Parameters assessment of the inductively-coupled circuit for wireless power transfer

In this paper, a wireless power transfer model through the example of inductively-coupled coils of irregular shape in software package COMSOL Multiphysics is studied. Circuit parameters, such as inductance, coil resistance and self-capacitance were defined through electromagnetic energy by the finite-element method. The study was carried out according to Helmholtz equation. Spatial distribution of current per unit depending on frequency and the coupling coefficient for analysis of resonant frequency and spatial distribution of the vector magnetic potential at different distances between coils were presented. The resulting algorithm allows simulating the wireless power transfer between the inductively coupled coils of irregular shape with the assessment of the optimal parameters

Polynomial reconstruction of electric charge distribution on the conductive plate caused by external electric field

The paper proposes an original method of calculating the charge distribution on the surface of the conductive plate introduced into the external electrostatic field. The authors managed to obtain the polynomials which allow to solve the integral equation that establishes the relationship between charge distribution of the conductive plate and the potential distribution of the external field and the potential on the surface of the plate. The proposed algorithms solutions are valid in the presence of axial symmetry of the field and the plate. Examples of calculation of conductor charge distribution in the presence of external field by using a polynomial expansion have been presented. The comparisons of results calculated by the polynomial method and by known analytical solutions have been given

Capacity calculation of the electrotechnical scheme of discharge gap replacement of the ozonizer in the COMSOL environment

The investigation of electrostatic field distribution of ozonator electrode gap caused by inhomogeneous permittivity has been done. The paper presents calculation of electrostatic field energy of ozonator electrode gap. From the energy of electrostatic field distribution the information about electrode system capacity has been extracted. For calculation of the electrostatic field and capacity of electrode system the software integrated environment COMSOL has been used

Regularising algorithm of parameter identification of electric charge equivalent circuit

A new algorithm of parameter identification of equivalent circuit for electrical charge replacement is suggested. The approach is based on the solution of integral equation of the I type with respect to the function of indicial admittance, by which then determination of replacement circuit parameters is carried out. Application of smoothing splines and original regulating algorithm including kernel setting error of integration equation permits to obtain a stable algorithm of parameter identification. The investigation of algorithm shows high calculating efficiency and sufficient accuracy of parameter identification

Investigation of wave processes of transformer windings as a distributed electric circuit

Relevance of the work is caused by the need of localize the defects of the transformer windings at diagnosis. It is possible to determine the spatial location of the defects along the transformer winding only if the transformer is considered as a system with distributed parameters. The main aim of the study is to present the transformer winding in the form of distributed electric circuit and to study the transients of such a circuit in two cases, at grounded winding end and at open winding end; to determine primary and steady voltage distribution along the transformer windings at the grounded winding end and open winding end; to search for possible ways to overcome the overvoltage arising along the transformer winding. The aim of the work is to obtain as well the transitional function caused by a unit action, in order to use subsequently the Duhamel integral which allows obtaining the transients in the transformer winding at an arbitrary external action. The methods used in the study are based on equations solution in partial derivatives using Laplace transform and transfer functions. It uses the initial conditions along the transformer winding and the boundary conditions at the ends of the windings. The final solution of partial differential equations is represented as a series, that is in the form of wave superposition with different frequencies. The results. The obtained transfer function of the voltage distribution along transformer windings allows determining: primary and steady stress distribution; over-voltage transient conditions; optimal values of distributed capacitance, and minimizing overvoltage along the winding

Investigation of wave processes of transformer windings as a distributed electric circuit

Relevance of the work is caused by the need of localize the defects of the transformer windings at diagnosis. It is possible to determine the spatial location of the defects along the transformer winding only if the transformer is considered as a system with distributed parameters. The main aim of the study is to present the transformer winding in the form of distributed electric circuit and to study the transients of such a circuit in two cases, at grounded winding end and at open winding end; to determine primary and steady voltage distribution along the transformer windings at the grounded winding end and open winding end; to search for possible ways to overcome the overvoltage arising along the transformer winding. The aim of the work is to obtain as well the transitional function caused by a unit action, in order to use subsequently the Duhamel integral which allows obtaining the transients in the transformer winding at an arbitrary external action. The methods used in the study are based on equations solution in partial derivatives using Laplace transform and transfer functions. It uses the initial conditions along the transformer winding and the boundary conditions at the ends of the windings. The final solution of partial differential equations is represented as a series, that is in the form of wave superposition with different frequencies. The results. The obtained transfer function of the voltage distribution along transformer windings allows determining: primary and steady stress distribution; over-voltage transient conditions; optimal values of distributed capacitance, and minimizing overvoltage along the winding

