CN114242425B - Hysteresis loss solving method for traction transformer considering iron core magnetic circuit grading - Google Patents
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Abstract
The invention discloses a method for solving hysteresis loss of a traction transformer considering iron core magnetic circuit grading, which is characterized in that a silicon steel sheet electromagnetic induction equation based on a nonlinear medium relation is constructed, the characteristic of negligible non-same-frequency physical quantity power integral is introduced, and a simplified calculation model suitable for the hysteresis loss of the iron core of the traction transformer for magnetic core magnetic circuit grading is provided aiming at the characteristic of different magnetic effects of magnetic circuits of iron core stages. The hysteresis loss evaluation method has the beneficial effects that the hysteresis loss evaluation method which is more in line with the physical properties and the operation conditions of materials is provided, and necessary data guarantee can be provided for the production optimization design and the service performance evaluation of the traction transformer.
Description
Technical Field
The invention belongs to the field of electromagnetic analysis and numerical calculation of electrical equipment, and particularly relates to a hysteresis loss solving method of a traction transformer considering iron core magnetic circuit grading.
Background
The traction transformer is used as key equipment in a traction power supply system, has the operation characteristics of large short-time impact load and long dead time, and has important engineering value for the evaluation and optimization research of the iron core energy consumption. The hysteresis loss is taken as an important component of the core energy consumption, and the method for solving the hysteresis loss of the traction transformer, which is accurate enough, has urgent engineering significance for further researching the core loss of the traction transformer.
The closed path that magnetic flux passes through in permanent magnets, ferromagnetic materials, and electromagnets is called the magnetic circuit, and the primary purpose of magnetic circuit analysis is to determine the relationship between the exciting magnetomotive force and the magnetic flux it produces. Due to the winding process of the transformer, the geometrical dimensions of iron cores of all levels of the traction transformer are different, so that the magnetic circuit magnetic resistances of all levels of the iron cores are different, and the magnetic field intensity and the magnetic flux density of all levels of the iron cores of the transformer are unevenly distributed. In the traditional calculation formula, the hysteresis loss calculation usually regards the iron core as a uniform whole, the numerical value is proportional to the average magnetic field intensity and the average magnetic flux density, the formula cannot explain the uneven magnetic field distribution caused by magnetic circuit grading, the calculation error is larger, the hysteresis loss at a certain point of the iron core cannot be accurately described, and the requirement of traction transformer development on higher loss calculation precision cannot be met. Therefore, it is important to provide a hysteresis loss calculation formula considering the magnetic circuit classification of the traction transformer.
Disclosure of Invention
The invention aims to provide a traction transformer hysteresis loss solving method considering iron core magnetic circuit grading, which is realized by the following technical means:
1) Due to the existence of power electronic equipment and the nonlinear characteristics of the iron core, the exciting current has obvious low harmonic components, and the exciting current function meets the Dirichlet full condition, and is subjected to Fourier transformation for analyzing the harmonic characteristics, and the expansion is as follows:
exciting current I h Is decomposed into DC components
And fundamental and each subharmonic a orthogonal to each other n cosnωt and b n sinnωt. Since the power supply system current has no direct current component and has periodic property, the above formula can be simplified as follows in one period of exciting current:
in which I n Characterised by the amplitudes of the fundamental and each subharmonic of the excitation current, I n The numerical value is obtained according to the Fourier decomposition property:
because the power grid is a balanced three-phase system, even harmonics are offset in the balanced three-phase system, and the even harmonics of exciting current can be ignored approximately, so that the expression is simplified as follows:
in the formula, k is {0,1,2,3 and … }, since the harmonic amplitude is inversely proportional to the harmonic frequency, the higher harmonic amplitude is smaller, and only the fundamental wave and third harmonic effect are considered in calculation, the expression is further simplified to obtain:
I h (t)≈I 1 cosωt+I 3 cos3ωt
considering that the hysteresis loss calculation needs to integrate the product of the magnetic field intensity and the magnetic flux density, the magnetic field intensity frequency is equal to the exciting current frequency, the magnetic flux density frequency is equal to the exciting voltage frequency, the exciting voltage is constant at standard power frequency, and the product integration of the physical quantities of different frequencies is equal to zero, so that the fundamental component of the exciting current for determining the magnetic field intensity can be considered in the hysteresis loss calculation, and the exciting current expression is further simplified into:
I h (t)≈I 1 cosωt
wherein I is h (t) represents the excitation current of the traction transformer, I 1 、I n The fundamental component amplitude and the n-order harmonic component amplitude of exciting current after Fourier decomposition are represented respectively, and omega is angular frequency, which satisfies the following conditions: ω=2pi f, f is excitation frequency, t is time;
2) Because the silicon steel sheet adopted by the traction transformer iron core is cold-rolled orientation type, the silicon steel sheet is consistent with the direction with optimal magnetic conductivity in the core column, the iron yoke and the corner in the winding process, and the full current law can be realized
Scalar to->
Wherein H is the magnetic field intensity, N is the number of turns of the coil, and L is the length of the magnetic circuit where the geometric center of the cross section of the traction transformer core is located.
