On Developing the Theory of Resonant Processes in Power Transformer Windings. Part 2. Frequency Responses of a Circuit with Four PI Sections

2021 ◽  
pp. 28-34
Author(s):  
Vasily S. LARIN ◽  
Keyword(s):  
Equivalent Circuit ◽  
Natural Frequencies ◽  
Power Transformer ◽  
Natural Oscillation ◽  
Frequency Responses ◽  
Voltage Resonance ◽  
A Chain ◽  
Oscillation Frequencies ◽  
Analytical Expressions ◽  
Resonant Processes

In the first part of the article, the results from theoretical studies of frequency responses in a homogeneous chain circuit containing two PI sections were presented, and conclusions about the voltage resonance occurrence conditions were drawn. A circuit with two PI sections has a single independent node and one natural frequency, whereas transformer windings are more complex oscillatory circuits and have a much larger number of natural oscillation frequencies. The second part of the article presents the results from studies of frequency responses in a homogeneous chain circuit containing four PI sections, three independent nodes, and three natural oscillation frequencies. Analytical expressions for the admittances of the equivalent circuit individual parts and the voltages at the intermediate nodes, as well as expressions for the natural frequencies are obtained. Using an analysis of the frequency dependences of the admittances of the equivalent circuit parts, the conditions and frequency ranges under which voltage resonance may occur at the first, second, and third natural frequencies, are shown. It has been demonstrated that for the considered resonant circuits there is a critical frequency above which the conditions for voltage resonance are not satisfied. Formulas for impedances and voltages at intermediate nodes of a chain circuit with an arbitrary number of PI sections are given.

On Developing the Theory of Resonant Processes in Power Transformer Windings. Part 1. Frequency Responses of a Circuit with Two PI Sections

2021 ◽  
pp. 49-55
Author(s):  
Vasily S. LARIN ◽  
Keyword(s):  
Equivalent Circuit ◽  
Power Transformer ◽  
Short Circuit ◽  
Natural Oscillation ◽  
Power Transformers ◽  
Equivalent Circuits ◽  
Electrical Networks ◽  
Frequency Responses ◽  
Voltage Resonance ◽  
Resonant Processes

The windings of power transformers are complex oscillatory circuits. Their natural oscillation frequencies range from a few to hundreds of kHz, and under unfavorable conditions they can coincide with the frequencies of transient voltage oscillations in electrical networks caused by switching operations and short circuit faults in cable lines. The development of electrical networks and the technical solutions that have come in wide use in recent years give rise to a situation in which the cases of damage to the insulation of power transformer windings resulting from resonant overvoltages are increasingly more frequently encountered during operation. An attempt is made in the article to develop the theory of resonant processes in the windings of power transformers. The frequency responses of voltages and currents in a simplified equivalent circuit of the power transformer winding are considered. Analytical expressions are obtained for the admittances of individual parts of equivalent circuits and the voltages at intermediate nodes, as well as expressions for the natural frequencies of the equivalent circuits considered. It is shown that the resonant growth of voltage in the transformer windings results from the voltage resonance caused by the presence of the winding’s own inductance and capacitance with respect to ground. By analyzing the frequency dependences of the admittances of equivalent circuit individual parts, the conditions and frequency ranges in which voltage resonance may occur are shown.

scholarly journals Analysis of the natural frequencies of oscillations of a planar truss with an arbitrary number of panels

Vestnik MGSU ◽  
2019 ◽  
pp. 284-292
Author(s):  
Mikhail N. Kirsanov ◽  
Dmitriy V. Tinkov
Keyword(s):  
Analytical Solutions ◽  
Natural Frequencies ◽  
Average Frequency ◽  
Natural Oscillation ◽  
Frequency Function ◽  
Compliance Matrix ◽  
Frequency Spectra ◽  
Alternative Approach ◽  
Inertia Properties ◽  
Oscillation Frequencies