Calculation of equivalent distributed circuit parameters of transformer winding under the influence of skin effect

The relevance of the work. When producing, transporting and processing geo assets it is necessary to use various electrical equipment, the transformers are the main electricity distributors in this case. In some cases it is necessary to route the local electric network to recharge electric power devices for processing geo assets, the transformers are the main units of network as well. The efficiency of electric networks depends on the state of distribution devices - transformers, especially on the condition of the transformer windings. Therefore, the investigation and modeling of the transformer windings are the important tasks. When presenting the transformer windings as a distributed system we need to know the values of the equivalent circuit elements. In the presence of high-frequency currents in the circuit, the skin effect - the dependence of current on frequency - influences the value of equivalent circuit elements. The relevance of the research is caused by the fact that the pulse of nanosecond duration, which spectrum is filled with high frequencies, is used as the probe pulse for the transformer. The high frequency of the probe pulse provokes the skin effect, which, in its turn, contributes the frequency dependence to the value of the equivalent circuit parameters. The aim of the research is to introduce the probe nanosecond pulse as a superposition of harmonics with different frequencies; to calculate inductance and resistance for the transformer equivalent circuit for each harmonic with fixed frequency, which forms the pulse; to determine the values of inductance and capacitance using the calculation of electromagnet field and taking into account the skin effect as well. The energy of the calculated electromagnetic field allows retrieving the inductance and resistance values of the investigated system at the fixed frequency. To get the resulting equivalent circuit, which consists of combined circuits obtained for each frequency separately, using superposition principle; to obtain voltage and currents of equivalent circuit of transformer windings and compare them with the experimental data. Methods of investigation. The method of investigation is based on superposition principle, which means the expansion of arbitrary functions to the Fourier series. The authors use as well the numerical solution of partial differential equations - Helmholtz equation using the finite element method. To implement the method of finite elements the mathematical package COMSOL Multyhpisics was used. Results: The authors carried out frequency decomposition of nanosecond pulse and use it as a probe signal for transformer diagnosis. Based on the solution of Maxwell's equations the spatial distribution pattern of the magnetic vector potential was obtained. To extract the information on the inductance and resistance of the equivalent circuit of transformer winding the vector magnetic potential was used. The authors obtained the graphs of inductance and resistance of equivalent circuit for the transformer winding as a function of frequency. Graphical dependences are given for a variety of cross-sections of the transformer windings: round and rectangular shapes. Based on the obtained graphical dependencies the equivalent circuits of the transformer windings with the possible influence of the skin effect were built. The paper introduces the calculations of voltage and current of transformer windings using the equivalent circuit, as well as the results of comparison of theoretical calculations with the experimental ones

Calculation of equivalent distributed circuit parameters of transformer winding under the influence of skin effect

The relevance of the work. When producing, transporting and processing geo assets it is necessary to use various electrical equipment, the transformers are the main electricity distributors in this case. In some cases it is necessary to route the local electric network to recharge electric power devices for processing geo assets, the transformers are the main units of network as well. The efficiency of electric networks depends on the state of distribution devices - transformers, especially on the condition of the transformer windings. Therefore, the investigation and modeling of the transformer windings are the important tasks. When presenting the transformer windings as a distributed system we need to know the values of the equivalent circuit elements. In the presence of high-frequency currents in the circuit, the skin effect - the dependence of current on frequency - influences the value of equivalent circuit elements. The relevance of the research is caused by the fact that the pulse of nanosecond duration, which spectrum is filled with high frequencies, is used as the probe pulse for the transformer. The high frequency of the probe pulse provokes the skin effect, which, in its turn, contributes the frequency dependence to the value of the equivalent circuit parameters. The aim of the research is to introduce the probe nanosecond pulse as a superposition of harmonics with different frequencies; to calculate inductance and resistance for the transformer equivalent circuit for each harmonic with fixed frequency, which forms the pulse; to determine the values of inductance and capacitance using the calculation of electromagnet field and taking into account the skin effect as well. The energy of the calculated electromagnetic field allows retrieving the inductance and resistance values of the investigated system at the fixed frequency. To get the resulting equivalent circuit, which consists of combined circuits obtained for each frequency separately, using superposition principle; to obtain voltage and currents of equivalent circuit of transformer windings and compare them with the experimental data. Methods of investigation. The method of investigation is based on superposition principle, which means the expansion of arbitrary functions to the Fourier series. The authors use as well the numerical solution of partial differential equations - Helmholtz equation using the finite element method. To implement the method of finite elements the mathematical package COMSOL Multyhpisics was used. Results: The authors carried out frequency decomposition of nanosecond pulse and use it as a probe signal for transformer diagnosis. Based on the solution of Maxwell's equations the spatial distribution pattern of the magnetic vector potential was obtained. To extract the information on the inductance and resistance of the equivalent circuit of transformer winding the vector magnetic potential was used. The authors obtained the graphs of inductance and resistance of equivalent circuit for the transformer winding as a function of frequency. Graphical dependences are given for a variety of cross-sections of the transformer windings: round and rectangular shapes. Based on the obtained graphical dependencies the equivalent circuits of the transformer windings with the possible influence of the skin effect were built. The paper introduces the calculations of voltage and current of transformer windings using the equivalent circuit, as well as the results of comparison of theoretical calculations with the experimental ones

