Fig 4 - uploaded by Muhammad Kabir
Content may be subject to copyright.
Source publication
Recently, Loewner Matrix (LM) method was introduced for generating time-domain macromodels based on frequency-domain measured parameters. For the case of lumped systems the order is clearly defined and is detected easily by the LM algorithm. However, for the case of distributed systems with a theoretically infinite number of poles, an optimum order...
Context in source publication
Context 1
... the superscript ‘( i )’ represents the iteration number in the increasing step. One can choose a large value ( 5 ∼ 10 ) to accelerate the convergence. We choose ST = 5 . Macromodel in (7) is extracted for each m ( i +1) and the stability and passivity is checked. Once stability and/or passivity is violated in the increasing scheme (9), the order is reduced by 1 until the macromodel becomes stable and passive. This order selection scheme makes the regular LM algorithm robust and more suitable for the distributed high-speed modules to extract the macromodel from limited number of data points. LM method is a system identification method. So this algorithm extracts a stable macromodel from the S -parameter data for a real-world system [7,11,12]. The passivity of the macromodel is checked by using the S parameter Generalized Hamiltonian Matrix ( S -GHM) theorem [14]. The Loewner Matrix method does not guarantee passivity by construction. However, in practice, we have been unable to find an example where passivity is violated for m = m init . Furthermore, the proposed method quits at m > m init before a passivity violation occurs. The method in [15] can be used to correct a passivity violation, however this has never been necessary in our experience. V. S IMULATION R ESULTS In this section, two numerical examples have been provided to illustrate the benefits of the proposed approach over the regular LM approach [11,12] and VF approach [1] for macromodeling of 3-D high-speed modules. Example 1 : This example is a 10 port microstrip line structure on a Rogers TMM 10 ( r = 9 . 2 ) substrate of dimension: L = 80 mm , W = 55 . 5 mm , h = 1 mm as shown in Fig. 3. Copper conductors with 1.5 mm width, 0.017 mm thickness and 0.01 μm roughness are used as the microstrip. At the end of each line there is 0.64 mm to 1 mm coaxial converter with 1 mm separation in the connection of two different sizes of the coaxial conductors. All the ports are terminated by 1 mm coaxial terminations. Example 2 : This example is a combination of microstrip and stripline structures on the same substrate as Example 1 of height, h = 1 . 5 mm as shown in Fig. 4. Copper conductors with similar specifications are used as strip and microstrip. 6 via holes with radius, r = 0 . 5 mm and height, h = 0 . 7 mm connect the microstrips (port 11 – port 14 etc.) through the strips situated at 0 . 7 mm from the top of the substrate surface. All the ports are terminated by 0.64 mm coaxial terminations. The example high-speed modules are characterized by the multiport S -parameter sampled data within the corresponding bandwidth (Example 1: 0–10 GHz and Example 2: 0–8 GHz) from which the macromodels are extracted using the the proposed, the regular LM and VF approach. To illustrate the accuracy, the macromodels are evaluated at 200 discrete sample points within the corresponding bandwidth. Table I shows required frequency samples to obtain the macromodel ( N ), the order of the extracted macromodels, overall RMS errors and the CPU times. Logarithmic RMS errors for the S -parameter matrices at each frequency points are plotted in Fig. 5 and 6. For Example 1, the regular LM approach is able to extract the macromodel at the index of largest singular value drop. However, the macromodel extracted by this approach violates stability for Example 2. The index of the 2 nd large drop ...
Similar publications
A method of forecasting the balance of electricity consumption of urban development objects, civil purposes using discrete macromodels is proposed. We consider the power supply system (PSS) of the district, which is characterised by power supply from general-purpose power grids, as well as having its own generation of electricity from renewable ene...
In this paper, a novel idea of reducing numerical complexity of finite difference method using multiple macromodels is presented. The efficiency of the macromodeling technique depends on the number of ports of a model. To enhance the efficiency of the algorithm the field samples at the boundary of the macromodel are replaced with amplitudes of disc...
In this paper the problems of electric power systems creation and their further adaptation to modern software tools is considered. The discrete mathematical macromodels of electric power systems and approaches for their creation based on experimental data are discussed.
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...
![Proposed technique for estimating lP\documentclass[12pt]{minimal}...](publication/376187646/figure/fig3/AS:11431281225963425@1709003029271/Proposed-technique-for-estimating-lPdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
The structural behavior prediction of the multistory reinforced concrete (RC) buildings with masonry infill walls (MIWs) during earthquakes is challenging. This paper presents a nonlinear macromodeling strategy seeking a simple, reliable, and low-cost computational analysis of the multistory RC buildings that comprise MIWs, and that use frames or s...
