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0Розглянута схема здшснюе регулювання по вщхиленню, принцип якого полягае в наступному:
- на регулятор подаеться сигнал ввдхилення ре-гульовано! величини вiд заданого значения.
- регулятор по сигналу вщхилення змiнюе ре-гулююча напруга U таким чином, щоб зменшити вь дхилення Д h0 (Т).
Необхвдно ввдзначити, що зазначений регулятор змiнюе регулюючу дiю незалежно ввд причини, що викликала помилку регулювання i оцiнюе поми-лку за рiвнем постшно! складово! в комплексному сигнала Тому додатково введена корекщя по пос-тiйнiй складово! у виглядi U0. Це постшна складова сигналу акселлерометра при нормальних значеннях температури.
Повнiстю помилка регулювання в данш сис-темi принципово не усуваеться повшстю, так як ре-гулюючий вплив формуеться тiльки помилкою. По-милка буде тим ближче наближатися до нульового значення чим точнiше буде реалiзована схема порi-вняння.
Схема працюе в такий спосiб: при шдвищенш температури середовища змiнюються геометричнi розмiри п'езоелемента 5 (дивись рис.6.), Що веде до зростання значення постшно! складово! у вихщ-ному сигналь Вихвдний сигнал надходить на один вхщ суматора на пряму, а на шший вх1д суматора через фазовий фшьтр, який затримуе його на 180°.
Шсля сумматора постшна складова потрапляе на штегратор з часом iнтегрування порiвнянним зi швидк1стю змiни температури. Щдсилювач потуж-ностi управляе п'езоелементом 4 прикладаючи до його електродiв напругу U, полярнiсть яко! проти-лежно спрямована полярностi поляризацп. П'езо-елемент-привщ вiдпрацьовуючи керуючий вплив прагне виконати вимоги рiвняння (1).
Запропонований метод корекци адитивно! складово! викликано! змшами температури е актуа-льним. Вш дозволяе пiдвищити точнiсть вимiрю-вань виконуваних п'езоелектричними акселерометрами i розширити !х сферу застосування щодо ви-мог до температури середовища при проведенш вимiрювань прискорень та вiбрацiй.
Список лiтератури
1. Антоненко А.М. Влияние доменной структуры на электромеханические свойства сегнетоке-рамики ЦТС и МНВТ/ А.М.Антоненко, А.Ю.Куд-зин, М.Г.Гавшин//. Физика твердого тела, 1997, том 39, №5.-М.: ФТИ им. А.Ф.Иоффе.
2. Гориш А.В. Пьезоэлектрическое приборостроение / А.В. Гориш, В.П. Дудкевич, М.Ф. Куприянов и др. - Т.1. Физика сегнетоэлектрической керамики. - М.: Издат. предпр. ред. жур. ^«Радиотехника», 1999. - 368 с.
ТРЁХФАЗНАЯ ИНДУКЦИОННАЯ МАШИНА ТРЕХЗОННОЙ КОНСТРУКЦИИ ДЛЯ МГД-
ПЕРЕМЕШИВАТЕЛЯ
Тяпин А.А.
Аспирант, ФГАОУВО Сибирский Федеральный Университет
THREE-PHASE INDUCTION MACHINE OF A THREE-ZONE DESIGN FOR MHD STIRRER
Tyapin A.A.
Postgraduate student, Siberian Federal University, Krasnoyarsk, Russia
Аннотация
Линейно-индукционные МГД-машины с низкочастотным инвертором электропитания образуют комплекс электромагнитного перемешивания жидкого алюминия в плавильных печах. В статье рассматриваются результаты расчета линейной индукционной машины с классификационными характеристиками, характерными для трехзонных индукторов продольного магнитного поля с питанием от трехфазного преобразователя частоты. Для расчета рабочих параметров линейной индукционной МГД-машины использована нелинейная многофазная модель магнитной цепи. В результате итерационного расчета получено распределение интегральных магнитных потоков в зубцовой зоне плоского индуктора построены векторные диаграммы параметров электромагнитного режима. Исследование показало возможность оптимизации режима малополюсной индукционной машины для получения наилучшего распределения токов в обмотках и эквивалентной линейной токовой нагрузки. По результатам анализа сформулированы основные задачи и последовательность этапов их решения при разработке энергоэффективных индукционных МГД-машин.