Introduction. Analytical solutions for problems of structural mechanics are not only an alternative approach to solving problems of strength, reliability and dynamics of structures, but also the possibility for simple performance evaluations and optimization of structures. Frequency analysis of planar trusses, most often used in construction and engineering, is an important part of the study of structures. Objectives - development of a three-parameter induction algorithm for deriving the analytical dependence of the natural oscillation frequencies of the truss on the number of panels. Materials and methods. A flat, statically definable truss with one additional external link and double braces has been considered. The inertia properties of the truss are modeled by point masses located in the nodes of the lower straight truss belt. Each mass is assumed to have only one vertical degree of freedom. The stiffness of all truss rods is assumed to be the same. The task is to obtain analytical dependences of the oscillation frequencies of the proposed truss model on the number of panels. The derivation of the desired formulas is performed by the method of induction in three stages - according to the numbers of rows and columns of the compliance matrix, calculated using the Maxwell - Mohr formula and the number of panels. To find common members of the obtained sequences of coefficients, an apparatus was used to compile and solve the recurrent equations of the Maple computer mathematics system. The task of determining frequencies has been reduced to the eigenvalue problem of a bisymmetric matrix. Results. For the elements of the compliance matrix, general formulas have been found, according to which the frequency equations are compiled and solved. It is shown that in the frequency spectra of trusses with different numbers of panels there is always one common frequency (middle frequency) located in the middle of the spectrum. An expression is found for the maximum value of the average oscillation frequency as a function of the height of the truss. Conclusions. The proposed truss scheme, despite its external static indeterminacy and the lattice, which does not allow for the calculation of forces by such methods as the method of cutting nodes and the cross section method, allows analytical solutions for the natural frequencies of loads in the nodes. The obtained formulas have a rather simple form, and some general properties, such as frequency coincidences for different numbers of panels and the presence of an analytically calculated maximum of the average frequency function of the truss height, make this solution convenient for practical structural evaluations.

scholarly journals INVESTIGATIONS OF VIBRO-OSCILLATORY PROVESSES OF THE SUPPORTS OF OFFSHORE STATIONARY PLATFORMS UNDER CONDITIONS OF ALTERNATING VORTEX FORMATION CAUSED BY THE INFLUENCE OF SEA CURRENTS

Akustika ◽  
2019 ◽  
Vol 32 ◽  
pp. 262-266
Author(s):  
Ivan Starokon ◽  
Y.M. Ovsannikov ◽  
A.O. Golovachev
Keyword(s):  
Frequency Spectrum ◽  
Characteristic Frequency ◽  
Natural Frequencies ◽  
Vortex Formation ◽  
Natural Oscillation ◽  
Hydrodynamic Forces ◽  
Offshore Platforms ◽  
Wide Range ◽  
Sea Currents ◽  
Oscillation Frequencies

Offshore platforms are exposed to hydrodynamic forces of a non-stationary nature and a wide range of frequencies. At present, hydrodynamic forces arising under conditions of alternating vortex formation have been poorly studied. These vortices are caused by the sea currents flowing around the supports of the platform, and due to their periodic occurrence and disruption, they lead to oscillations of the supports of the offshore platforms. Of particular danger are the cases of resonance of the characteristic frequency spectrum of hydrodynamic forces from the "occurrence-stalling" of vortices and natural frequencies of elements of the platform supports, which can cause fatigue damages in the platform supports. The article developed a method for determining the natural oscillation frequencies of marine supports and described the conditions under which the occurrence of alternating vortex formation is possible.

On the problem of finding the parameters of elastic fastening of the edges of a rectangular plate

2006 ◽  
Vol 4 ◽  
pp. 11-22
Author(s):  
A.M. Akhtyamov ◽  
A.V. Muftakhov
Keyword(s):  
Rectangular Plate ◽  
Natural Frequencies ◽  
Natural Oscillation ◽  
The Other ◽  
Elastic Beams ◽  
Oscillation Frequencies

A rectangular plate is considered, in which two opposite edges are freely supported, while the other two are supported by elastic beams. In this paper, we prove the duality of the solution of the problem of finding the rigidities of elastic beams by all natural oscillation frequencies of a plate. It is also shown that if the rank of a certain matrix is equal to 9, then both solutions of the problem of finding the rigidities of elastic beams can be found from 9 natural frequencies.

