Linear Electrical Networks Research Articles

Reciprocity and Inter-Reciprocity: A Tutorial— Part I: Linear Time-Invariant Networks

We present a systematic treatment of reciprocity and inter-reciprocity in linear electrical networks, without resorting to the use of Tellegen’s theorem. In Part I, we discuss linear time-invariant networks. In Part II, we expand the applications of our approach to linear periodically time-varying (LPTV) networks.

Phase Analysis of N-Port Electrical Networks under Interconnections

We study a class of matrix operations induced by n-port electrical network interconnections. We first introduce the frequency-dependent phases of n-port linear time-invariant electrical networks in light of the recently defined phases of complex matrices. Such phase characterization can be used to, for example, qualify the well-known notion of passivity for networks in a way. Next, the intrinsic phase properties of networks under interconnections are investigated. A scalable phase-preserving criterion is presented. The criterion involves only the phase properties of individual sub-networks, under the matrix operations featured by interconnections, whereby the phase range of the newly integrated network can be verified effectively, and more importantly maintains the scalability of analysis.

Bandgap coupling effects between hybrid nonlinear synchronized switch damping and linear two-order resonant bandgaps in piezoelectric meta-structures

To study the bandgap coupling mechanism between nonlinear synchronized switch damping and multi-order linear resonant bandgaps, this research investigates the bandgap coupling effects between nonlinear synchronized switch damping and two-order linear resonant bandgaps in a piezoelectric meta-structure. Specifically, a piezoelectric meta-structure with hybrid nonlinear synchronized switch damping and linear two-order linear resonant electrical networks is proposed. Based on the theoretical modeling of the proposed structure, band structure and vibration transmittance performance are investigated in detail. Results showed that bandgap coupling can produce a trade-off with the advantages of both nonlinear and two-order linear resonant bandgaps. The new bandgap can not only realize broadband wave attenuation performance but also greatly enhance the attenuation in selected frequency bands, which provides a new method to manipulate elastic waves in a diversified way.

A threshold mechanism ensures minimum-path flow in lightning discharge

A well-known property of linear resistive electrical networks is that the current distribution minimizes the total dissipated power. When the circuit includes resistors with nonlinear monotonic characteristic, the current distribution minimizes in general a different functional. We show that, if the nonlinear characteristic is a threshold-like function and the current generator is concentrated in a single point, as in the case of lightning or dielectric discharge, then the current flow is concentrated along a single path, which is a minimum path to the ground with respect to the threshold. We also propose a dynamic model that explains and qualitatively reproduces the lightning transient behavior: initial generation of several plasma branches and subsequent dismissal of all branches but the one reaching the ground first, which is the optimal one.

THE CONTROLLABILITY OF RLCM NETWORKS OVER F(Z) AND THEIR APPLICATIONS

The subject of controllability and observability in dynamical systems is one of the important matters in linear system theory. The controllability and observability in linear systems over the field R of real numbers were heavily studied. If the values of resistors, capacitors, and self and mutual inductors in RLCM network are regard as independently variable parameters, the network is defined over the field F(z) of all rational functions in its physical parameters. Some concepts and results on the reducibility and separability for this network are presented; the controllability problem and application are discussed. In this paper the author will discuss the relationship between separability and reducibility of RLCM networks over F(z), the relationship between the physical structure of RLCM networks and structural controllability. The results in this paper can be used to study different kinds of linear or nonlinear electrical networks (over F(z)).

Comparative Analysis of State-Space and Companion-Circuit Methodologies for the Periodic Steady-State Solution in Time-Domain of Nonlinear Electric Networks

This contribution reports a comparative analysis of two methodologies for the periodic steady-state solution of linear and nonlinear electric networks in time-domain (TD), based on the state-space model (SSM) and companion-circuit analysis (CCA), respectively. Newton methods based on a conventional Numerical Differentiation (ND) and Enhanced Numerical Differentiation (END) process, respectively, are used. The application of the SSM and CCA methods is done using a brute force (BF) procedure and the application of advanced numerical techniques (ANT) through the forward Euler (FE) and Trapezoidal Rule (TR) numerical integration, respectively. Related analysis show comparisons of each case study in terms of efficiency, speed, computer effort and accuracy between the combination of SSM, CCA and Newton methods, e.g. SSM-ND, SSM-END, CCA-ND and CCA-END, respectively. In particular, the analysis is shown in terms of time-steps per period, CPU processing time and a computer platform used are shown. The results obtained from the reported case studies are validated against the Power Systems Computer Aided Design/Electromagnetic Transients Program including Direct Current (PSCAD/EMTDC®) response, widely accepted by the power industry.