Abstract
Linear induction MHD machines with a low-frequency power supply inverter form a complex of electromagnetic stirring of liquid aluminum in smelting furnaces. The article discusses the results of the calculation of a linear induction machine with classification characteristics characteristic of three-zone inductors of a longitudinal magnetic field with three-phase power. To calculate the operating parameters of a linear induction MHD machine, a nonlinear multiphase model of a magnetic circuit was used. As a result of an iterative calculation, the distribution of the integral magnetic fluxes in the tooth zone of a flat inductor is obtained, and vector diagrams of electromagnetic regime parameters are constructed. The study showed the possibility and identified the main directions for optimizing the mode of a low-pole induction machine to obtain the best distribution of currents in the windings
and the equivalent linear current load. According to the results of the analysis, the main tasks and the sequence of stages of their solution were formulated when developing energy-efficient induction MHD machines.
Ключевые слова: Индукционная МГД-машина, индуктор продольного магнитного поля, электромагнитный перемешиватель алюминия, многофазная модель магнитной цепи, векторная диаграмма магнитных потоков, трехфазная система электропитания, преобразователь частоты.
Keywords: Induction MHD machine, longitudinal magnetic field inductor, electromagnetic stirrer, multiphase magnetic circuit model, magnetic fluxes vector diagram, three-phase power supply system, frequency converter.
For mixing metal melts in furnaces, linear induction machines of transverse and longitudinal magnetic fields are used. The cost price of each technical solution along with the technological and energy efficiency of induction machines and power sources is a decisive factor in the decision to modernize production or to develop design solutions for new construction of smelting furnaces. As induction machines for stirring aluminum alloys in mixers and furnaces, in addition to transverse field inductors, high-tech shortened inductors of the longitudinal field are used [1, p.12]. Among the simplest flat induction MHD machines, two constructive solutions can be distinguished, which determine the type of machine, according to the number of force inducing windings (inducing zones).
These design features appropriately characterize the polarity of the inductor and the magnitude of the synchronous velocity of the traveling magnetic field in the melt. The following designations are used as design parameters in the description:
2p is the number of poles of the inductor;
Z is the number of teeth of the core;
q is the number of grooves of the core per pole and phase;
a is the phase zone of the inductor;
m is the number of phases of a multiphase winding inductor;
A working gap.
The classical induction MHD machine of a longitudinal magnetic field can have three or four windings (a three-zone or four-zone inductor). In addition, the power supply of induction machines can be provided in a three-phase or two-phase version. Thus, when developing inductors and evaluating their effectiveness, four main options for constructing shortened low-pole induction machines (IMs) of a longitudinal magnetic field should be considered.
1. Three-zone inductor with a three-phase power supply.
2p = 1, Z = 4, q = 1, m = 3, a = 60°.
2. Three-zone inductor with a two-phase power supply.
2p = 3/2, Z = 4, q = 1, m = 2, a = 90°.
3. Four-zone inductor with a three-phase power supply.
2p = 4/3, Z = 5, q = 1, m = 3, a = 60°.
4. Four-zone inductor with two-phase power supply.
2p = 2, Z = 5, q = 1, m = 2, a = 90°.
The traditional approach to the development of linear induction machines for metallurgical purposes contains a number of stages, among which one can single out engineering calculation, development of winding switching circuits, mathematical modeling and optimization of the electromagnetic field, modeling of melt hydrodynamics and thermal calculation, design, manufacturing and testing. Each stage is implemented in a specific sequence using the appropriate mathematical, software, hardware, technical and other equipment [2, p.77]. Already on the basis of engineering calculation, the basic characteristics of the machine are determined, which are then refined by the results of mathematical modeling of the field [3, p.27].
However, despite careful calculation and the transition to the use of powerful software environments, some integral parameters, such as magnetic flux of the dentate zone, are not readily available for perception, evaluation, and timely adjustment [4, p.65]. Therefore, it seems appropriate, prior to the stage of mathematical modeling of the electromagnetic field, to refer to the calculation and modeling of the induction device by the methods of the theory of circuits. By reviewing the magnetic and electrical equivalent circuit of the inductor and generating the appropriate mathematical models, you can get additional information about the modes, which will allow you to more consciously refer to the evaluation of the differential parameters of a linear induction machine and outline ways to achieve the best result.
This article discusses some results of the calculation of the electromagnetic mode of a linear induction machine, which has the classification features of a three-zone inductor of a longitudinal magnetic field with a three-phase power supply [5, p.87]. A sketch of the construction of a flat induction MHD machine in a three-zone version is shown in Fig. 1. Three groups of windings 1, denoted w1, w2, w3, with two-acting disk coils of insulated copper bus are located on a steel laminated core 2 (sections S1 and S2) and are separated by teeth 3, which act as magnetic field concentrators. In the inductor circuit implemented AXZCBY, similar to the scheme AXYBCZ. The difference in the coding of the circuit characterizes the inverted phase of the inductor [5, 6], and the practical difference between the modes concerns only a change in the order of the phase alternation.