Evaluation of the Power Transformer Magnetizing Inrush Currents

2020 ◽  
Vol 10 (10) ◽  
pp. 20-32
Author(s):  
Aleksey A. KUVSHINOV ◽  
◽  
Vera V. VAKHNINA ◽  
Aleksey N. CHERNENKO ◽  
◽  
...  
Keyword(s):  
Power Transformer ◽  
Geomagnetic Disturbance ◽  
Inrush Current ◽  
Geomagnetic Disturbances ◽  
Disturbance Intensity ◽  
Magnetizing Inrush Current ◽  
Inrush Currents ◽  
Instantaneous Values ◽  
The Mathematical Model ◽  
Analytical Expressions

The mathematical model of a shell-core power transformer’s magnetization branch is substantiated. By using the model, analytical expressions for the magnetizing current instantaneous values under the conditions of geomagnetic disturbances can be obtained. Quantitative assessments of the magnetizing inrush current amplitudes and durations versus the geomagnetic disturbance intensity are obtained. The dynamics of the power transformer magnetic system saturation transient and changes in the magnetization inrush current amplitudes and durations after a sudden occurrence of geomagnetic disturbances are shown. The error of estimating the magnetizing inrush current amplitudes under geomagnetic disturbances is determined based on comparison with experimental data.

Determination of Damping Factors Based on the Measured Frequency Responses of Power Transformer Windings. Part 2. Analysis of Measurement Results

2021 ◽  
Vol 2 (2) ◽  
pp. 22-28
Author(s):  
Vasily S. LARIN ◽  
◽  
Daniil A. MATVEEV ◽  
Keyword(s):  
Active Component ◽  
Power Transformer ◽  
Neutral Current ◽  
Response Analysis ◽  
Reactive Component ◽  
Input Admittance ◽  
Equivalent Scheme ◽  
Frequency Responses ◽  
Measurement Results ◽  
Transformer Winding

In the first part of the article, based on the results of theoretical studies performed for a simplified transformer winding equivalent scheme, it was shown that the damping factors can be estimated from the width of the resonant peaks of the frequency responses of the module and the reactive component of the voltage at the midpoint of the equivalent scheme, as well as the active component of the input admittance and neutral current of the considered resonant scheme. In this part of the article, the practical possibility of applying the obtained theoretical relations between the damping factors and the width of resonant peaks in relation to the frequency responses of power transformer windings is considered. The results of calculations of the damping factors at the two power transformers made by using the fitting of the free component of transient voltage and by determining the width of the resonance peaks of the active component of winding neutral current and the voltage transfer function, corresponding to intermediate points of the winding. It is shown that the evaluation of the values of the winding damping factors can be performed as a byproduct of transformer condition assessment by frequency response analysis (FRA).

An Approach on Tuning the Frequency of a Rotating Blade

2008 ◽  
Author(s):  
L. Duong ◽  
K. Kazerounian ◽  
K. D. Murphy
Keyword(s):  
Structural Integrity ◽  
Natural Frequencies ◽  
Vibration Mode ◽  
The Finite Element Method ◽  
Turbine Engine ◽  
Rotating Blade ◽  
Alternative Approach ◽  
Excited Vibration ◽  
Speed Range ◽  
Analytical Expressions

In a gas turbine engine, the forced vibration of a turbine blade under resonant conditions is undesirable and may lead to premature high cycle fatigue failure. From the aspect of structural integrity, this demonstrates that it is extremely important to tune the excited vibration mode out of the operating speed range. This leads to the question: Is it possible to perform structural perturbations, namely to the mass and stiffness, in such a way that only the eigenvalue of choice significantly changes — while causing little or no change in the other natural frequencies? This is focus of the present paper. Due to the complexity of the blade structure, it is difficult to obtain an analytical solution from the eigenvalue perturbation theory. Nevertheless, the derived analytical expressions provide guidance from which the finite element method may successfully be applied as an alternative approach. This tuning approach is validated experimentally.

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