Synthesis, Design, and Synchronization Analysis of Coupled Linear Electrical Networks

Synchronization has been associated with fundamental brain functions, especially learning behavior. Electrical circuits are candidates to mimic the tremendous information processing ability of simple brain structures due to their inherent massive parallelism. This paper gives an electrical interpretation of synchronization behavior in linear, identical subsystems with diffusive couplings. We consider a general linear state-space model for which we synthesize a minimal, generic electrical circuit. A conductance models the couplings between subsystems to form the overall electrical system. To investigate the synchronization behavior, we show how a beneficial placement of transformers decouples the overall circuit and consequently obtain many smaller circuits that are easier to examine. It is shown that the asymptotic stability in these decoupled circuits leads to a vanishing synchronization error over time. Based on this observation, we are able to formulate a synchronization condition that is entirely dependent on electrical quantities. One benefit, among others, is that by the notion of passivity, the asymptotic stability in some electrical circuits becomes evident without any further calculation. Lastly, we apply these insights to a network of interconnected Chua circuits mimicking neuron populations and their synaptic coupling structure. We investigate different topologies, such as a two ring-topology with a bridge-synapse connection.

Designing Electronic Systems Using SystemC and SystemC–AMS

Current trends in the design of electronic systems is the use of embedded systems based on systems on a chip (System-on-Chip (SoC)) or (VLSI SoC). The paper discusses the design features of electronic systems on a chip using the SystemC design and verification language. For the joint design and simulation of digital systems hardware and software, seven modeling levels are presented and discussed: executable specification, disabled functional model, temporary functional model, transaction-level model, behavioral hardware model, accurate hardware model, register transfer model. The SystemC design methodology with functional verification is presented, which reduces development time.The architecture of the SystemC language and its main components are shown. The expansion of SystemC–AMS for analog and mixed analog-digital signals and its use cases in the design of electronic systems are considered. Computing models are discussed: temporary data stream (TDF), linear signal stream (LSF) and electric linear networks (ELN). The architecture of the SystemC–AMS language standard is shown and examples of its application are given. It is shown that the design languages SystemC and SystemC–AMS are widely used by leading developers of computer-aided design systems for electronic devices.

Modeling of Combined Metal Oxide Varistors and Ferrite Core Filters to Augment Avionic Safety During Lightning Transients

The performance of ferrite core filters to suppress lightning induced fast transients in the avionic environment is discussed in this article. A dynamic model of a ferrite core filter has been formulated by the combination of a rate independent hysteresis model based on the Preisach theory, and a linear electrical network. The dynamic model is numerically implemented by a finite difference scheme in the time domain. The model is parametric with respect to the geometrical dimension of the ferrite core, which is useful in the design process. The combination of a ferrite core filter with a metal oxide varistor is considered in order to study the practical effect in terms of protection level, size of the core, and the lifetime of varistor. The pin injection tests procedure, reported in the specific standards for avionic diagnostic is simulated, and the predicted results are experimentally verified.

Rational Polynomial Chaos Expansions for the Stochastic Macromodeling of Network Responses

This paper introduces rational polynomial chaos expansions for the stochastic modeling of the frequency-domain responses of linear electrical networks. The proposed method models stochastic network responses as a ratio of polynomial chaos expansions, rather than the standard single polynomial expansion. This approach is motivated by the fact that network responses are best represented by rational functions of both frequency and parameters. In particular, it is proven that the rational stochastic model is exact for lumped networks. The model coefficients are computed via an iterative re-weighted linear least-square regression. Several application examples, concerning both lumped and a distributed systems, illustrate and validate the advocated methodology.

SystemC-AMS Thermal Modeling for the Co-simulation of Functional and Extra-Functional Properties

Temperature is a critical property of smart systems, due to its impact on reliability and to its inter-dependence with power consumption. Unfortunately, the current design flows evaluate thermal evolution ex-post on offline power traces. This does not allow to consider temperature as a dimension in the design loop, and it misses all the complex inter-dependencies with design choices and power evolution. In this article, by adopting the functional language SystemC-AMS (Analog Mixed Signal), we propose a method to enable thermal/power/functional co-simulation. The system thermal model is built by using state-of-the-art circuit equivalent models, by exploiting the support for electrical linear networks intrinsic of SystemC-AMS. The experimental results will show that the choice of SystemC-AMS is a winning strategy for building a simultaneous simulation of multiple functional and extra-functional properties of a system. The generated code exposes an accuracy comparable to that of the reference thermal simulator HotSpot. Additionally, the initial overhead due to the general purpose nature of SystemC-AMS is compensated by the surprisingly high performance of transient simulation, with speedups as high as two orders of magnitude.

