Linear Multiport Research Articles

Frequency domain behavior of S‐parameters piecewise‐linear fitting in a digital‐wave framework

AbstractThis 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+.

Applications of matroids in electric network theory

Professor Masao Iri and his students had several important contributions to the theory of electric networks. He was one of the first scientists who recognized the importance of matroid theory in studying the qualitative problems of electric networks. The present paper summarizes his results in the unique solvability of linear active networks and in the existence of hybrid immittance description of linear multiports. Then his contribution to the proper understanding of the duality principle in electric network theory is presented. The last two sections are strongly related to the concept of genericity, leading to some results in the qualitatively reliable synthesis of linear multiports and to some considerations on how to store the data of linear multiports. These applications require a broad spectrum of matroidal tools, like the matroid partition, intersection and parity algorithms and concepts like matroid union, gammoids, transversal matroids, etc. The survey is extended by some more recent results showing that the influence of Professor Iri to the younger generation of researchers will be thriving for many years to come.

Automated Framework for Time-Domain Piecewise-Linear Fitting Method Based on Digital Wave Processing of $S$ -Parameters

An innovative full time-domain macromodeling technique for general, linear multiport systems is described. The methodology is defined in a digital wave framework and time-domain simulations are performed via an efficient method called Segment Fast Convolution (SFC). It is based on a piecewise-constant (PWC) model of the impulse response of scattering parameters, computed starting from a piecewise-linear fitting of their step response (PWLFIT). Such step response is directly available from time-domain reflectometer measurements (TDR/TDT) or equivalent simulations. The model-building phase is performed in a fast automated framework and an analytic formulation of computational efficiency of the SFC with respect to the standard time-domain convolution is given. Two application examples are used to verify the PWLFIT performance and to perform a comparison with macromodeling methods defined in the frequency-domain, such as Vector Fitting (VF).

Hybrid Description and the Spectrum of Linear Multiports

We introduce the spectrum as the set of the possible numbers of nullator–norator pairs for embedding a linear time-invariant resistive multiport into a uniquely solvable network, and as a generalization of an earlier result of Iri and Tomizawa we find the minimum of the spectrum using the matroid intersection algorithm (this number is zero if and only if the multiport has at least one hybrid description). We show that the maximum value can also be determined in polynomial time, using the linear matroid parity algorithm. We also prove that the spectrum is an invariant, that is, a multiport has the same spectrum as its negative, inverse, dual and adjoint. The proof clearly shows the difference between the two types of voltage–current symmetries for nonreciprocal networks. Most of the results remain valid for generalized multiports where the number of linearly independent equations need not be equal to the number of the ports.

Energy-Based Representation of Multiport Circuits and Antennas Suitable for Near- and Far-Field Syntheses

An energy-based representation suitable for the electromagnetic characterization of linear multiport circuits and antenna systems, as well as for the optimization of antenna beamforming and near-field radio frequency-focusing performances, is presented. Radiation, ohmic, dielectric power loss, and reactive power storage are described in the concise matrix form, yielding respective eigenmodes and eigenfields that rigorously account for all energetic processes. Noteworthy eigenfields orthogonality properties are illustrated and employed to maximize (minimize) active power flow through, or to synthesize fields on, arbitrary surfaces. Applications of the proposed formulation include multiport antennas, microwave-sensing devices, microwave hyperthermia applicators, through-the-wall imaging, and wireless power transfer systems. The numerical results illustrate features and performance advantages of the proposed formulation as applied to multiport antennas and lead to define some basic design guidelines.

Collective phases of identical particles interfering on linear multiports

We introduce collective geometric phases of bosons and fermions interfering on a linear unitary multiport, where each phase depends on the internal states of identical particles (i.e., not affected by the multiport) and corresponds to a cycle of the symmetric group. We show that quantum interference of $N$ particles in generic pure internal states, i.e., with no pair being orthogonal, is governed by $(N-1)(N-2)/2$ independent triad phases (each involving only three particles). The deterministic distinguishability, preventing quantum interference with two or three particles, allows for the genuine $(N\ge 4)$-particle phase (interference) on a multiport: setting each particle to be deterministically distinguishable from all others except two by their internal states allows for a novel (circle-dance) interference of $N\ge 4$ particles governed by a collective $N$-particle phase, while simultaneously preventing the $R$-particle interference for $3\le R\le N-1$. The genuine $N$-particle interference manifests the $N$th order quantum correlations between identical particles at a multiport output, it does not appear in the marginal probability for a subset of the particles, e.g., it cannot be detected if at least one of the particles is lost. This means that the collective phases are not detectable by the usual "quantumness" criteria based on the second-order quantum correlations. The results can be useful for quantum computation, quantum information, and other quantum technologies with single photons. \end{abstract}

Multiple photon effective Hamiltonians in linear quantum optical networks

Abstract We give an alternative derivation for the explicit formula of the effective Hamiltonian describing the evolution of the quantum state of any number of photons entering a linear optics multiport. The description is based on the effective Hamiltonian of the optical system for a single photon and comes from relating the evolution in the Lie group that describes the unitary evolution matrices in the Hilbert space of the photon states to the evolution in the Lie algebra of the Hamiltonians for one and multiple photons. Our derivation complements previous results with a point of view which reminds of the Schrodinger picture. These group theory results allow us to prove that quantum optical linear networks obey different proportionality rules which relate the expected photon number at each input and output ports. In particular, we show that for a uniform photon input the average photon number at the output is preserved for any linear multiport.

