Electrical circuit analysis using four dimensional complex numbers, in the form of quaternions

Techniques for reconstructing or estimating unmeasured circuit variables and parameters in power electronic circuits are investigated. In numerous applications in power electronic systems, measurements of circuit variables are required for closed-loop control purposes or for diagnostic purposes, but are expensive or impossible to obtain directly. The method for state estimation relies on observer theory and is based on an intrinsic Lyapunov function, termed the energy in the increment. For cases where circuit parameters are not completely known, an adaptive estimation algorithm is developed for estimating the unknown circuit parameters along with the unmeasured circuit variables. The exponential stability of the combined parameter and state estimation scheme is established using averaging and singular perturbation analysis. Averaging analysis is used in off-line calculations to determine the feasibility of estimating certain parameters and to determine a basis from which to choose parameter update gains. Examples with an up-down converter illustrate the results. A hardware implementation of the state estimation scheme is also documented along with a description of a hardware implementation for parameter estimation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Understanding fully developed turbulence: Some new ideas

Introduction to Circuit Analysis and Design

This chapter introduces the analysis and design of linear circuits. It reviews basic direct‐current (DC) analysis using traditional circuits theorems with application to circuit analysis and design. Using node voltage instead of element voltages as circuit variables can greatly reduce the number of equations that must be treated simultaneously. The chapter treats the analysis and design of circuits containing active devices such as transistors or operational amplifiers (op amps). It also discusses alternating current (AC) circuits. Those ac circuits that are excited by a single frequency, for example, power systems, can be easily and effectively analyzed using sinusoidal steady‐state techniques. Linear circuits are often characterized by their behavior to sinusoids, in particular, how they process signals versus frequency. Audio, communication, instrumentation, and control systems all require signal processing that depends at least in part on their frequency response.

Linear algebra is considered an essential mathematical theory that has many engineering applications. While many theorem provers support linear spaces, they only consider finite dimensional spaces. In addition, available libraries only deal with real vectors, whereas complex vectors are extremely useful in many fields of engineering. In this paper, we propose a new linear space formalization which covers both finite and infinite dimensional complex vector spaces, implemented in HOL-Light. We give the definition of a linear space and prove many properties about its operations, e.g., addition and scalar multiplication. We also formalize a number of related fundamental concepts such as linearity, hermitian operation, self-adjoint, and inner product space. Using the developed linear algebra library, we were able to implement basic definitions about quantum mechanics and use them to verify a quantum beam splitter, an optical device that has many applications in quantum computing.

This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description of the basics of the theory of Cherednik and Hecke algebras of varieties started in arXiv:math/0406499, as well as a new Section 4, which reviews the developments in this theory since 2004 with references to the relevant literature. Let $G$ be a finite group of linear transformations of a finite dimensional complex vector space $V$. To this data one can attach a family of algebras $H_{t,c}(V,G)$, parametrized by complex numbers $t$ and conjugation invariant functions $c$ on the set of complex reflections in $G$, which are called rational Cherednik algebras. These algebras have been studied for over 15 years and revealed a rich structure and deep connections with algebraic geometry, representation theory, and combinatorics. In this paper, we define global analogs of Cherednik algebras, attached to any smooth algebraic or analytic variety $X$ with a finite group $G$ of automorphisms. We show that many interesting properties of Cherednik algebras (such as the PBW theorem, universal deformation property, relation to Calogero-Moser spaces, action on quasiinvariants) still hold in the global case, and give several interesting examples. Then we define the KZ functor for global Cherednik algebras, and use it to define (in the case $\pi_2(X)\otimes \Bbb Q=0$) a flat deformation of the orbifold fundamental group of the orbifold $X/G$, which we call the Hecke algebra of $X/G$. This includes usual, affine, and double affine Hecke algebras for Weyl groups, Hecke algebras of complex reflection groups, as well as many new examples.

Techniques for reconstructing or estimating unmeasured circuit variables and parameters in power electronic circuits are investigated. In numerous applications in power electronic systems, measurements of circuit variables (currents and voltages) are required for closed-loop control purposes or for diagnostic purposes, but are expensive or impossible to obtain directly. Examples are certain inductor currents, voltage at high-impedance nodes where probes cannot be applied or circuit variables in electrically isolated portions of a circuit. The approach taken is based on observer theory, and so the resulting algorithms may be implemented in real time. A natural scheme for implementing an observer is given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Few researches pay attention to prediction about analog circuits. The few methods lack the correlation with circuit analysis during extracting and calculating features so that FI (fault indicator) calculation often lack rationality, thus affecting prognostic performance. To solve the above problem, this paper proposes a novel prediction method about single components of analog circuits based on complex field modeling. Aiming at the feature that faults of single components hold the largest number in analog circuits, the method starts with circuit structure, analyzes transfer function of circuits, and implements complex field modeling. Then, by an established parameter scanning model related to complex field, it analyzes the relationship between parameter variation and degeneration of single components in the model in order to obtain a more reasonable FI feature set via calculation. According to the obtained FI feature set, it establishes a novel model about degeneration trend of analog circuits' single components. At last, it uses particle filter (PF) to update parameters for the model and predicts remaining useful performance (RUP) of analog circuits' single components. Since calculation about the FI feature set is more reasonable, accuracy of prediction is improved to some extent. Finally, the foregoing conclusions are verified by experiments.

