Norton's Theorem Research Articles

Online Measurement Method and System of Excitation Impedance of Current Transformers Based on Norton's Theorem and Differential Method to Measure Difference of Two Currents.

An online measurement method is proposed in this paper, and a system is established for detecting the excitation impedance of current transformers (CTs) based on Norton's theorem. The theorem is carried out by connecting a resistance and an inductance at the secondary side port of the CT to get the equations for calculating the impedance. The iterative method is used to solve the equations, and the solution is revised to consider the nonlinearity of the core. The main variable in the equations is the variation of the secondary current with the resistance or inductance. To obtain the secondary current variation accurately, which is less than 1‱ of the current, a differential method is proposed, which is based on charging two capacitors and measuring the difference of their voltages instead of measuring each current separately first and then obtaining the current variation by subtraction. This is equivalent to saving two currents first and then measuring the current difference. The differential method avoids the problem of error amplification in the process of measuring two currents separately first and then subtracting them to obtain the current variation and solves the problem that two currents do not appear simultaneously. The results verify the correctness and accuracy of the proposed method and system. The acquisition of the excitation impedance is the basis for obtaining the working characteristics of CT cores, including magnetic and loss characteristics, as well as the error of CTs.

A Comprehensive Framework for the Thévenin–Norton Theorem Using Homogeneous Circuit Models

The homogeneous description of a linear, uncoupled circuit is based on the assignment to each device of a triad <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(p: q: s)$ </tex-math></inline-formula> , where the parameters are defined up to a nonzero multiplicative constant and characterize a voltage-current relation of the form <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$pv-qi=s$ </tex-math></inline-formula> . Given a one-port, the open-circuit and short-circuit network determinants, to be denoted as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p_{e}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q_{e}$ </tex-math></inline-formula> , are polynomial functions of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> - and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -parameters of the individual devices. With this formalism, we may state the Thévenin-Norton theorem in a uniform manner by saying that, for any given set of parameter values, if at least one of the functions <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p_{e}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q_{e}$ </tex-math></inline-formula> does not vanish then the voltage-current behavior at the port is characterized by a homogeneous triad <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(p_{e}: q_{e}: s_{e})$ </tex-math></inline-formula> . In particular, the assumptions <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p_{e} \neq 0$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q_{e} \neq 0$ </tex-math></inline-formula> , respectively, characterize the existence of the Thévenin and the Norton equivalents, but the formulation proposed above avoids the need to make an a priori distinction between one form and another. The excitation parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s_{e}$ </tex-math></inline-formula> can be computed by inserting any admissible load at the port, but also analytically, in terms of the topology of the underlying digraph. The results hold without the need to specify whether each circuit element is a source or a passive device, much less to assume whether they are voltage- or current-controlled.

Effects of Source and Load Impedance on the Insertion Loss of Expansion Chamber Hydraulic Noise Suppressors

Currently, the acoustical performance of hydraulic noise attenuators is usually measured in terms of insertion loss (IL) and transmission loss (TL). Compared with the TL, experimental measurements of IL appear to be easier, however, the acquisition of source and load impedance for theoretical IL seems to be time-consuming and costly. Considering that the analogy between electrical system variables and fluidic ones is complete, well-established electrical circuit representations could be used for hydraulic systems using expansion chamber configurations as the hydraulic suppressors. Utilizing the Thevenin theorem and Norton theorem in the electrical network theory, two types of pressure oscillation source representations are equivalent insofar as the suppressors are concerned and then the expression for IL could be simplified. Finally, through the experimental measurements of the IL, the most suitable electrical representation would be selected. By implementing this method, the measurements of source impedance and load impedance tend to be avoided, which appears to be an attractive approach.

