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Macromodeling of Distributed Networks From Frequency-Domain Data Using the Loewner Matrix Approach
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Recently, Loewner matrix (LM)-based methods were introduced for generating time-domain macromodels based on frequency-domain measured parameters. These methods were shown to be very efficient and accurate for lumped systems with a large number of ports; however, they were not suitable for distributed transmission-line networks. In this paper, an LM-based approach is proposed for modeling distributed networks. The new method was shown to be efficient and accurate for large-scale distributed networks.
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... The frequency samples do not need to be equally spaced; it is recommended to select more samples in regions where the frequency response is more dynamic. Note that, considering that the baseband scattering parameters are non Hermitian symmetric, they are divided in very differently way compared to the conventional scattering parameters [19]. Then, to construct the baseband LM model, the Loewner matrix for S(s) and sS(s) must be formulated. ...
... In the existing LM method [19], which focuses on modeling physical systems, both the original data set and its complex conjugate at negative frequencies are utilized to ensure that the final model satisfies Hermitian symmetry. This practice is essential because, for real-valued time-domain signals, the frequency-domain representation must exhibit Hermitian symmetry. ...
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.
... Curve-fitting methods used in frequency domain admittance characterization [43], [44], [45] can have potential applications in fast frequency sweep of s-parameters data obtained using full-wave EM simulation. The commonly used curve-fitting techniques in power-system applications are the vector-fitting [14], matrix pencil method [12], [13], and the Loewner matrix method [46]. Among these techniques, the Loewner matrix method can be most suitable for multiport s-matrix characterization over a wide band of frequency. ...
This paper employs a fully adaptive and semi-adaptive frequency sweep algorithm using the Loewner matrix-based state model for the electromagnetic simulation. The proposed algorithms use two Loewner matrix models with different or the same orders with small frequency perturbation for adaptive frequency selection. The error between the two models is calculated in each iteration, and the next frequency points are selected to minimize maximum error. With the help of memory, the algorithm terminates when the error between the model and the simulation result is reached within the specified error tolerance. In the fully adaptive frequency sweep algorithm, the method starts with the minimum and maximum frequency of simulation. In the semi-adaptive algorithm, a novel approach has been proposed to determine the initial number of frequency points necessary for system interpolation based on the electrical size of the structure. The proposed algorithms have been compared with the Stoer-Bulirsch algorithm and Pradovera's minimal sampling algorithm for electromagnetic simulation. Four examples are presented using MATLAB R2024b. The results show that the proposed methods offer better performance in terms of speed, accuracy and the requirement of the minimum number of frequency samples. The proposed method shows remarkable consistency with full-wave simulation data, and the algorithm can be effectively applicable to electromagnetic simulations.
... Most of linear models and identification are based on a representation that can describe the system dynamics in the whole frequency range (e.g., the state space representation or transfer function input-output models). For a selectivefrequency range, Loewner interpolation was commonly used in black-box modeling of large MIMO microwave structures that show the capability for rational interpolation of frequency data [97,98]. In 2015, a tangential interpolation framework based on Loewner matrix pencil was first used in power systems for modeling frequency-dependent network equivalents that can describe electromagnetic transients (EMT) [39,99]. ...
Modern power grids are fast evolving with the increasing volatile renewable generation, distributed energy resources (DERs) and time-varying operating conditions. The DERs include rooftop photovoltaic (PV), small wind turbines, energy storages, flexible loads, electric vehicles (EVs), etc. The grid control is confronted with low inertia, uncertainty and nonlinearity that challenge the operation security, efficacy and efficiency. The ongoing digitization of power grids provides opportunities to address the challenges with data-driven and control. This paper provides a comprehensive review of emerging data-driven dynamical modeling and control methods and their various applications in power grid. Future trends are also discussed based on advances in data-driven control.
... There exist other attractive curve fitting methods, such as Matrix Pencil [56], Loewner Matrix 363 [57], and Polynomial Frequency Partitioning method (Noda's method) [4]. The Matrix Pencil and 364 ...
This paper presents a qualitative overview of the main developments carried out by the Brazilian Electrical Energy Research Center and the Brazilian Power System Operator to provide tools to assembly Frequency Dependent Network Equivalents. These equivalents have fundamental importance in the electromagnetic transient studies in the Brazilian Interconnected Power System. Applications for these studies in offline and real-time simulation environments are also presented, using the ATP and the HYPERSIM, respectively.
