RL Circuit Research Articles

Application of Fractional Modified Taylor Wavelets in the Dynamical Analysis of Fractional Electrical Circuits under Generalized Caputo Fractional Derivative

Abstract This study examines the application of fractional calculus in the analysis and modeling of electrical circuits of fractional order, highlighting its potential to explain the behaviour of complex electrical circuits accurately. In the domain of electrical circuits, fractional differential equations are employed in the analysis and simulation of systems that consist of resistors, capacitors and inductors. In the present paper, a novel approach utilizing fractional order modified Taylor wavelets is implemented to solve the fractional model of RL, LC, RC and RLC electrical circuits under generalized Caputo fractional derivative which offers precise and flexible modeling of non-locality and hereditary characteristics in complex systems. Furthermore, an additional parameter $\sigma$ (time scale parameter) is incorporated in fractional circuit dynamics to maintain the physical dimensionality. The considered wavelets with the collocation technique offer an efficient solution by converting the fractional model of electrical circuits into a set of algebraic equations which are further solved by using the Newton iteration method. Moreover, this study discusses the significance of Ulam-Hyers stability, emphasizing its role in ensuring stable and reliable circuit performance. The impact of fractional order on the dynamics of the electric circuit model is presented by tables and graphs. The approximate solutions obtained by the proposed method are well comparable with exact solutions and some other existing wavelet-based techniques. The residual errors are also evaluated under various model parameters for fractional orders. Furthermore, the graphs illustrate that the error progressively decreases as the number of wavelets basis increases.

Machine learning classification of permeable conducting spheres in air and seawater using electromagnetic pulses

Abstract This paper presents machine learning classification on simulated data of permeable conducting spheres in air and seawater irradiated by low frequency electromagnetic pulses. Classification accuracy greater than 90% was achieved. The simulated data were generated using an analytical model of a magnetic dipole in air and seawater placed 1.5–3.5 m above the center of the sphere in 50 cm increments. The spheres had radii of 40 cm and 50 cm and were of permeable materials, such as steel, and non-permeable materials, such as aluminum. A series RL circuit was analytically modeled as the transmitter coil, and an RLC circuit as the receiver coil. Additive white Gaussian noise was added to the simulated data to test the robustness of the machine learning algorithms to noise. Multiple machine learning algorithms were used for classification including a perceptron and multiclass logistic regression, which are linear models, and a neural network, 1D convolutional neural network (CNN), and 2D CNN, which are nonlinear models. Feature maps are plotted for the CNNs and provide explainability of the salient parts of the time signature and spectrogram data used for classification. The pulses investigated, which expand the literature, include a two-sided decaying exponential, Heaviside step-off, triangular, Gaussian, rectangular, modulated Gaussian, raised cosine, and rectangular down-chirp. Propagation effects, including dispersion and frequency dependent attenuation, are encapsulated by the analytical model, which was verified using finite element modeling. The results in this paper show that machine learning methods are a viable alternative to inversion of electromagnetic induction (EMI) data for metallic sphere classification, with the advantage of real-time classification without the use of a physics-based model. The nonlinear machine learning algorithms used in this work were able to accurately classify metallic spheres in seawater even with significant pulse distortion caused by dispersion and frequency dependent attenuation. This paper presents the first effort towards the use of machine learning to classify metallic objects in seawater based on EMI sensing.

Investigation of electromagnetic pulse scattering for metallic object classification using machine learning

ABSTRACT This paper presents a metallic object classification method using various electromagnetic pulses. Each electromagnetic pulse irradiated eight metallic objects placed at increasing distances from 10 mm to 40 mm relative to the electromagnetic sensing system. The electromagnetic sensing system consisted of two RL circuits placed in close proximity. Objects were classified using linear (perceptron and multiclass logistic regression) and non-linear (neural network, 1D convolutional neural network (CNN) and 2D CNN) machine learning classifiers. The machine learning classifiers were trained on experimental data collected in an electromagnetically shielded laboratory. A 10-fold cross-validation mean classification accuracy of 99.4 ± 0.3% for the 1D CNN classifier, and 92.9 ± 1.2% for the 2D CNN classifier, was achieved using a rectangular chirp electromagnetic pulse. The rectangular chirp pulse outperformed two-sided decaying exponential, Gaussian, triangular, raised cosine, and rectangular pulses. All pulses had equal energy. While the rectangular chirp performed best overall, other pulses more accurately distinguished between some objects.