Crosstalk Analysis of Printed Circuit Board Traces With Right-Angled Bent Corners via Time Domain Hybrid Method

Discontinuous impedance exists when bent corners are introduced into the traces of printed circuit board (PCB). By combining transmission line (TL) equations, higher order finite-difference time-domain (FDTD) method, state variable method, and Norton's theorem, an efficient time domain hybrid method is proposed to realize the fast crosstalk simulation of the traces with right-angled bent corners (RABCs) in this work. First, the whole structure of the traces is decomposed into a cascade network, which consists of multiple equal length parallel microstrip lines (EL-PMSLs) and the RABC structures. Then, the crosstalk responses on these EL-PMSLs can be quickly solved by applying TL equations combined with the higher order FDTD method. With the application of Norton's theorem, the equivalent circuit model of whole PCB traces is constructed. Finally, the equivalent circuits of the RABC structures are extracted, and the port voltages of these RABCs are calculated via state variable method and fed back to these EL-PMSLs as boundaries, thus the round-trip transmission of interference signals on the PCB traces is realized. Numerous crosstalk simulations of PCB traces with typical RABC structures, as well as extracted from a practical PCB, are presented and compared with the results of commercial software CST to verify the correctness and efficiency of the proposed method.

Time Domain Hybrid Method for the Coupling Analysis of Oblique Transmission Line Network Excited by Ambient Wave

Efficient numerical methods which analyse the coupling within an oblique transmission line network (OTLN) excited by electromagnetic wave are still rare in the literature. A novel time domain hybrid method is proposed in this article, in which finite-difference time-domain (FDTD) method, transmission line (TL) equations, Norton's theorem, and interpolation techniques are organically combined. In this method, the terminal loads and connection nodes of OTLN are first shifted to the center lines of FDTD grids, where the loads and nodes are located. Then the OTLN is divided into several independent oblique transmission lines according to the connecting nodes. Finally, the FDTD method combined with TL equations and interpolation techniques is applied to compute the transient responses on each oblique transmission line, and the Norton's theorem is utilized to build the equivalent circuit models of connecting nodes to realize the transmissions of interference signals on these oblique transmission lines. Numerical simulations of OTLN on the perfect conductor (PEC) ground and in a shielded cavity excited by ambient wave are employed to exhibit the accuracy and efficiency of the proposed method. Because the structure of OTLN does not need to be meshed, the proposed method outperforms the conventional FDTD method in both memory usage and computation time.

Models of Optimal Allocation Information Resources on the Nodes of a Distributed Enterprise Information Processing System Using the Decomposition Approximation Method

In this article, using the decomposition approximation method based on the Norton theorem, the authors developed mathematical models of the functioning a distributed information processing system based on a local computer network of a two- level client-server and a three- level client-server architecture with an arbitrary function distribution of service time applications in the network nodes. A developed conceptual model of an equivalent two-node exponential queuing network and the basic mathematical relationships for calculating the intensity of service in the composition center, as well as expressions for calculating the average system response time to user requests are also presented. For the given mathematical models, using the heuristic algorithm developed earlier, the problem optimal allocation of information resources across nodes the computer network was solved by the criterion of the minimum average system response time to user requests. The results of numerical experiments are given, the analysis the results of which allows us to state about the effectiveness the developed models.

Norton Equivalent Circuit for Pulsed Photoconductive Antennas–Part I: Theoretical Model

A novel equivalent circuit for pulsed photoconductive sources is introduced for describing the coupling between the photoconductive gap and the antenna. The proposed circuit effectively describes the mechanism of feeding the antenna by the semiconductor when this latter is illuminated by a laser operating in a pulsed mode. Starting from the classical continuity equation, which models the free carriers’ density with respect to the laser power pump and the semiconductor features, a Norton equivalent circuit in the frequency domain is derived. According to the Norton theorem, the equivalent source representation is decoupled from the antenna. In particular, for photoconductive antennas (PCAs), the Norton circuit takes into account of the electrical and optical properties of the semiconductor material, the features of the laser excitation, as well as the geometrical dimensions of the gap. The presence of the electrodes around the gap is part of the antenna and, therefore, it is taken into account in the antenna impedance. The proposed circuit allows the analysis of the coupling between the photoconductive source and the antenna, providing a tool to analyze and design PCAs.