Modeling Circuits With Arbitrary Topologies and Active Linear Multiports Using Wave Digital Filters

The wave digital filter (WDF) technique derives digital filters from analog prototypes that classically have been restricted to passive circuits with series/parallel topologies. Since most audio circuits contain active elements (e.g., op-amps) and complex topologies, WDFs have only had limited use in virtual analog modeling. In this paper, we extend the WDF approach to accommodate the unbounded class of non-series/parallel junctions which may absorb linear multiports. We present four modified-nodal-analysis-based scattering matrix derivations for these junctions, using parametric waves with voltage, power, and current waves as particular cases. Three derivations afford implementations whose cost in multiplies are lower than multiplying by the scattering matrix. Negative port resistances may be needed in WDF modeling of active circuits, restricting the WDF to voltage or current waves. We propose two techniques for localizing this restriction. Case studies on the Baxandall tone circuit and a “Frequency Booster” guitar pedal demonstrate the proposed techniques in action.

A Linear Multiport Network Approach for Elastically Coupled Underactuated Grippers

This paper presents a framework based on multiport network theory for modeling underactuated grippers where the actuators produce finger motion by deforming an elastic transmission mechanism. If the transmission is synthesized from compliant components joined together with series (equal force) or parallel (equal displacement) connections, the resulting multiport immittance (stiffness) matrix for the entire transmission can be used to deduce how the object will behave in the grasp. To illustrate this, a three-fingered gripper is presented in which each finger is driven by one of two linear two-port spring networks. The multiport approach predicts contact force distribution with good fidelity even with asymmetric objects. The parallel-connected configuration exhibited object rotation and was more prone to object ejection than the series-connected case, which balanced the contact forces evenly.

Polynomial Chaos-Based Macromodeling of General Linear Multiport Systems for Time-Domain Analysis

In this paper, we present a new effective methodology to build stochastic macromodels for the time-domain analysis of generic linear multiport systems. The proposed technique allows one to calculate a stable and passive polynomial chaos (PC)-based macromodel of a system under stochastic variations. The Galerkin projections method and a PC-based model of the system scattering parameters are used along with the Vector Fitting algorithm, leading to an accurate and efficient description of the system variability features. The proposed technique results to be very versatile and, thus, is well suited to be applied to many different and complex modern electrical systems (e.g., interconnections and filters). The accuracy and computational efficiency of the proposed technique are verified through comparison with the standard Monte Carlo analysis for two pertinent numerical examples, showing a maximum simulation speedup of 765 times.

Brune’s algorithm for lumped element equivalent circuit synthesis of linear lossy multiports

An algorithm based on Brune’s multiport synthesis method is presented for the synthesis of lumped element equivalent circuits of linear passive reciprocal multiports, applicable also to lossy multiports. Such multiports are represented by positive real symmetric impedance- or admittance matrices. The developed algorithm is an iterative algorithm which extracts in each step a part of the given positive real symmetric matrix (PRSM), synthesizes a circuit representing this part and leaves a reduced PRSM behind for the next iteration. This synthesis process results in a cascaded network model.

Adjoint of a Linear Multiport Element Based on Generalized Duality

In this brief, a direct and universal method of finding the adjoint of a multiport element is proposed. This is achieved by expressing the characteristic equations of the element as a hybrid-transmission matrix, which is based on the concept of generalized duality, and deriving a simple relation between the hybrid-transmission matrices of the original and adjoint element. The proposed method unifies the existing transpose relations based on conventional admittance, impedance, and hybrid parameter matrices and can be further generalized to cases which existing methods cannot conveniently deal, or even deal, with. Once the characteristic equations of the element are given, we can obtain the adjoint (if it does exist) immediately by using the proposed method without any additional operations. The proofs of relevant theorems and some applications of the proposed method are given and the sensitivity formula based on the proposed method is also derived.

Polynomial chaos‐based macromodeling of multiport systems using an input–output approach

SummaryAn innovative technique to build stochastic frequency‐domain macromodels of generic linear multiport systems is presented. The proposed method calculates a macromodel of the system transfer function including its statistical properties, making it tailored for variability analysis. The combination of the modeling power of the Vector Fitting algorithm with the Polynomial Chaos expansion applied at an input–output level allows to accurately and efficiently describe the system variability features. Thanks to its versatility and automated order selection, the proposed technique is suitable to be applied to a large range of complex modern electrical systems (e.g., filters and interconnections) and can tackle the case of correlated random variables. The performance in terms of accuracy and computational efficiency of the proposed method is compared with respect to the standard Monte Carlo analysis for two pertinent numerical examples. Copyright © 2014 John Wiley & Sons, Ltd.