Holomorphic embedding as analytical technique for calculating electric grids of oil and gas deposits

Для того чтобы обеспечить требуемый уровень надежного функционирования объектов нефтедобычи, необходимо обращать внимание на условия эксплуатации энергосистемы. Это важно, если принять во внимание, что для добычи нефти и газа необходимы достаточно мощные источники энергии, которые существенно влияют на режим работы электросети. При этом актуальным является задача расчета установившихся режимов электрической сети, питающей объекты нефтедобычи. Расчеты установившихся режимов имеют большое практическое значение для обеспечения эффективного и безопасного управления режимами работы нефтяных и газовых предприятий, являются важными при проектировании электрических сетей, питающих объекты нефтегазовых предприятий. Однако применение классических итерационных методов расчета установившихся режимов, таких как метод Гаусса-Зайделя и Ньютона-Рафсона, не всегда позволяет найти правильное решение системы нелинейных уравнений, описывающих установившиеся режимы работы сети, так как сходимость данных методов зависит от начальных приближений. В работе предлагается аналитический метод расчета установившихся режимов электрических сетей, описываемых нелинейными уравнениями. Метод основан на аппроксимации Паде и методе возмущений. Приводятся преимущества метода перед известным методом итераций Гаусса-Зайделя и Ньютона-Рафсона. Приводятся примеры решения задач электроэнергетических сетей, и обсуждаются недостатки предлагаемого метода. Рассматриваются проблемы устойчивости. Цель: применить аналитический метод голоморфного погружения для расчета двух и трехузловой энергетической схемы; сравнить возможности метода с другими альтернативными методами; исследовать ограничения метода голоморфного погружения и показать область его работы. Методы: разложение Тейлора, аналитическое продолжение, решения алгебраических уравнений рекуррентным методом, бесконечные дроби. Результаты. Рассмотрены примеры использования метода голоморфного погружения для двух и трех PQ узловых схем, и показаны недостатки метода голоморфного погружения. Проведено сравнение метода голоморфного погружения с альтернативными методами. Выводы. Аналитический метод голоморфного погружения обладает рядом преимуществ: физической наглядностью, простотой алгоритмической реализации, заключающейся в рекуррентных соотношениях для коэффициентов разложения искомой функции в ряд Тейлора. Разложенная в ряд функция является голоморфной, что позволяет осуществлять ее аналитическое продолжение и получить желаемую точность решения.In order to ensure the required level of reliable operation of oil production, it is necessary to pay attention to the operating conditions of the power system. This is important if we take into account that for oil and gas extraction the powerful sources of energy that significantly affect the mode of operation of the power grid are needed. In this case, the urgent task is to calculate the steady-state modes of the oil production electric network. The calculations of the established modes are of great practical importance to ensure efficient and safe management of the operating modes of oil and gas enterprises, and are important in the design of electrical networks for oil and gas enterprises. However, the application of classical iterative methods for calculating steady-state regimes, such as the Gauss-Seidel and Newton-Raphson methods, does not always allow finding the right solution, since the convergence of these methods depends on initial conditions. The method is based on the Padé approximation and the perturbation method. The paper demonstrates the disadvantages and the advantages of the proposed method over the well-known Gauss-Seidel and Newton-Raphson iteration method and the examples of solving the problems of electric power chains. The problems of sustainability are considered. The aim of the research is to apply the analytical method of holomorphic embedding to calculate two and three nodal energy schemes; compare the capabilities of the method with other alternative methods; investigate the limitations of the holomorphic embedding method and show the area of its work. Methods: Taylor expansion, analytic continuation, solving algebraic equations by the recurrent method, infinite fractions. Results. The authors gave the examples of using the holomorphic embedding method for two and three PQ nodal circuits, and showed the shortcomings of the holomorphic embedding method. The holomorphic embedding method is compared with alternative methods. Conclusions. The analytical method of holomorphic embedding has several advantages: physical visibility; the simplicity of the algorithmic implementation consisting in recurrence relations for the coefficients of the expansion of the desired function in a Taylor series. The function laid out in a series is holomorphic, which allows analytic continuation of a function to obtain the desired accuracy of solution