Citations
... Since power systems are considered as distributed and if the frequency bandwidth of interest does not cover all frequencies, then it would not be possible to observe this drop. To solve this problem different approaches have been proposed [11,16]. Nevertheless, this investigation takes into consideration a singular values energy-based criterion, as in [12]. ...
... The result of the frequency responses for the SSA, ERA and LBFI models are presented in Figure 5. For the sake of brevity, the figure only depicts the response for 4 outputs associated with 4 inputs, that is, the comparison of frequency responses 16] for 4-output and 4-input in the Kundur system by means of all three approaches. It is noteworthy to observe that both SSA and LBFI-based responses are closely similar, identifying the same two frequency peaks in the spectrum, meanwhile the ERA response with a reduced order of m = 8 is not able to precisely capture the energy of the second peak. ...
This paper proposes a new approach to identify a data‐based power system linear model by means of frequency interpolations, aiming to obtain a suitable system representation for selective‐modal analysis purposes. The key idea behind the identification process is the Loewner‐based frequency interpolation carried out by the Loewner matrices. The proposed approach demonstrates that the Loewner‐based frequency interpolation is able to fit a linear model that can be used for small‐signal analysis studies, since it provides the state‐space representation, the frequency response, and modal information (frequency, damping, and mode‐shape). Then, a selective modal analysis is accomplished over two test cases (Kundur and New England–New York power grids) by employing the identified linear model provided by the Loewner‐based frequency interpolation method. The attained results confirm the outstanding performance of the proposal which is validated against the small‐signal analysis and compared with the eigensystem realisation algorithm, overcoming the absolute error of the model identified with the traditional eigensystem realisation algorithm approach by at least 37 times, and properly capturing the modal information in a frequency band of concern.
... Thereafter, the overall stochastic macromodel is derived via a closed form relationship of the root macromodels. In this work, the robust Loewner matrix interpolation technique is used to generate the root macromodels [10]. ...
... The matrices in (9) and (10) are complex by construction. The equivalent real matrices, L r , σL r , F r and W r , can be computed by using a similarity transformation [10]. The LM pencil defined as (xL r − σL r ) where x ∈ {γ i }∪{μ j }, is usually singular and the regular part can be obtained using singular value decomposition on the pencil as [10]: ...
... The equivalent real matrices, L r , σL r , F r and W r , can be computed by using a similarity transformation [10]. The LM pencil defined as (xL r − σL r ) where x ∈ {γ i }∪{μ j }, is usually singular and the regular part can be obtained using singular value decomposition on the pencil as [10]: ...
In this paper, a novel stochastic macromodeling technique for the variability analysis of complex electromagnetic (EM) structures is proposed. This work combines a pseudo spectral approach with the Loewner matrix interpolation technique to generate the polynomial chaos macromodel from the stochastic S-parameters of the structure. The major benefit of the proposed strategy is that by exploiting the non-intrusive nature of the pseudo spectral approach, the stochastic macromodel can be generated directly from a small number of deterministic EM full-wave simulations. This enables the utilization of the robustness and versatility of conventional deterministic full-wave techniques without the need for the cumbersome stochastic Galerkin formulation.
... Thereafter, the overall stochastic macromodel is derived via a closed form relationship of the root macromodels. In this work, the robust Loewner matrix interpolation technique is used to generate the root macromodels [10]. ...
... The matrices in (9) and (10) are complex by construction. The equivalent real matrices, L r , σL r , F r and W r , can be computed by using a similarity transformation [10]. The LM pencil defined as (xL r − σL r ) where x ∈ {γ i }∪{μ j }, is usually singular and the regular part can be obtained using singular value decomposition on the pencil as [10]: ...
... The equivalent real matrices, L r , σL r , F r and W r , can be computed by using a similarity transformation [10]. The LM pencil defined as (xL r − σL r ) where x ∈ {γ i }∪{μ j }, is usually singular and the regular part can be obtained using singular value decomposition on the pencil as [10]: ...
In this paper, a novel stochastic macromodeling technique for the variability analysis of complex electromagnetic (EM) structures is proposed. This work combines a pseudo spectral approach with the Loewner matrix interpolation technique to generate the polynomial chaos macromodel from the stochastic S-parameters of the structure. The major benefit of the proposed strategy is that by exploiting the non-intrusive nature of the pseudo spectral approach, the stochastic macromodel can be generated directly from a small number of deterministic EM full-wave simulations. This enables the utilization of the robustness and versatility of conventional deterministic full-wave techniques without the need for the cumbersome stochastic Galerkin formulation.