Machine learning classification of metallic objects using pulse induction electromagnetic data

This paper presents the classification of metallic objects using total and scattered pulse induction electromagnetic data, with a classification accuracy greater than 90%. Machine learning classification is applied to raw electromagnetic induction (EMI) data without the use of a physics-based model. The EMI method is applied to 8 metallic objects placed at increasing distances from 10–55 mm to the EMI sensing system. The EMI sensing system consists of two RL circuits placed in close proximity. Metallic objects are classified using linear algorithms including a perceptron and multiclass logistic regression, and nonlinear algorithms including a neural network, a 1D and 2D convolutional neural network (CNN). EMI data was collected using an experiment in an electromagnetically shielded laboratory. Feature maps are presented that explain the salient components of the EMI data used by the 1D and 2D CNN.

Конденсаторная установка для сельской распределительной сети 0,4 кВ: особенности построения

Reactive power compensation based on a capacitor plant connected to a rural 0.4 kV distribution network, when using non-specialized contactors, is accompanied by the occurrence of large starting currents of capacitors. In addition, the installation of a conductor connection with load current transformers requires structural modification of the capacitor plant. In order to eliminate these disadvantages, it is proposed to use a capacitor bank controlled by a time relay and protected from inrush currents by three RL circuits. In this case, the “on-off” mode of the capacitor bank must be switched to by an internal signal of the capacitor bank setting the time interval “on-off”. Using a mathematical model, as well as a three-phase physical model with a power of 1.2 kV, the authors studied the behavior of transient and steady-state currents at various values of the parameters of RL circuits.The initial invariant parameters of the model were the reactive resistances of capacitors and inductors. The nominal value of the phase current of the capacitor bank was 2.14 A. The variable parameters were the values of the active resistances of the RL circuits, which took the values of 0, 10, 20, 30 and ∞. The oscillogram of stationary and transient current was studied. As a result, it was found that in order to match the capacitor bank to the criterion of using circuit breakers and contactors of the 0.4 kV network, the resistance value of the RL circuit resistor of each phase of a three-phase capacitor bank should be tenfold higher than the reactance of the current-limiting coil of the RL circuit and be five times less than the reactance value of the power capacitor of the capacitor bank phase. In rural 0.4 kV electric networks, a single 25 kV capacitor plant with simple autonomous control can be installed to compensate for reactive power. Several capacitor plants can compensate for reactive loads of 50 and 75 kVAr.

Examination of Gas Active Arc Extinguishing and Lightning Protection for Transmission Lines