Meanwhile, considering that the magnetic path lengths of geometric centers of cross sections of all levels of the traction transformer iron core are different, calculating the magnetic field intensities of all levels of the iron core respectively, and substituting the excitation current expression into the magnetic field intensity expression of all levels of the traction transformer iron core to obtain the magnetic field intensity expression of all levels of the traction transformer iron core:
wherein H is i (t) represents the magnetic field intensity of the ith stage of the traction transformer core, a and b respectively represent the magnetic path length and the magnetic path width of the traction transformer core, R i Represents the radius of the arc section of the ith magnetic circuit;
3) The electromagnetic induction electromotive force formula of the coil is as follows
In E 1 Is the effective value of the primary side induced voltage of the transformer, phi m Represents a maximum magnetic flux; due to lower primary side voltage drop of the transformer, E is present 1 And (3) approximately equal to U, wherein U is an effective value of the excitation voltage of the transformer. The two formulas are combined, and the relationship between the magnetic induction intensity and the magnetic flux is +.>
Substituting to obtain:
wherein B is m Represents the maximum value of the magnetic flux density of the traction transformer iron core, S represents the cross section area of a magnetic circuit, and w and d respectively represent the widths of all stages of the traction transformer iron core and the thickness of the silicon steel sheet;
4) Since magnetomotive force f=ni of magnetic circuits of all stages of the iron core of the traction transformer is equal, the cold-rolled oriented silicon steel sheet has unidirectional magnetic permeability, insulating layers exist among all stages of the iron core, and at the layer of the magnetic circuits, all stages of the iron core can be regarded as being connected in parallel in topology. Parallel magnetic circuit, magnetic flux and magnetic resistance R m Inversely proportional, the reluctance expression is:
mu is the magnetic permeability of the iron core material of the transformer, and as the materials of all levels of magnetic circuits are the same and the cross sectional areas are equal, the magnetic fluxes of all levels of the iron core can be regarded as being inversely proportional to the length of the magnetic circuit where the geometric center of the iron core is positioned, namely:
in B of i (t) is the magnetic flux density of each stage of the iron core, R 1 Representing the radius of the stage 1 magnetic circuit arc. The primary magnetic flux is considered to lag the exciting current phase due to the shortest magnetic circuit and the largest magnetic flux of the first-stage magnetic circuit
The first order magnetic circuit flux density can be expressed as:
further, substituting the first-stage magnetic circuit magnetic flux density expression into each-stage magnetic circuit magnetic flux density relation to obtain each-stage magnetic flux density expression of the traction transformer:
the magnetic flux density is only a fundamental frequency component because the magnetic flux density is determined by exciting voltage, and the transformer exciting voltage is generally power frequency voltage;
5) Substituting the above formula into the definition of the hysteresis loss of the electromagnetism to obtain the calculation formula of the average hysteresis loss P of the traction transformer considering the grading of the iron core magnetic circuit in the high humidity environment:
the invention has the beneficial effects that a hysteresis loss calculation mode of the traction transformer which is more in line with material physical properties and operation conditions and considers magnetic circuit grading is provided, and necessary data guarantee can be provided for production optimization design and service performance evaluation of the traction transformer.
Drawings
Fig. 1 is a schematic diagram of the magnetic circuit classification of the traction transformer core according to the present invention.