Holomorphic embedding as analytical technique for calculating electric grids of oil and gas deposits

Для того чтобы обеспечить требуемый уровень надежного функционирования объектов нефтедобычи, необходимо обращать внимание на условия эксплуатации энергосистемы. Это важно, если принять во внимание, что для добычи нефти и газа необходимы достаточно мощные источники энергии, которые существенно влияют на режим работы электросети. При этом актуальным является задача расчета установившихся режимов электрической сети, питающей объекты нефтедобычи. Расчеты установившихся режимов имеют большое практическое значение для обеспечения эффективного и безопасного управления режимами работы нефтяных и газовых предприятий, являются важными при проектировании электрических сетей, питающих объекты нефтегазовых предприятий. Однако применение классических итерационных методов расчета установившихся режимов, таких как метод Гаусса-Зайделя и Ньютона-Рафсона, не всегда позволяет найти правильное решение системы нелинейных уравнений, описывающих установившиеся режимы работы сети, так как сходимость данных методов зависит от начальных приближений. В работе предлагается аналитический метод расчета установившихся режимов электрических сетей, описываемых нелинейными уравнениями. Метод основан на аппроксимации Паде и методе возмущений. Приводятся преимущества метода перед известным методом итераций Гаусса-Зайделя и Ньютона-Рафсона. Приводятся примеры решения задач электроэнергетических сетей, и обсуждаются недостатки предлагаемого метода. Рассматриваются проблемы устойчивости. Цель: применить аналитический метод голоморфного погружения для расчета двух и трехузловой энергетической схемы; сравнить возможности метода с другими альтернативными методами; исследовать ограничения метода голоморфного погружения и показать область его работы. Методы: разложение Тейлора, аналитическое продолжение, решения алгебраических уравнений рекуррентным методом, бесконечные дроби. Результаты. Рассмотрены примеры использования метода голоморфного погружения для двух и трех PQ узловых схем, и показаны недостатки метода голоморфного погружения. Проведено сравнение метода голоморфного погружения с альтернативными методами. Выводы. Аналитический метод голоморфного погружения обладает рядом преимуществ: физической наглядностью, простотой алгоритмической реализации, заключающейся в рекуррентных соотношениях для коэффициентов разложения искомой функции в ряд Тейлора. Разложенная в ряд функция является голоморфной, что позволяет осуществлять ее аналитическое продолжение и получить желаемую точность решения.In order to ensure the required level of reliable operation of oil production, it is necessary to pay attention to the operating conditions of the power system. This is important if we take into account that for oil and gas extraction the powerful sources of energy that significantly affect the mode of operation of the power grid are needed. In this case, the urgent task is to calculate the steady-state modes of the oil production electric network. The calculations of the established modes are of great practical importance to ensure efficient and safe management of the operating modes of oil and gas enterprises, and are important in the design of electrical networks for oil and gas enterprises. However, the application of classical iterative methods for calculating steady-state regimes, such as the Gauss-Seidel and Newton-Raphson methods, does not always allow finding the right solution, since the convergence of these methods depends on initial conditions. The method is based on the Padé approximation and the perturbation method. The paper demonstrates the disadvantages and the advantages of the proposed method over the well-known Gauss-Seidel and Newton-Raphson iteration method and the examples of solving the problems of electric power chains. The problems of sustainability are considered. The aim of the research is to apply the analytical method of holomorphic embedding to calculate two and three nodal energy schemes; compare the capabilities of the method with other alternative methods; investigate the limitations of the holomorphic embedding method and show the area of its work. Methods: Taylor expansion, analytic continuation, solving algebraic equations by the recurrent method, infinite fractions. Results. The authors gave the examples of using the holomorphic embedding method for two and three PQ nodal circuits, and showed the shortcomings of the holomorphic embedding method. The holomorphic embedding method is compared with alternative methods. Conclusions. The analytical method of holomorphic embedding has several advantages: physical visibility; the simplicity of the algorithmic implementation consisting in recurrence relations for the coefficients of the expansion of the desired function in a Taylor series. The function laid out in a series is holomorphic, which allows analytic continuation of a function to obtain the desired accuracy of solution