Fig. 1. Sketch of a linear induction machine design
For metallurgical inductors of medium size, the characteristic operating parameters of the inverter can be as follows. Line voltages up to 0.4 kV are ensured by smooth acceleration of the frequency converter at steady-state currents up to 300 Amps with asymmetry up to 50%. A characteristic feature of induction machines of the longitudinal field should be considered extremely low values of the natural power factor [6]. Most induction machines have a significant inductance. Moreover, with a peak power consumption of 250 kVA inductor of average size, the active power consumed for losses and cravings, often does not exceed 45-50 kW. When cos^ = 0.05-0.1, the inductive load of the frequency converter in the distribution network of an industrial enterprise serves as a source of a huge number of higher harmonics in the spectrum up to the fiftieth
and even higher [3, 5, 6]. Thus, it becomes quite obvious that it is necessary to use an electromagnetic melting mixing separating transformer acting as a filter at the input of the complex. In this case, the frequency converter serves as a means of compensating reactive power.
The simplest options for including inductive windings of a linear MHD machine in a triangle and in a star are shown in fig. 2, a, b. In a typical variant of connecting windings to a three-phase inverter, induction machines operate at a frequency of about 1 Hz [6, 8]. When an inductor is connected to a frequency inverter with different phase designations, the letters UWV, the IM switching circuit can be labeled in another UuwWVv coding corresponding to the reverse phase sequence.
The specified information should be taken into account when completing the equipment with inverters of foreign production. To visualize the electromagnetic mode of MI during the development of induction equipment, vector diagrams are used, which can be considered a practical way of representing the operational parameters in phase coordinates [7, 10]. For currents and voltages of MI, phase coordinates can be obtained automatically by mathematical modeling of the known dependence u(t) = Ldi(t)/dt, which corresponds to Ohm's law for an inductive element.
In the phase domain, data series with a description of operating parameters can be decomposed into trajectories (Fig. 3, a) and portraits (Fig. 3, b) of electromag-
netic mode parameters, reflecting the relationship between the operating characteristic and its derivative, according to an expression with two coordinates:
x = f (x,t) . (1)
An example of combining a vector diagram of currents in the windings of a three-zone three-phase linear induction machine and the phase coordinates of the steady state is shown in Fig. 3.
It is easy to understand from the diagram why the three-phase system of currents shown in fig. 1 inductor turns out to be unbalanced. Judging by the structure of the phase curves (Fig. 3, a), the three-phase power flux propagates from the source to the receiver along a cyclic trajectory with a period T = 1s.
At the same time, in any cross-section, the instantaneous voltages and currents are not only non-zero, but the instantaneous power is also obviously not zero. The results of the study of the phase characteristics of electrical devices are described in more detail in [7, 9]. In addition to visualizing calculations using vector diagrams, the phase space method can be successfully applied to reflect the spectral composition of higher voltage harmonics in inductors, as well as higher current harmonics due to saturation, which are usually referred to zero sequence currents [4, 5, 15].
However, the approach to the study of the phase characteristics of induction machines requires the construction of special circuit or mathematical models. The main components of the models, as a rule, are the standard four-pole elements of the theory of circuits, called controlled sources [7, 9].
Below, in fig. 5 shows a fragment of a detailed circuit model generated using the mentioned element base. Moreover, for constructing a high-order volume model, the principle of analogy of electric and magnetic circuits was used, and in the circuit model, magnetic
analogs of controlled electric voltage sources controlled by current were used. The calculation of the electromagnetic state of IM for steady state is usually carried out for an equivalent sinusoid, and using the results of engineering calculations, the electrical and magnetic states can be estimated separately by creating and comparing the equivalent circuits of electrical and magnetic circuits. As a rule, in the presence of automated computing systems, numerical calculation is used, characterized by high efficiency [9, 12, 16]. To assess the nature of the distribution of magnetic fluxes, you can make a simplified magnetic circuit of the device, without specifying the magnetic poles. A sketch of the cross section of the induction machine with an approximate distribution of the working integral magnetic fluxes of the dentate zone is shown in Fig. 4, a. The vector diagram of the magnetizing forces of the windings is shown in Fig. 4, b. The indicated values of the magnetomotive forces vectors Fi, F2, F3 can be optimized to obtain both uniform and extremely non-uniform distribution of dentate flows in a circular raster of field distribution of a linear machine.
a b
Fig. 4. Magnetic flux distribution and diagram of the magnetizing forces
Integral values of tooth flows ®i, ®2, ®3, ®4, a nonlinear multiphase magnetic circuit presented in shown in fig. 4, b can be calculated using the model of fig. 5. When generating the model, the controlled
sources of magnetizing forces &1F1, k2F2, k3F3, are used as magnetizing forces, the mode of which is set by the equivalent sinusoid over the set of harmonics depending on the degree of saturation of the steel. Moreover, the equivalent vector of magnetomotive forces k2F2, phase B, is inverted by an angle n to obtain a given phase shift a = +n/3 with respect to the first phase. The complex of the equivalent sine wave k3F3 is set with the natural phase +2n/3 in direct sequence.