Fig. 2 is a parallel topology of the stages of the traction transformer core of the present invention.
Detailed Description
The following describes the implementation procedure of the present invention in further detail with reference to the accompanying drawings. Due to the existence of power electronic equipment and the nonlinear characteristics of the iron core, the exciting current has obvious low harmonic components, and the exciting current function meets the Dirichlet full condition, and is subjected to Fourier transformation for analyzing the harmonic characteristics, and the expansion is as follows:
exciting current I h Is decomposed into DC components
And fundamental and each subharmonic a orthogonal to each other n cosnωt and b n sinnωt. Since the power supply system current has no direct current component and has periodic property, the above formula can be simplified as follows in one period of exciting current:
in which I n Characterised by the amplitudes of the fundamental and each subharmonic of the excitation current, I n The numerical value is obtained according to the Fourier decomposition property:
because the power grid is a balanced three-phase system, even harmonics are offset in the balanced three-phase system, and the even harmonics of exciting current can be ignored approximately, so that the expression is simplified as follows:
in the formula, k is {0,1,2,3 and … }, since the harmonic amplitude is inversely proportional to the harmonic frequency, the higher harmonic amplitude is smaller, and only the fundamental wave and third harmonic effect are considered in calculation, the expression is further simplified to obtain:
I h (t)≈I 1 cosωt+I 3 cos3ωt
considering that the hysteresis loss calculation needs to integrate the product of the magnetic field intensity and the magnetic flux density, the magnetic field intensity frequency is equal to the exciting current frequency, the magnetic flux density frequency is equal to the exciting voltage frequency, the exciting voltage is constant at standard power frequency, and the product integration of the physical quantities of different frequencies is equal to zero, so that the fundamental wave component of the exciting current for determining the magnetic field intensity can be considered in the hysteresis loss calculation, and the exciting current expression is further simplified into I h (t)≈I 1 cosωt
Wherein I is h (t) represents the excitation current of the traction transformer, I 1 、I n The fundamental component amplitude and the n-order harmonic component amplitude of exciting current after Fourier decomposition are represented respectively, and omega is angular frequency, which satisfies the following conditions: ω=2pi f, f is the excitation frequency, and t is time.
Because the silicon steel sheet adopted by the traction transformer iron core is cold-rolled orientation type, the silicon steel sheet is consistent with the direction with optimal magnetic conductivity in the core column, the iron yoke and the corner in the winding process, and the full current law can be realized
Scalar to->
Wherein H is the magnetic field intensity, N is the number of turns of the coil, and L is the length of the magnetic circuit where the geometric center of the cross section of the traction transformer core is located.
Meanwhile, the magnetic field intensity of each stage of the iron core is calculated respectively by considering the fact that the magnetic path lengths of the geometric centers of the cross sections of each stage of the iron core of the traction transformer are different.
Fig. 1 is a schematic diagram of the magnetic circuit of the traction transformer core according to the present invention, in which 8 stages are taken as an example, and it can be seen from the figure that each stage of magnetic circuit is composed of four rectangles and four quarter circles, and the magnetic circuit length can be regarded as the sum of twice the magnetic circuit length of the core, twice the magnetic circuit width of the core and the circumference, so that the excitation current expression is substituted to obtain the expression of the magnetic field intensity of each stage of the traction transformer core:
wherein H is i (t) represents the magnetic field intensity of the ith stage of the traction transformer core, a and b respectively represent the magnetic path length and the magnetic path width of the traction transformer core, R i Representing the radius of the arc of the ith stage of magnetic circuit.
The electromagnetic induction electromotive force formula of the coil is as follows
In E 1 Is the effective value of the primary side induced voltage of the transformer, phi m Represents a maximum magnetic flux; due to lower primary side voltage drop of the transformer, E is present 1 And (3) approximately equal to U, wherein U is an effective value of the excitation voltage of the transformer. The two formulas are combined, and the relationship between the magnetic induction intensity and the magnetic flux is +.>
Substituting to obtain:
wherein B is m Represents the maximum value of the magnetic flux density of the traction transformer core, S represents the cross-sectional area of a magnetic circuit, and w and d represent the widths of each stage of the traction transformer core and the thickness of the silicon steel sheet respectively.