The study investigated the gas active arc extinguishing and lightning protection for transmission lines. Lightning is one of the most significant sources of over voltages in overhead transmission lines. The lightning over voltages could lead to failure of the devices connected to the transmission line. Fundamental constraint on the reliability of an electrical power transmission system is the effectiveness of its protective system. The role of the protective system is to safeguard transmission system components from the effects of electrical overstress. Shield wire and surge arresters are an important means of lightning protection in transmission and distribution systems. Therefore, it is necessary to analyze the influence of over voltages caused due to lighting. The requirements of system security for lightning protection technology reach the high standard that lightning strike tripping and wire breakage are not allowed. Both the power supply side and the demand side have very high requirements for lightning protection of transmission lines, but the lightning accident rate remains high. This situation poses a great threat to the reliability and safety of power network, it is urgent to solve this problem. The gas active interrupter principle is put forward. From two latitudes of lightning overvoltage and power frequency over current, the effective control of lightning accident rate caused by large probability and multiple lightning strike is realized. This paper is aimed at evaluation of protection level of transmission line by using gas active arc extinguishing, shield wire and line surge arresters against lightning overvoltage. When lightning strikes, the internal short gap of the main body and the air gap break down, and the lightning arc enters the arc-extinguishing body through the breakdown channel. Arc is compressed by force to form internal and external gradients temperature in order to produce jet airflow. Jet airflow acts on the subsequent power frequency follow current arc, a model coupling an arc and a compressed jet airflow in a multi-fracture compression airflow arc-extinguishing structure is established and simulated. The increase in voltage due to lightning are investigated by using the MATLAB software. Direct and induced, types of lightning strokes are considered. The model of a 33kV three-phase overhead power transmission line is developed considering the RL circuit. The simulations are performed based on time domain analysis. Also, numerical analysis is made for 133kV tower structure transmitting power over 220km from 132 KV Sub-transmission Lines in port Harcourt region. Based on the simulation study this research study gas active arc extinguishing and lightning protection for transmission lines, are very important in computer application like Electromagnetic Transient Propagation (EMTP) and Power System Computer Aided Design (PSCAD) which are used in future study. These soft wares provide an alternative way to bring the precise result in study of lightning protection level of high voltage transmission line.

Data-Driven-Based Vector Space Decomposition Modeling of Multiphase Induction Machines

For contemporary variable-speed electric drives, the accuracy of the machine's mathematical model is critical for optimal control performance. Basically, phase variables of multiphase machines are preferably decomposed into multiple orthogonal subspaces based on vector space decomposition (VSD). In the available literature, identifying the correlation between states governed by the dynamic equations and the parameter estimate of different subspaces of multiphase IM remains scarce, especially under unbalanced conditions, where the effect of secondary subspaces sounds influential. Most available literature has relied on simple RL circuit representation to model these secondary subspaces. To this end, this paper presents an effective data-driven-based space harmonic model for n-phase IMs using sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover the IM governing equations. Moreover, the proposed approach is computationally efficient, and it precisely identifies both the electrical and mechanical dynamics of all subspaces of an IM using a single transient startup run. Additionally, the derived model can be reformulated into the standard canonical form of the induction machine model to easily extract the parameters of all subspaces based on online measurements. Eventually, the proposed modeling approach is experimentally validated using a 1.5 Hp asymmetrical six-phase induction machine.

Сравнительный анализ математических описаний и методов широтно-импульсной модуляции

The article considers the mathematical description and comparison of the most common methods of pulse-width modulation (PWM) for a three-phase voltage inverter, namely vector PWM, PWM with third harmonic premodulation and classical PWM. In contrast to the classical mathematical description of PWM, a description is presented that is close to the implementation of modulation using modern microcontroller technology. The classic representation of PWM is based on comparing two signals. In this representation, the modulation pulse occurs at the moment when the modulating function is greater in value than the reference signal. When implementing PWM by a microcontroller, the pulse is placed within the period symmetrically with respect to the middle of the interval, while the pulse width is determined by the average value of the voltage modulating function. With a different mathematical description of PWM, the pulse is located differently within the modulation interval. There are presented the results of modeling and compares the mathematical descriptions of PWM methods according to the criterion of minimum dispersion of the current flowing in the load. As a result of the comparison, it was revealed that the same, differently described PWM algorithm gives different results. A comparison of vector PWM with a modified vector PWM was made. The modification consists in applying a dynamic pulse shift algorithm in such a way as to obtain a minimum of current dispersion in the load. Comparison of PWM methods was carried out at different frequencies of the reference signal; a static RL circuit was used as a load when simulating a three-phase voltage inverter. The results obtained allow us to recommend a pulse shift algorithm for implementing PWM in voltage inverters operating at relatively low PWM frequencies. Minimization of the current dispersion makes it possible to obtain improved energy and vibration-noise characteristics of the electric drive.