Revisited generalized substitution theorem and its consequences for circuit analysis

SummaryThe Substitution Theorem (ST) is generally perceived as a mere theoretical curiosity. In this paper, a formerly derived generalized ST (GST) is carefully revised, which leads to both a Weak Revisited GST (RGST) and a Strong RGST (characterized by noticeably relaxed hypotheses with respect to the GST). Then, despite the common opinion about the ST, such RGSTs are showed to be powerful analytical tools to generalize, make rigorous and rigorously prove several classic results of Circuit Theory, namely: the Substitution Theorem for Multiterminal Circuits, the Source‐Shift Theorem, the Thévenin–Norton Theorem, the Miller Theorem alongside its Dual, and the Augmentation Principle. More specifically, the Substitution Theorem for Multiterminal Circuits is extended to an arbitrary set of sources, possibly including nullors. The Source‐Shift Theorem is rigorously derived, and possible related ambiguities are removed. Also, all possible hybrid forms of the Thévenin–Norton Theorem for multiports are individuated, and a precise operative procedure for calculating the relevant entities is provided for all cases. Furthermore, the Miller Theorem and its Dual are extended to an arbitrary number of variables and to multiports. As to the Augmentation Principle, the constraint regarding the linearity of the augmenting resistors is removed. Finally, thoroughly worked examples are given in which the aforementioned noteworthy consequences of the RGSTs are proved to be efficient tools for analysis by inspection of linear and nonlinear circuits. Among the other things, systematic pencil‐and‐paper procedures for DC‐point and input‐output (or driving‐point) characteristic calculation in nonlinear networks are derived and applied to circuits with considerably complex topology. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Indirect Calculation Methods for Open Circuit Voltages

Open circuit voltage (OCV) of electrical devices is an issue in various fields, whose numerical evaluation needs careful treatment. The open-circuited structure is ill-conditioned because of the singular electric field at the comers, and the TEM component of the electric field has to be extracted before integrated to give the voltage in the direct method of obtaining the OCV. This paper introduces the indirect methods to calculate the OCV, the admittance matrix method and the Norton theorem method. Both methods are based on the short-circuited structure which is well-conditioned. The explicit expressions of the OCV are derived in terms of the admittance matrix elements in the admittance matrix method, and in terms of the short circuit current and the antenna impedance of the electrical device under consideration in the Norton theorem method. These two methods are equivalent in theory, but the admittance matrix method is suitable for the nearby transmitter cases while the Norton theorem method is suitable for the distant transmitter cases. Several examples are given to show the usefulness of the present theory.

NORTON'S THEOREM FOR BATCH ROUTING QUEUEING NETWORKS

This paper shows that the aggregation and decomposition result known as Norton's theorem for queueing networks can be extended to a general class of batch routing queueing networks with product-form solution that allows for multiple components to simultaneously release and receive (batches of) customers.

Inverse matrix modification lemma and Thévenin theorem for compensated network solutions

The close relationship between the inverse matrix modification lemma (IMML) and the Thévenin theorem is investigated. Making reference to the modified network solution by compensation, the IMML formula is demonstrated using the Thévenin theorem (or its dual Norton theorem) and superposition. Such equivalence allows either approach to be applied to network modification cases with the same computation burden. However, the Thévenin–Norton approach leads to a reduced number of operations compared to IMML in those contingency cases involving solid connections (bus join). Examples of the compensation procedure based on the Thévenin–Norton approach are given with reference to IEEE and to actual large-scale power networks.

Norton's equivalent for queueing networks comprised of quasireversible components linked by state-dependent routing

This paper considers queueing networks comprised of quasireversible components linked by state-dependent routing. It is shown that the product form property is valid despite the state-dependent nature of the routing process. From this observation Norton's theorem for queueing networks is extended to include queueing networks with non-product form components and state-dependent routing, such as most notably including blocking of customers. The methods used to prove the results provide a combination of two approaches to obtain product form results: local balance and quasireversibility.

Norton's equivalent for batch routing queueing networks with independently routing customers

This paper shows that the aggregation and decomposition result known as Norton's theorem for queueing networks can be extended to discrete–time and batch routing queueing networks under the assumption that customers in a batch route independently

Steady-state analysis of rate-controlled data transport in ATM Local Area Networks

This paper presents a steady-state analysis of rate-based flow control operating with the regular window mechanism of the transport protocol in ATM local area network (LAN) or local high-speed network environments. The rate control schemes are divided into single-threshold and two-threshold cases. By applying Norton's theorem for queuing model simplification and deriving a computation algorithm, the analytical results for steady-state performance of these schemes are found to be computationally efficient and yield accurate approximations. In the numerical results, the rate control schemes have been shown to have significant effects in reducing congestion levels at the receiving entity as well as in the high-speed networks. We also conclude that the two-threshold rate control scheme can be more effective in alleviating network congestion conditions than single-threshold control, while keeping the control overhead (in terms of the rate adjustment frequency) at a low level, provided that threshold values are selected appropriately. The total throughput could be affected, but only in a limited fashion.