Decomposition of a class of linear electrical networks for calculation of total power

A new method of decomposition of linear electrical networks for calculating total power is presented. By an iterative procedure, on the basis of decomposition of the electrical network, the branches containing power sources (voltage and/or current) are selected first until the remaining network becomes passive. Then, one calculates the power dissipated by the decomposed networks supplied by the corresponding power sources. Total power dissipated by impedances of the starting networks is equal to the sum of powers dissipated by impedances of the decomposed networks, under the condition that all impedances Z of the network are either resistive \({\text{Im}}(Z)=0\) or reactive \({\text{Re}}(Z)=0\) or mutually equal.

A 1 MHz BW 34.2 fJ/step Continuous Time Delta Sigma Modulator With an Integrated Mixer for Cardiac Ultrasound.

Fully digitized 2D ultrasound transducer arrays require one ADC per channel with a beamforming architecture consuming low power. We give design considerations for per-channel digitization and beamforming, and present the design and measurements of a continuous time delta-sigma modulator (CTDSM) for cardiac ultrasound applications. By integrating a mixer into the modulator frontend, the phase and frequency of the input signal can be shifted, thereby enabling both improved conversion efficiency and narrowband beamforming. To minimize the power consumption, we propose an optimization methodology using a simulated annealing framework combined with a C++ simulator solving linear electrical networks. The 3rd order single-bit feedback type modulator, implemented in a 65 nm CMOS process, achieves an SNR/SNDR of 67.8/67.4 dB across 1 MHz bandwidth consuming 131 [Formula: see text] of power. The achieved figure of merit of 34.2 fJ/step is comparable with state-of-the-art feedforward type multi-bit designs. We further demonstrate the influence to the dynamic range when performing dynamic receive beamforming on recorded delta-sigma modulated bit-stream sequences.

Periodic structure with interconnected nonlinear electrical networks

This article aims at investigating the filtering abilities of periodic structures with nonlinear interconnected synchronized switch damping on inductor electrical networks. Periodic structures without electrical networks themselves naturally have the function of filtering since the structure response breaks into pass bands and stop bands when the structure is excited by an external force with multiple or varying frequencies. Introduction of linear electrical networks in the periodic structure makes stop bands of the structure wider than that of the structure without electrical networks. However, nonlinear piezoelectric electrical networks may have better effect on the mechanical wave attenuation than linear piezoelectric electrical networks in terms of frequency band. Therefore, this article proposes a piezoelectric periodic structure with nonlinear interconnected synchronized switch damping on short-circuit/synchronized switch damping on inductor electrical network. A transfer matrix formulation including the interconnected electrical network is also proposed for deriving the characteristics of elastic wave propagation. The results show that the proposed technique permits enhancing the damping abilities in particular frequency bands compared to electrically independent periodic cells, which, combined with structural tailoring, would allow achieving high damping performance.

Energy-optimal current distribution in a complex linear electrical network with pulse or periodic voltage and current signals. Optimal control

Abstract The article presents that in the circuits of electrical signals belonging to the L1-impulses space or periodic signals space, real distribution of electrical currents occurs which does not meet the principle of minimum energy losses. The paper presents a solution of this problem by using the control system in the form of current-dependent voltage sources entering it into a meshes set of a complex RLC network. It has been shown that the control is energy-neutral.

Sensitivity analysis and model order reduction for random linear dynamical systems

We consider linear dynamical systems defined by differential algebraic equations. The associated input–output behaviour is given by a transfer function in the frequency domain. Physical parameters of the dynamical system are replaced by random variables to quantify uncertainties. We analyse the sensitivity of the transfer function with respect to the random variables. Total sensitivity coefficients are computed by a nonintrusive and by an intrusive method based on the expansions in series of the polynomial chaos. In addition, a reduction of the state space is applied in the intrusive method. Due to the sensitivities, we perform a model order reduction within the random space by changing unessential random variables back to constants. The error of this reduction is analysed. We present numerical simulations of a test example modelling a linear electric network.