Dynamic simulation of rail potential considering rail skin effect

The existing simulation of rail potential only considers the equivalent DC resistance and 1st-order RL circuit of a rail. However, the rail current contains a large number of harmonic components, which greatly affects the calculation of rail potential. To accurately calculate the rail potential, the rail is equivalent to a tubular conductor considering the skin effect. The improved equivalent circuit of reflux system is established by vector fitting method (VF), and the 8th-order foster equivalent circuit of rail is built. The dynamic simulation model considering multi-trains, multi-stations, and over voltage protection device (OVPD) actions is established. By measuring the actual internal impedance of the rail, the parameters of the equivalent tubular conductor model of the rail are fitted with the least square method. The maximum error between VF method and finite element simulation, field test results is 6.531%, and 7.271%, respectively. Compared with the equivalent of the 1st-order RL circuit and DC resistance, the rail potential results of the modified model are significantly improved, the rail potential is evaluated more accurately.

Transfer Function and Z-Transform of an Electrical System in MATLAB/Simulink

This paper presents the transfer function (T.F) and Z-Transform (Z.T) of an Circuits in MATLAB/Simulink. We have dealt with the electrical system, which consists of circuits in which there is a capacitor, resistance, and capacitance. Simulink is useful for obtaining the response of systems to input functions that are more complicated than step, impulse, ramp, or sine functions. The aims of this paper is to describe the solution of the equations of motion with One Degree and Two Degree Freedom of the equations of motion of the circuits by using transfer function equations in MATLAB/Simulink. We followed two applied mathematical methods to obtain the T.F and Z.F of a circuits in MATLAB/Simulink. Different graphics showed a close match result. Using these methods, researchers can effectively solve the RL and RLC circuit problems while presenting the results in excellent graphical form. The numerical approximation provides an acceptable result using the analytical one as a benchmark when we compare simulation results with analytical results.

Fault current analysis and active current limiting strategy for hybrid cascaded LCC/MMC HVDC system

The hybrid cascaded LCC/MMC high voltage direct current (HC-HVDC) system is a prospective topology that can avoid the commutation failure problem of LCC-HVDC while ensuring bulk power transmission capability. However, the complex topology results in different DC fault characteristics at the receiving end, which brings challenges to fault detection and current limiting technology. As a result, it is of great theoretical importance to investigate the fault characteristics of the HC-HVDC system and propose the corresponding fault current limiting strategy. Firstly, this paper simplifies the equivalent discharging circuit of the hybrid cascaded LCC/MMC inverter into a first-order RL circuit by assuming the firing-advance angle is constant after fault, and proposes the fault current expressions by Laplace transform. Secondly, by changing the insertion sequence of submodules, an active fault current limiting strategy based on the submodule sorting algorithm (SMS-FCL) is proposed, which can accomplish fault current suppression, reactive power and DC voltage support, and energy loss reduction simultaneously. Moreover, the proposed SMS-FCL can cooperate with other FCLs to further suppress the fault current, and the cooperation prerequisites are analyzed. Finally, an LCC-LCC/MMC HVDC system is built in PSCAD/EMTDC to validate the accuracy of the proposed fault current expressions and the effectiveness of the current limiting strategy.