The mathematics of product form queuing networks

Markov processes that have a product form solution have become an important computer performance modeling tool. The fact that such a simple solution exists for seemingly complex Markov processes is surprising at first encounter and can be established by showing that balance equations are satisfied. In this article we attempt to provide insight as to why such a solution form exists and demonstrate that product form and companion results, such as the arrival theorem and Norton's theorem, are consequences of four properties satisfied by queues that satisfy partial balance. Notions of reverse processes, reversibility, and quasireversibility are developed to establish the four properties

Electromagnetic coupling through a wire-penetrated small aperture in and infinite conducting plane

The electromagnetic coupling of an incident plane wave through an electrically small aperture, penetrated by a wire, in an infinite conducting screen is considered. An equivalent-circuit model for the analysis of the wire-aperture coupling is developed using Norton's equivalence theorem. The method of moments is used to find the equivalent network parameters in the model. Numerical results are presented for several cases of interest.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A generalization of Norton's theorem for queueing networks

A general framework for aggregation and decomposition of product form queueing networks with state dependent routing and servicing is presented. By analogy with electrical circuit theory, the stations are grouped into clusters of subnetworks such that the process decomposes into a global process and a local process. Moreover, the local process factorizes into the subnetworks. The global process and the local processes can be analyzed separately as if they were independent. The global process describes the behaviour of the queuing network in which each cluster is aggregated into a single station, whereas the local process describes the behaviour of the subnetworks as if they are not part of the queueing network. The decomposition and aggregation method formalized in this paper allows us to first analyze the global behaviour of the queueing network and subsequently analyze the local behaviour of the subnetworks of interest or to aggregate clusters into single stations without affecting the behaviour of the rest of the queueing network. Conditions are provided such that: As a special application, Norton's theorem for queueing networks is extended to queueing networks with state dependent routing such as due to capacity constraints at stations or at clusters of stations and state dependent servicing such as due to service delays for clusters of stations.

Model reduction of general queueing networks

Abstract A method of model reduction using the universal maximum entropy algorithm in conjunction with Norton's theorem for queueing networks is proposed for the analysis of large G-type distributed closed queueing networks. This method is called Norton-Maximum Entropy (N-ME) and has an advantage over the direct application of universal maximum entropy whereby the parametric study of a subset of queueing centres of interest can be done repeatedly without solving the entire network. The complexity of the original system is reduced into a flow equivalent two-stage load dependent queue. The marginal probability density function and various performance parameters of the two-stage load dependent queue can be obtained by using the maximum entropy analysis of the GE(n)/GE(n)/I/N queue.

New method of modeling electronic circuits coupled with 3D electromagnetic finite element models

New zero-dimensional or scalar electromagnetic finite elements, that have the time integral of electric scalar potential as their nodal variable are presented. There are three zero-dimensional element types, representing resistors, capacitors, and inductors. These elements can be easily combined with two- or three-dimensional elements, with three components of magnetic vector potential and the time integral of electric scalar potential as nodal variables. Constant current sources are directly modeled by inhomogeneous Neumann excitations, and constant voltage sources are modeled by use of Norton's theorem. By the addition of dependent current and voltage sources, electronic circuits can be modeled. Example finite-element analyses include an R-L circuit, a transistor circuit driving a wire loop modeled with three-dimensional finite elements, and a circuit impedance on the secondary of a saturable three-dimensional transformer model. >

Approximate analysis of product-form type queueing networks with blocking and deadlock

A numerical procedure for analyzing exactly closed exponential queueing networks with finite queues is presented first. Due to the finiteness of these queues, blocking and deadlock may occur. Deadlocks are assumed to be detected and resolved instantaneously. The numerical procedure is then incorporated in an approximation algorithm for analyzing closed exponential queueing networks of the product-form type, in which some of the queues are finite. These finite queues are assumed to be linked together to form a single subnetwork. The approximation algorithm is based on a variant of Norton's theorem. Comparisons between the approximate results and exact numerical results were carried out and the relative error was observed to be small.