The Loewner matrix (LM) is a powerful macromodeling technique that uses sampled data in the frequency domain for representing dynamical systems in a descriptor form. As stability cannot be enforced during the construction process, a postprocessing step is required to ensure a stable macromodel, which reduces its accuracy. In this article, two techniques are incorporated and evaluated with the LM framework in this postprocessing phase. The first one is an efficient sign-pole-flipping technique for stability enforcement. The other one is a matrix updating technique applied after the stability enforcement of the model. The proposed approaches are evaluated using data in the scattering parameter form for printed circuit board via interconnects examples and an electromagnetic shielding structure and compared to the state-of-the-art stability enforcement algorithms. For completeness, we will also compare the results with respect to the macromodels created by the vector fitting algorithm. It was found that by the proposed flipping and matrix updating approaches, an error reduction by a factor between 2 and 20 was achieved with respect to the previously reported methods.
This paper describes PWLFIT+, an extension to the frequency domain of PWLFIT, a new paradigm in time-domain macromodeling for linear multiport systems, based on a piecewise-linear (PWL) behavioral representation of the S-parameters step response. While the impulse response of each S-parameter is approximated as sum of delayed rectangles (rect) functions, its spectrum is interpolated as sum of the corresponding delayed cardinal sine (sinc) functions. Exploiting this correspondence, the model building is performed by an iterative procedure where the PWL macromodels can be determined in order to meet defined accuracy goals on the spectrum. At runtime, waves at macromodels ports are calculated using the Segment Fast Convolution (SFC) algorithm within the Digital Wave Simulator (DWS) framework. The proposed method is characterized by its simplicity, stability, speed and scalability, all features that are emphasized when it is used in the DWS framework. After an analysis of the excellent numerical features of SFC in the Z-domain, clearly differentiated with respect conventional macromodeling methods based on poles and residues, two suitable application examples are presented to demonstrate the unique features of PWLFIT+. K E Y W O R D S impulse response, S-parameters, step response, time-and frequency-domain macromodeling
Macromodeling techniques using Loewner Matrix (LM) interpolation were proposed recently as a way to generate time-domain macromodels based on simulated Y-parameters. These approaches scale very well with respect to the number of ports as well as number of poles in the system. However, these methods become less efficient in terms of accuracy and passivity for Y-parameters obtained using Electromagnetic (EM) simulators.
In this paper, we propose a LM based interpolation technique that is applicable for large scale distributed systems described by full-wave Y-parameters. An algorithm to approximate and extract the port impedance matrix, D, directly from the data is proposed. Additionally, an order selection scheme is proposed that results in an accurate macromodel while maintaining passivity. The efficiency and accuracy of the proposed approach is illustrated using comparisons to a standard technique.
Loewner Matrix interpolation based macromodeling methods can generate an accurate macromodel for large and complex structures. However, there is less flexibility in the choice of the macromodel order due to the need to ensure passivity. In this paper, an algorithm is proposed to obtain a reduced order model based on Loewner Matrix interpolation while still maintaining the passivity of the macromodel. This is achieved by first relaxing the passivity requirement during the order selection process, then a new passivity enforcement scheme with significantly improved speed and convergence properties is applied. The proposed approach results in a significantly reduced macromodel size while maintaining accuracy and passivity conditions. The
efficiency of the new approach is shown in the examples.
In this paper, a method for obtaining a reduced order macromodel based on measured/simulated admittance parameters is presented. Loewner Matrix based macromodeling may require an higher than optimal system order in order to preserve the passivity of the macromodel. In this paper, a modified order selection scheme, combined with the Hamiltonian-Symplectic Matrix Pencil perturbation methodology, is used in order to obtain a reduced order passive Loewner Matrix based macromodel based on measured/simulated admittance parameters. The efficiency of the new approach is shown in the examples.
In this paper, a Stroud cubature based stochastic collocation approach for the efficient statistical analysis of
microwave networks is proposed. The key advantage of this approach is that the number of Stroud collocation nodes scales optimally (i.e. linearly) with the number of random dimensions thereby requiring substantially smaller number of full-wave simulations than classical collocation approaches based on tensor product grids and sparse grids. The validity of the proposed work is demonstrated using a numerical example.