Use of compensation theorem for the robustness assessment of electromagnetic devices optimal design

Purpose – The purpose of this paper is to present the use of the compensation theorem (CT), well known in the analysis of linear electric networks, to compute sensitivity of the performance functions used in the robust design or tolerance analysis of electromagnetic devices. Design/methodology/approach – The CT is first illustrated in the case of a simple field analysis problem. Then, using numerical simulations, the effectiveness of compensation approach for assessing impact of the small modification of material properties is shown. The numerical simulations are performed with a finite elements code based on an integral formulation. Findings – The complexity of additional computations to assess the effect of small variations involved in sensitivity analysis can be reduced. Research limitations/implications – The method can be applied only to linear systems; in addition, although compensation applies to any variations, the reduction of computational complexity is achieved only for small variations, giving localized effects. Practical implications – The method proposed in the paper can speed up the computations of sensitivity arrays in the robust design and tolerance analysis of electromagnetic device, when numerical methods are applied. Originality/value – The use of CT in field computations is not new, but its adoption in the sensitivity computation is new to the best knowledge of authors.

A novel method of network function analysis based on the Andreani–Mattisson extension to the Cochrun–Grabel algorithm

SummaryKnowing the Network Function (NF) pertaining to a certain input–output couple individuated inside a linear electrical network (LEN) is crucial, as it provides a synthetic and effective description of the input–output relationship and allows one to derive several key features of the subject system as well as to devise proper strategies to modify them.From a mathematical point of view, an NF may be expressed as the product of a real number, called the gain constant (GC), by a real rational function of the complex variables, whose numerator and denominator are called the zero polynomial (ZP) and the characteristic polynomial (CP), respectively, and may be obtained by calculating the determinants of proper matrices. However, when a design‐oriented symbolic hand analysis is needed, calculating the said polynomials in this manner may turn out to be tediously cumbersome; and above all, such a route may fail to illuminate the way that their coefficients depend on the circuit elements. The same may be said of numerical computer methods, which are suited to providing a global description of the NF in the frequency domain rather than analytical expressions of its coefficients. In this paper, a method is proposed that permits one to obtain the ZP without calculating any determinants. With respect to other methods that achieve the same goal, the present one does not require prior calculation of the CP or a nullator‐norator pair (NNP) to be compulsorily inserted in the subject circuit. Instead, it is based on applying the Andreani‐Mattisson extension (AME) of the Cochrun–Grabel algorithm (CGA) for the calculation of the CP to a circuit obtained from the assigned one by inserting in the latter a proper dependent source. The nature of such new circuit allows the worker to employ the classical tools of LEN theory so as to make the analysis as simple as possible and often performable by straightforward inspection, without the need for possibly involved equilibrium equations. Moreover, it is shown how the same technique can be applied to a properly modified version of the original network in order to calculate the CP as well. In this case (not compulsorily but if judged convenient for simplifying the calculations), a NNP may be employed in conjunction with the dependent source. The proposed method once again allows one to exploit the classic tools of LEN theory so as to considerably, if not drastically, simplify the analysis associated with the direct application of the AME to the original network. Finally, the calculation of the GC (that completes the knowledge of the whole NF under study) as well as circuit degeneracy are discussed and practical examples of application of the proposed techniques are given. Copyright © 2013 John Wiley & Sons, Ltd.

Feedback network analysis: an eclectic approach

SummaryBased on new results in circuit theory, a novel approach to feedback in linear electrical networks is proposed that generalizes the elementary one and circumvents some drawbacks and limitations that may be encountered with other general methods. It applies to all linear circuits and, differently from other approaches, does not demand two‐port models or the feedback to be referred to some controlled source. Also, it evades the necessity to construct an equivalent network or to use null double injection conditions. Instead, it allows the feedback to be detected virtually in any point of the network and seems to be more effective at ‘opening the loop’ and agevolating an analysis by inspection on less complex subcircuits. Though simple and readily applicable, the present approach is rigorous, general and provides exact formulas for overall transfer functions and immittances as well as a general model of feedback. Moreover, it allows some of the most relevant achievements of other general approaches to be generalized and rederived as particular cases. Further results, such as a generalization of the virtual short‐circuit method as well as a geometric interpretation of its limits of validity and of closed‐loop −3 dB band limits, are achieved. Practical examples of application and comparison with other methods are also given. Copyright © 2013 John Wiley & Sons, Ltd.

Linear Passive Networks With Ideal Switches: Consistent Initial Conditions and State Discontinuities

This paper studies linear passive electrical networks with ideal switches. We employ the so-called linear switched systems framework in which these circuits can be analyzed for any given switch configuration. After providing a complete characterization of admissible inputs and consistent initial states with respect to a switch configuration, the paper introduces a new state reinitialization rule that is based on energy minimization at the time of switching. This new rule is proven to be equivalent to the classical methods of Laplace transform and charge/flux conservation principle. Also we illustrate the new rule on typical examples that have been treated in the literature.