Formation of Probing Signals of Piezoelectric Transducers for Ultrasonic Testing

Introduction. Reducing the duration of the probing pulse at the output of a multilayer radiator is an urgent task of acoustic non-destructive testing. This not only improves the resolution of the system and the accuracy of determining the coordinates of defects, but also reduces the length of the dead zone. The most versatile method for achieving short signal duration is mechanical damping. The use of RL circuits connected to the electrical side of a piezoelectric transducer (PET) for this purpose has been studied to a lesser extent. Of interest is a comparative study of the potential possibilities of the two indicated methods for obtaining a short signal.Aim. To carry out a comparative study of two options for reducing the duration of the probing signal in order to establish their preferential use in the practice of ultrasonic testing.Materials and methods. To determine the boundaries of the preferred application of one of the methods in comparison with the other, a mathematical apparatus is used based on the use of integral calculus, as well as numerical calculation methods. When constructing a mathematical model of piezoelectric transducers operating in a pulsed mode, the method of analog circuits is used in combination with the spectral method based on Fourier transforms. Numerical calculations were performed in the MathCad environment.Results. It was established that the use of an electrical corrective circuit with optimal parameters makes it possible, across a wide range of changes in the values of the specific acoustic resistance of the protector, to achieve a shorter duration of probing signals at the output of the probe than in the case of using a damped probe with values of the specific acoustic resistance of the damper zд latitude 10⋅106 Pa⋅s/m. At zд >10⋅106 Pa⋅s/m, preference should be given to mechanical damping of the piezoelectric element. It was found that the amplitude of the signals at the output of the PET with a corrective circuit connected thereto exceeds the amplitude of the signal when the piezoelectric element is damped.Conclusion. The results obtained allow an a-priori evaluation and comparison of PET capabilities using two methods for creating a short probing signal, as well as a justified selection of materials for creating a protector across a wide range of specific acoustic resistances. The correctly selected parameters of the structural elements of the probe makes it possible to improve the resolution of radiation-reception systems, reduce the length of the dead zone, and increase the accuracy of determining the coordinates of defects. This ultimately improves the quality of acoustic testing of materials and products.

Eddy currents damping understood as Zener viscoelastic damping

This paper identifies and explains the eddy currents damping originated in a solid that vibrates in a magnetic field as a Zener viscoelastic model damping. For that, a coupled mechanical-electrical model is proposed for the eddy currents damping. The mechanical model is studied as a single degree-of-freedom system, and the electrical model is assumed to be a closed RL circuit. The coupling between them is given by the magnetic force originated by the motional inductance. The motion equation is obtained for the coupled system for two different excitation types: harmonic force and harmonic base motion. These motion equations include derivatives of the displacement with respect to time up to the third order. An equivalence between these motion equations and the ones derived for a viscoelastic system modelled with a Zener model is found. This allows understanding the parameters of the eddy currents model as the ones of a viscoelastic Zener model. The term related with the third order derivative is associated with relaxation, a characteristic property of viscoelastic systems. This equivalence between Zener and eddy currents damping explains forces proportional to the third order derivative that other authors have experimentally found in vibrating systems with eddy currents. The proposed model is validated both from an analytical comparison with another model of the literature and from experimental results previously published. Finally, a parametric study is carried out to show up the influence of the magnetic field and electrical properties of the model on the frequency and time responses of the system.

The Impact of Current in Resistance and Inductance of Rl Circuit in the Applications of Stochastic Differential Equations Using Caputo-Fabrizio Derivative

In this research article, the circuit thus designed here, can also be implemented in electrical substations, where in the reactors are used to control the flow of current, thereby regulating the inductance and resistance values. The optimum values of Inductance and Resistance found through this circuit, facilitates optimum flow of electricity, reduces the power loss due to inductance, also eases the burden on reactors in regulating power supply.

The Impact of Current in Resistance and Inductance of Rl Circuit in the Applications of Stochastic Differential Equations Using Caputo-Fabrizio Derivative

In this research article, the circuit thus designed here, can also be implemented in electrical substations, where in the reactors are used to control the flow of current, thereby regulating the inductance and resistance values. The optimum values of Inductance and Resistance found through this circuit, facilitates optimum flow of electricity, reduces the power loss due to inductance, also eases the burden on reactors in regulating power supply.