In the paper, two combinations (consecutive and integrated) of vector quantization methods (self-orga- nizing map and neural gas) and multidimensional scaling (MDS) have been investigated and compared. The vector quantization is used to reduce the number of dataset items. The dataset with a smaller number of items is analyzed by multidimensional scaling in order to reduce the number of features of data (dimensionality of space) and to map them onto the plane, i.e., to visualize. Some ways of the initialization (at random, on a line, by PCs and by variances) of two- dimensional vectors in MDS have been investigated. Two ways of assignment of two-dimensional vectors in the integ- rated combinations of MDS and vector quantization methods have been examined, too.

A method for the nonlinear electric circuit analysis with the use of the orthogonal components of instantaneous admittance and resistance is proposed. The mechanism of the generation of the admittance orthogonal components of a nonlinear electric circuit is demonstrated. The analytical expressions for the components of instantaneous admittance and resistance are given and analyzed. The balance equations for the components of instantaneous admittance and resistance are analyzed. On their basis, the harmonic components of the current of the nonlinear electric circuit consisting of an active linear and nonlinear resistance, connected in series, are determined. The accuracy and adequacy of the developed method are proved by the comparative analysis of the current harmonic components calculated with the use of the proposed method and the ones obtained as a result of the numerical calculation of the researched circuit mathematical model. The advantages of the proposed method include its versatility in the use, the possibility to assess the circuit parameters influence on the current spectrum composition, good adaptation to the automation of the calculations in the frequency domain, the ability to obtain the predicted result independently of the degree of the approximating polynomial and the number of the analyzed harmonics.

Circuit Analysis and Design aims to accomplish the four vital objectives of a foundational course in the majority of electrical and computer engineering curricula: (1) It should introduce the fundamental principles of circuit analysis and equip the student with the skills necessary to analyze any planar, linear circuit, including those driven by dc or ac sources, or by more complicated waveforms such as pulses and exponentials. (2) It should start the student on the journey of circuit design. (3) It should guide the student into the seemingly magical world of domain transformations—such as the Laplace and Fourier transforms, not only as circuit analysis tools, but also as mathematical languages that are “spoken” by many fields of science and engineering. (4) It should expand the student’s technical horizon by introducing him/her to some of the many allied fields of science and technology.

In this paper a novel topology of switched capacitor based multilevel inverters is proposed. In this topology, to increase the output voltage levels and to add the voltage boosting capability, several switched capacitor units are cascaded in series. Having voltage boosting capability and generating maximum number of output voltage levels using minimum number of circuit components, is a merit for this topology. To highlight the attributes of the proposed topology, it is compared with other recently proposed multilevel inverters on number of circuit components such as power electronic switches, diodes and capacitors, voltage boosting capability, total standing voltage, cost and efficiency. The comparison results verify the superiority of the proposed topology over other proposed topologies on number of circuit components, and number of output voltage levels. It is also clear that, this topology has lower voltage stress on power electronic switches, which makes it an appropriate solution for industrial applications. Finally, to show the capabilities of the proposed topology, and to verify the accurate operation of the topology, circuit analysis as well as the experimental results are provided.

a.c. circuit analysis

This paper proposes a step-by-step optimization algorithm for optimum design of Series-Series (SS) compensated Inductive Power Transfer (IPT) systems with spiral coils. In the proposed solution, optimized parameters and Zero Voltage Switching (ZVS) frequency area are derived considering bifurcation condition. The suggested strategy optimizes the net efficiency of the SS network for a specific output power. To achieve the objective, the problem is first characterized by a three degree of freedom optimizing strategy and then, circuit parameters are calculated based on the circuit analysis. The system is optimized and simulated in MATLAB and PSIM softwares, respectively. Finally, the proposed method is verified by a laboratory IPT system with operating frequency of about 64 kHz.

This paper presents a simplified equivalent circuit model (ECM) for the post‐arc sheath stage in DC vacuum circuit breakers and outlines the method for calculating circuit parameters using the particle‐in‐cell Monte Carlo collision (PIC‐MCC) method. The post‐arc sheath is modelled as a parallel combination of a variable resistor and a variable capacitor. Based on a specific DC experimental case, the PIC‐MCC simulation provides essential parameters of charged particles and electric field data for calculating equivalent resistance and capacitance. The total current through the circuit components is calculated by applying the transient recovery voltage (TRV), and the resulting post‐arc current (PAC) waveform is compared with experimental results to validate the simplified ECM and the proposed parameter calculation approach. Further simulations study the impact of the initial drift velocity of charged particles on equivalent circuit parameters and PAC traces, showing that higher drift velocities increase capacitance and decrease resistance, leading to higher PAC peaks. The comparison with experimental data indicates that considering the drift velocity contributes to the simulation accuracy.