... To draw a real model from a complex from, a similarity transformation is performed over matrices (7) -(10) [43], so the Loewner matrices in real form becomes [L , L , F , W ] represented by: ...
This work examines two strategies for enhancing the rational approximation of Frequency-Dependent Network Equivalents (FDNE) using an 8-port FDNE featuring a frequency response marked by numerous peaks and valleys. Firstly, we employ Complex Vector Fitting (CVF), an alternative to the Vector Fitting (VF). CVF eliminates the constraint of complex conjugate pairs and was originally conceived for modeling baseband equivalents through scattering parameters. The implications of CVF for admittance (or impedance) matrix synthesis have not yet been previously reported in specialized literature. To enhance code performance and remove dependence on commercial software such as MATLAB®, VF and CVF were implemented in the C-language, utilizing a low-level linear algebra package and exploiting parallelism. We evaluated performance by varying the model order, number of ports, and frequency samples. The results confirm the feasibility of our approach, prompting a more in-depth exploration of the potential benefits regarding FDNE realization.
This paper presents SRPOEE which is a novel algorithm and automated Python-based open-source software (OSS) designed to synthesize the reduced-order passivity-enforced equivalent circuits of networks represented by scattering parameters (S-parameters). The proposed algorithm comprises three main modules Full Synthesis, Passive MOR, and Reduced Synthesis, each incorporating a combination of established and novel techniques. The Full synthesis module calculates a full-order passive fitted system in state-space form, via fitting the given S-parameters using the vector fitting (VF) method, and assesses/enforces passivity through the singularity half-size test matrix method. Subsequently, it synthesizes a full-order equivalent circuit, comprised of R/L/C circuit elements, based on the calculated full-order fitted system. The Passive MOR module represents the synthesized full-order equivalent circuits using modified nodal analysis (MNA) matrices, and subsequently reduces the order of these matrices through a model order reduction (MOR) technique known as the Block SAPOR technique. Finally, the Reduced Synthesis module constructs a reduced-order equivalent circuit composed of R/L/C circuit elements based on the reduced-order MNA matrices generated in the previous module. In addition to synthesizing the reduced- and full-order equivalent circuits, SROPEE also generates SPICE-compatible netlists for both the full- and reduced-order equivalent circuits, enabling further circuit analysis in SPICE circuit solvers. We evaluate SROPEE by comparing the network parameters of the synthesized reduced-order equivalent circuit with the original network parameters for a 4-port optical network operating in frequency range of 175 - 215 THz. Furthermore, we compare the computational costs of circuit analysis for synthesized full-and reduced-order equivalent circuits in terms of network order, matrix dimensions, compute-time, accuracy, memory utilization, and passivity considerations. The Python-based OSS may be deployed as a stand-alone tool, or integrated into existing process technology flows and CAD tools to expedite design automation.
With the rapid developments in very large-scale integration (VLSI)
technology, design and computer-aided design (CAD) techniques, at both
the chip and package level, the operating frequencies are fast reaching
the vicinity of gigahertz and switching times are getting to the
subnanosecond levels. The ever increasing quest for high-speed
applications is placing higher demands on interconnect performance and
highlighted the previously negligible effects of interconnects such as
ringing, signal delay, distortion, reflections, and crosstalk. In this
review paper various high-speed interconnect effects are briefly
discussed. In addition, recent advances in transmission line
macromodeling techniques are presented. Also, simulation of high-speed
interconnects using model-reduction-based algorithms is discussed in
detail
Recently, Loewner Matrix (LM) based methods were introduced for generating time-domain macromodels based on frequency domain measured parameters. These methods were shown to be very efficient and accurate for systems with a very large number of ports, however they were not suitable for distributed transmission line networks. In this paper, an LM based approach is proposed for modeling distributed networks. The new method was shown to be efficient and accurate for large-scale distributed networks.
We consider the stabilization problem for large-scale linear descriptor systems in continuous- and discrete-time. We suggest
a partial stabilization algorithm which preserves stable poles of the system while the unstable ones are moved to the left
half plane using state feedback. Our algorithm involves the matrix pencil disk function method to separate the finite from
the infinite generalized eigenvalues and the stable from the unstable eigenvalues. In order to stabilize the unstable poles,
either the generalized Bass algorithm or an algebraic Bernoulli equation can be used. Some numerical examples demonstrate
the behavior of our algorithm.