Since magnetomotive force f=ni of magnetic circuits of all stages of the iron core of the traction transformer is equal, the cold-rolled oriented silicon steel sheet has unidirectional magnetic permeability, insulating layers exist among all stages of the iron core, and at the layer of the magnetic circuits, all stages of the iron core can be regarded as being connected in parallel in topology. FIG. 2 is a parallel topology of the stages of the traction transformer core according to the present invention, from which the parallel magnetic circuit, flux and reluctance R are known m Inversely proportional, the reluctance expression is:
mu is the magnetic permeability of the iron core material of the transformer, and as the materials of all levels of magnetic circuits are the same and the cross sectional areas are equal, the magnetic fluxes of all levels of the iron core can be regarded as being inversely proportional to the length of the magnetic circuit where the geometric center of the iron core is positioned, namely:
in B of i (t) is the magnetic flux density of each stage of the iron core, R 1 Representing the radius of the stage 1 magnetic circuit arc. The primary magnetic flux is considered to lag the exciting current phase due to the shortest magnetic circuit and the largest magnetic flux of the first-stage magnetic circuit
The first order magnetic circuit flux density can be expressed as:
further, substituting the first-stage magnetic circuit magnetic flux density expression into each-stage magnetic circuit magnetic flux density relation to obtain each-stage magnetic flux density expression of the traction transformer:
the magnetic flux density is only a fundamental frequency component because the magnetic flux density is determined by the exciting voltage, which is typically the mains frequency voltage.
Substituting the above formula into the definition of hysteresis loss in electromagnetism, the calculation formula of the average hysteresis loss P of the traction transformer considering the grading of the iron core magnetic circuit is obtained:
the invention has the beneficial effects that a hysteresis loss calculation mode of the traction transformer which is more in line with material physical properties and operation conditions and considers magnetic circuit grading is provided, and necessary data guarantee can be provided for production optimization design and service performance evaluation of the traction transformer.
Claims (1)
1. A method for solving hysteresis loss of a traction transformer considering the grading of an iron core magnetic circuit is characterized in that the iron core is made of high-permeability cold-rolled grain-oriented silicon steel sheets, and comprises the following steps:
1) According to the hysteresis calculation principle, the exciting current for determining the magnetic field strength only considers the fundamental component and the periodic integral of the non-common frequency physical quantity is zero, so that an exciting current expression is obtained:
wherein I is h (t) is excitation current, I 1 、I n The fundamental component amplitude and the n-time component amplitude of exciting current after Fourier decomposition are respectively, ω is angular frequency, and the method meets the following conditions: ω=2pi f, f is excitation frequency, t is time;
2) According to the full current law, considering the magnetic circuit classification, the magnetic field intensity of each stage of the traction transformer iron core is expressed as:
wherein H is i (t) represents the magnetic field intensity of the ith stage of the traction transformer core, N represents the total number of turns of the exciting winding coil, L is the length of a magnetic circuit where the geometric center of the cross section of the traction transformer core is located, a and b respectively represent the length and the width of the magnetic circuit of the traction transformer core, R i Represents the radius of the arc section of the ith magnetic circuit;
3) According to the coil induced electromotive force formula and the approximate equal relation between the primary side induced electromotive force of the transformer and the exciting voltage, the expression of the maximum value of the magnetic flux density in the traction transformer core is obtained:
wherein B is m Represents the maximum value of the magnetic flux density phi of the traction transformer core m Representing the maximum value of magnetic flux, S representing the cross section area of a magnetic circuit, U representing the effective value of exciting voltage of the transformer, w and d respectively representing the widths of all stages of the traction transformer iron core and the thickness of the silicon steel sheet;
4) According to the parallel topology and the magnetic resistance proportional relation of each level of the magnetic circuit, the magnetic flux density calculation formula of each level of the iron core is obtained:
in B of i (t) is the magnetic flux density of each stage of the iron core, R 1 Represents the radius of the arc section of the level 1 magnetic circuit,
retarding the exciting current phase for the main magnetic flux;
5) According to (2) (4) and the definition of hysteresis loss by electromagnetism, the calculation formula of the average hysteresis loss P of the traction transformer considering the grading of the iron core magnetic circuit is obtained:
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