On the test of novel constitutive relation of capacitor for electrical circuit analysis: a fractal calculus-based approach

PurposeThe purpose of this paper is to test the capability to properly analyze the electrical circuits of a novel constitutive relation of capacitor.Design/methodology/approachFor ceteris paribus, the constitutive relations of the resistor and inductor have been reformulated by following the novel constitutive relation of capacitor. The responses of RL, RC, LC and RLC circuits defined on the fractal set described by these definitions have been derived by means of the fractal calculus and fractal Laplace transformation. A comparative Hamiltonian formalism-based analysis has been performed where the circuits described by the conventional and the formerly proposed revisited constitutive relations have also been considered.FindingsThis study has found that the novel constitutive relations give unreasonable results unlike the conventional ones. Like such previous revisited constitutive relations, an odd Hamiltonian has been obtained. On the other hand, the conventional constitutive relations give a reasonable Hamiltonian.Originality/valueTo the best of the author’s knowledge, for the first time, the analysis of fractal set defined electrical circuits by means of unconventional constitutive relations has been performed where the deficiency of the tested capacitive constitutive relation has been pointed out.

Innovative Approach for the Study of Tuning Circuit with Open Source Tools

In the new age of an online teaching-learning process, present instructional approach expects students to be an active learner, which necessitates a new instructional tool that can provide opportunity for active learning. The use of technology in teaching-learning has become a key factor which helps in conceptual understanding. Simulation being one of the valuable instructional tools renders an opportunity to visualize the concepts and facilitate to build the mental model. The present work proposes a study of electrical circuit experiments performed using open source tools such as XCOS and expEYES. These tools facilitate in developing higher order critical thinking and analysis skill. XCOS being one of the tools in Scilab provides the visual simulation environment for varying the input parameters of the model. The RL, RC, and RLC tuning circuits are virtually modeled by using the XCOS and the real circuits made with expEYES, which facilitates the study of response curves with ease. In the later case, the results obtained by using expEYES compared with the results of XCOS. In addition, the study explains how such response of simple linear circuits can be extended in different applications. Demonstrations and visualization of the response curve and phase flow of linear circuits will overlay the way to understand experimental and numerical results. This innovative approach can be used in the classroom demonstration of a harmonic oscillator and resonant circuits in advance level physics and engineering laboratories.

Анализ электровзрывобезопасности на основе динамических моделей разряда при коммутации электрической цепи

A mathematical model describing a switching arc discharge in an RL electric circuit based on the Mayr equation is developed and tested. The body of mathematics representing an arc discharge appearing in an RL circuit that takes into account the dynamic current-voltage arc characteristic and arc thermal processes is described. The results from calculating the arc thermal inertia constant for the switching period for a resistive circuit by several known methods, including the values obtained during simulation, are given. It has been found that the arc discharge model that takes into account the arc thermal inertia constant dynamics calculated from the instantaneous values of the average discharge power (taking the cathode losses into account) gives better convergence of the simulation results. An improved mathematical model of discharge in resistive and inductive electrical circuits is proposed, which makes it possible to refine the well-known method for quantifying their intrinsic safety. The convergence with the minimal igniting parameters obtained experimentally for similar circuits is analyzed.

Analysis on determining the solution of fourth-order fuzzy initial value problem with Laplace operator.

This study presents a new analytical method to extract the fuzzy solution of the fuzzy initial value problem (FIVP) of fourth-order fuzzy ordinary differential equations (FODEs) using the Laplace operator under the strongly generalized Hukuhara differentiability (SGH-differentiability). To this end, firstly the fourth-order derivative of the fuzzy valued function (FVF) according to the type of the SGH-differentiability is introduced, and then the relationships between the fourth-order derivative of the FVF and its Laplace transform are established. Furthermore, considering the types of differentiabilities and switching points, some fundamental theorems related to the Laplace transform of the fourth-order derivative of the FVF are stated and proved in detail and a method to solve FIVP by the fuzzy Laplace transform is presented in detail. An application of our proposed method in Resistance-Inductance circuit (RL circuit) is presented. Finally, FIVP's solution is graphically analyzed to visualize and support theoretical results.

Transient and Steady Current in a Series RL Circuit

Transient and Steady Current in a Series RL Circuit