromagnetic (EM) parameterized modeling of microwave passive components. A vector fitting technique is commonly used to obtain the TF coefficients from the EM responses utilized in conventional neuro-TF approaches. However, the use of this method can introduce an “order-changing” issue, where the TF orders vary across different regions of the design parameter space. Recently, an alternative neuro-TF method has been proposed, which employs a model-order reduction (MOR)-based TF. Instead of utilizing a vector fitting process, the MOR process acquires the poles/zeros (rational coefficients) during full-wave EM simulations. The MOR technique ensures a consistent TF order across different geometric parameter values. This article provides an extensive overview of the most recent developments in MOR-based neuro-TF techniques. In particular, the approach is examined in two formats: the rational format and the pole/zero format. The former technique provides a solution to the problem of “order-changing” in conventional vector fitting-based neuro-TF, but it still has problems with poles/zeros being mismatched. Thus, the latter format, an EM sensitivity-based approach, has been developed to guarantee the continuity of the poles/zeros across different geometric training samples to address the mismatch problem.
A widely applicable, general methodology for estimation of transfer function parameters from frequency response data is presented. The procedure is based on the solution of a linear least squares problem by the singular value decomposition (SVD). The condition of the problem is discussed and approaches referred to as shifting and scaling are introduced in order to reduce the condition number. To extend the application to practical cases with measurement errors and/or a large number of poles, a partitioned estimation method with Gauss-Seidel iterations is developed. An iterative improvement process with constraints on the poles is applied to increase the accuracy and to avoid the possibility of obtaining unstable poles. The application of the suggested method of estimation to the representation of transformers is presented with practical examples. Either transfer function or state equation representation can be obtained for transformers described by their terminal frequency responses.< >
Summary form only given, as follows.
The parameters of transmission lines with ground return are highly dependent on the frequency. Accurate modelling of this frequency dependence over the entire frequency range of the signals is of essential importance for the correct simulation of electromagnetic transient conditions. Closed mathematical solutions of the frequency-dependent line equations in the time domain are very difficult. Numerical approximation techniques are thus required for practical solutions. The oscillatory nature of the problem, however, makes ordinary numerical techniques very susceptible to instability and to accuracy errors. The, methods presented in this paper are aimed to overcome these numerical difficulties.
The paper presents a new approach to the calculation of transients on transmission lines with frequency-dependent parameters. Its purpose is to obtain significant computer-time savings by avoiding convolutions. This is achieved by approximating all line and ground distortions and also time variable characteristic admittances by exponential functions, i.e. solutions of linear differential equations. The method produces a simple Norton-type line equivalent which permits its incorporation into an existing system representation like the B.P.A. program for the calculation of transients. The program has been tested on systems of different degrees of complexity and proved to be superior, in terms of speed and accuracy, to other advanced methods.
This paper introduces two highly accurate transmission line
models. In the first one, particularly suitable for overhead lines, the
matrix transfer functions for both characteristic admittance and
propagation are fitted directly in the phase domain using the method of
vector fitting by optimal scaling. In the second model we use for
propagation a modal decomposition with a constant transformation matrix
and an optional phase domain correction term. Both models are
computationally highly efficient due to their time domain realizations
based an vector fitting
This paper introduces a fast and robust method for rational
fitting of frequency domain responses, well suited for both scalar and
vector transfer functions. Application of the new method results in
increased computational efficiency for transmission line models using
modal decomposition with frequency dependent transformation matrices.
This is due to the fact that the method allows the fitted elements of
each eigenvector to share the same set of poles, and that accurate
fitting can be achieved with a relatively low number of poles
A model to simulate the high-frequency behavior of a power transformer is presented. This model is based on the frequency characteristics of the transformer admittance matrix between its terminals over a given range of frequencies. The transformer admittance characteristics can be obtained from measurements or from detailed internal models based on the physical layout of the transformer. The elements of the nodal admittance matrix are approximated with rational functions consisting of real as well as complex conjugate poles and zeroes. These approximations are realized in the form of an RLC network in a format suitable for direct use with the ElectroMagnetics Transient Program (EMTP). The high-frequency transformer model can be used as a stand-alone linear model or as an add-on module of a more comprehensive model where iron core nonlinearities are represented in detail.< >
Practical Transfer Function Estimation and its Application to Transformers Transmission Lines in Electromagnetic Transient Simulations
- A O Soysal
- A Semlyen
A.O. Soysal and A. Semlyen, Practical Transfer Function Estimation and its Application to Transformers , JEEE Trans. PWRD, vol. 8, no. 3, July 1993, pp. 1627-1637. J.R. Transmission Lines in Electromagnetic Transient Simulations , IEEE Trans. PAS, vol. 101, no. 1, January 1982, pp. 147-157
Henriksen ( E n ) for providing the frequency responses for the transformer and the complex network equivalent, respectively
- T Dr
Dr. T. Henriksen ( E n ) for providing the frequency responses for
the transformer and the complex network equivalent, respectively.
11 REFERENCES