Large scale dynamical systems are a common framework for the modeling and control of many complex phenomena of scientific interest and industrial value, with examples of diverse origin that include signal propagation and interference in electric circuits, storm surge prediction before an advancing hurricane, vibration suppression in large structures, temperature control in various media, neurotransmission in the nervous system, and behavior of micro-electro- mechanical systems. Direct numerical simulation of underlying mathematical models is one of few available means for accurate prediction and control of these complex phenomena. The need for ever greater accuracy compels inclusion of greater detail in the model and potential coupling to other complex systems leading inevitably to very large-scale and complex dynamical models. Simulations in such large-scale settings can make untenable demands on computational resources and efficient model utilization becomes necessary. Model reduction is one response to this challenge, wherein one seeks a simpler (typically lower order) model that nearly replicates the behavior of the original model. When high fidelity is achieved with a reduced-order model, it can then be used reliably as an efficient surrogate to the original, perhaps replacing it as a component in larger simulations or in allied contexts such as development of simpler, faster controllers suitable for real time applications.
In today's technological world, physical and artificial processes are mainly described by mathematical models, which can be used for simulation or control. These processes are dynamical systems, as their future behavior depends on their past evolution. The weather and very large scale integration (VLSI) circuits are examples, the former physical and the latter artificial. In simulation (control) one seeks to predict (modify) the system behavior; however, simulation of the full model is often not feasible, necessitating simplification of it. Due to limited computational, accuracy, and storage capabilities, system approximation—the development of simplified models that capture the main features of the original dynamical systems—evolved. Simplified models are used in place of original complex models and result in simulation (control) with reduced computational complexity. This book deals with what may be called the curse of complexity, by addressing the approximation of dynamical systems described by a finite set of differential or difference equations together with a finite set of algebraic equations. Our goal is to present approximation methods related to the singular value decomposition (SVD), to Krylov or moment matching methods, and to combinations thereof, referred to as SVD-Krylov methods.
Part I addresses the above in more detail. Part II is devoted to a review of the necessary mathematical and system theoretic prerequisites. In particular, norms of vectors and (finite) matrices are introduced in Chapter 3, together with a detailed discussion of the SVD of matrices. The approximation problem in the induced 2-norm and its solution given by the Schmidt—Eckart—Young—Mirsky theorem are tackled next. This result is generalized to linear dynamical systems in Chapter 8, which covers Hankel-norm approximation. Elements of numerical linear algebra are also presented in Chapter 3. Chapter 4 presents some basic concepts from linear system theory. Its first section discusses the external description of linear systems in terms of convolution integrals or convolution sums. The section following treats the internal description of linear systems. This is a representation in terms of first-order ordinary differential or difference equations, depending on whether we are dealing with continuous- or discrete-time systems. The associated structural concepts of reachability and observability are analyzed. Gramians, which are important tools for system approximation, are introduced in this chapter and their properties are explored. The last section of Chapter 4 is concerned with the relationship between internal and external descriptions, which is known as the realization problem. Finally, aspects of the more general problem of rational interpolation are displayed.
In this paper we present a novel way of constructing generalized state space representations [E,A,B,C,D] of interpolants matching tangential interpolation data. The Loewner and shifted Loewner matrices are the key tools in this approach.
The so-called Vector Fitting algorithm computes a rational approximation of the transfer matrix for linear multiport elements, starting from input-output data. This rational form is compatible with standard circuit solvers, thus enabling circuit-based Signal Integrity simulation flows. In this work, we document the first parallel Vector Fitting implementation for shared memory architectures. The results show close-to-ideal speedup factors and demonstrate the feasibility of almost real-time generation of macromodels for highly complex interconnect structures, with up to 160 ports.
The Loewner matrix framework can identify the underlying system from given noise-free measurements either in the frequency,
or in the time domain[1, 2]. This paper provides an analysis of the effects of noise on the performance of the SVD implementation
of the Loewner matrix framework for different noise levels and proposes an improved approach which is able to identify an
approximation of the original system even for high levels of noise. Moreover, for frequency domain measurements, our framework
can handle systems with a large number of inputs and outputs while requiring small computational time and storage.