FDTD MODELING OF ARBITRARY LINEAR LUMPED NETWORKS USING PIECEWISE LINEAR RECURSIVE CONVOLUTION TECHNIQUE

Based on the piecewise linear recursive convolution (PLRC) technique, FDTD modeling of Arbitrary linear lumped networks is studied in this paper, including one-port networks and two-port networks. Their general FDTD iterative formulations are obtained. Firstly, the admittance parameters in Laplace domain of lumped network are written as a summation form of several rational fractions; then the time domain admittance parameters can be obtained by means of inverse Fourier transform technique. Finally the time domain results are directly incorporated into the MaxwellAmpere’s difference equation using the PLRC technique. It is worth pointing out that this approach preserves the second-order accuracy and the explicit nature of the conventional FDTD method. The proposed technique can be extended to model arbitrary linear multiport lumped networks. To show the validity of the proposed algorithm, we analyze two microstrip circuits including lumped networks. The results are compared with those obtained from the Z-transform technique and the good agreement is achieved.

Multiport S-Parameter Measurements of Linear Circuits With Open Ports

A simple technique to measure the S-parameters of a linear multiport with a two-port network analyzer is evaluated. Just two probes and a single specimen are required. All two-port combinations are measured with the other ports left open. Special care was taken to estimate the maximum measurement uncertainties and to identify critical parameter values. Results can be further improved by using a simplified weighting scheme or by correcting for open-port capacitances. The technique is especially suited for wafer probes.

An adjoint-based approach to computing time-domain sensitivity of multiport systems described by reduced-order models

This paper describes a new algorithm for obtaining time-domain sensitivity information of large linear networks via projection-based model-order reduction (MOR) techniques. The proposed algorithm enables adjoint-based time-domain sensitivity analysis techniques to be applied to a large network containing multiple linear multiport subnetworks. In addition, the proposed algorithm can also be used to obtain frequency-domain sensitivity information. The efficiency and accuracy of the proposed work is demonstrated through optimizing the time-domain waveforms of a high-speed interconnect circuit with nonlinear terminations.

Squeezing as an irreducible resource

We show that squeezing is an irreducible resource which remains invariant under transformations by linear optical elements. In particular, we give a decomposition of any optical circuit with linear input-output relations into a linear multiport interferometer followed by a unique set of single mode squeezers and then another multiport interferometer. Using this decomposition we derive a no-go theorem for superpositions of macroscopically distinct states from single-photon detection. Further, we demonstrate the equivalence between several schemes for randomly creating polarization-entangled states. Finally, we derive minimal quantum optical circuits for ideal quantum non-demolition coupling of quadrature-phase amplitudes.

A synthetic network using current conveyors of the open-circuit impedance matrix of a linear time-invariant multiport

Previous research appearing in the literature required that all the realizable z parameters of the open-circuit impedance matrix of a linear time-invariant multiport network must be real, rational, stable functions. Although different z parameters could be selected independently without necessarily to having common poles, this also required that all the z parameters be minimum phase. A synthetic network using second-generation current conveyors (CCIIs), third-generation current conveyors (CCIIIs) and differential difference current conveyors (DDCCs) is proposed here, with each transfer impedance parameter being either minimum or nonminimum phase. Additionally, it can be designed automatically by computers since the current conveyor circuits, the transfer functions and numerical methods are available.

The Time-Domain Vector Fitting Algorithm for Linear Macromodeling

The Time-Domain Vector Fitting (TD-VF) algorithm for macromodeling of linear multiport systems is presented. A rational approximation for the system matrix transfer function is easily derived from transient input/output port responses. Several validations illustrate the high accuracy of the method.

Matrix representations and parameterizations of lossless and reciprocal multiports

Stimulated by recent work on the synthesis of a certain class of surface acoustic wave (SAW) filters, we discuss the exceedingly classical subject of physical realizability criteria for lossless, and reciprocal linear multiports. We give an essentially coordinate-free formulation of the realizability conditions for passive, lossless and reciprocal multiports. For this sake we describe a linear multiport by its external behavior (the linear subspace generated by all admissible signal pairs at the ports) and analyze the properties of this space in terms of Gramians of its bases with respect to physically relevant metrics. The standard matrix representations (impedance matrix, scattering matrix, . . . ) arise by the introduction of various affine coordinates in the external behavior. Instead of discussing their well-known coordinate-dependent properties we focus on the so called P -matrix as an attractive non-standard example and study its properties and relations to other matrix representations. As a major result we show that the parameterization of lossless multiports as induced by the P -matrix is largely equivalent to the Arov-Dewilde-Dym parameterization of J-inner functions.