Air gaps and molten metal in the model are replaced by constant resistive magnetic resistances, which are calculated from the actual geometry of the inductor. The sections of the magnetic circuit and the teeth are replaced by nonlinear resistive resistances with a tabular description of the web-ampere characteristics of structural steel, with linear interpolation between uniquely defined points [10, 17].
A fragment of the spatial circuit model of a nonlinear multiphase magnetic circuit with windings w1, w2, w3 is shown in fig. 5.
Fig. 5. Fragment of the circuit model of a nonlinear magnetic circuit IM
The model of a multiphase magnetic circuit is constructed for a three-phase three-zone inductor of the simplest configuration (Fig. 1), designed for furnaces up to 30 tons with an air gap of about 500 mm, taking into account the dimensions and characteristics of the yoke and the teeth IM.
The order of construction of a detailed nonlinear model of a three-phase magnetic circuit and the determination of its parameters are considered in [8, 9, 12]. It is possible to briefly discuss the elemental base of the computational model, in which the controlled sources of the magnetic field kF serve as the key link. Their use was made possible thanks to the principle of the formal analogy of electric and magnetic circuits. As a prototype for sources of magnetomotive forces controlled by magnetic flux, an analogue is used - a voltage source controlled by a current, the matrix description of which
allows integrating it into software environments of widely used circuit simulation simulators.
The use of authoring software [7], effectively used for modeling electronic devices, allowed us to construct high-order nonlinear mathematical models into which circuit components were imported (Fig. 6). The integration of the matrix description of the controlled source (CS) into the algorithm of the modified node analysis [8, 9] is performed automatically according to the detailed description of the circuit model (Fig. 5) in the ASCII code, similar to some versions of the Ansys software environment [16].
Naturally, for each new circuit model of an induction installation, it is necessary to form an appropriate computational project [15], grouping the necessary decision and auxiliary modules in a separate crucial module, connecting the necessary libraries and using computer-aided computing, for example Fortran [12].
b c
Fig. 6 Models of Managed Sources
d
a
As a means of designing circuit models of magnetic circuits for induction devices, in addition to the resistive element RM [H-1], four active controlled ele-
ments FU, FO, OU, OO, are used, similar to the models of the standard element basis of electrical circuits. Four-pole primary links have the letter designation. FU - a source of magnetic voltage (Fig. 6, a), controlled by
a magnetic voltage; F® - a source of magnetic voltage (Fig. 6, b), controlled by a magnetic flux; ®U - source of magnetic flux (Fig. 6, c), controlled by magnetic voltage; ®® - a source of magnetic flux (Fig. 6, d), controlled by the flow.
Unfortunately, neither the described elements nor their electrical analogs have found wide application in practice, although it is their capabilities that provide direct access to the numerical values of the mode parameters when modeling the behavior of induction installations in phase space [7, 9]. Apparently the reason for their relatively rare use was the incomplete evidence of mathematical models of controlled sources obtained using component equations and some of the complexity of their conjugation with the traditional method of calculating electrical and magnetic circuits using topolog-ical equations (Kirchhoffs laws).
Below is an example of a matrix mathematical model of a controlled source of magnetomotive force in the basis of extended nodal equations. The structure of the description of the controlled source has the following form: [F® <num> g, h, r, n, k]. The indices i, j denote the input nodes of the CS, r, n are the output nodes of the CS, k is the transmission coefficient of the controlled source.
The mode of transmission of a controlled source is determined by the expression:
«2« = Vr(i) - Fn(i) = k • 41(t) , (2)
where: m(t) - is the output scalar magnetic voltage of the source [A], 0\(t) - (T) is the instantaneous value of the input magnetic flux of the source [Vb], V(t) - is the scalar magnetic potential of the node [A], k = Rm -is the transfer coefficient source is a transient magnetic resistance.
For nonlinear resistive elements, the description of the tabular weber-ampere characteristics is provided, allowing only a unique representation of nonlinearity, for example, based on the main magnetization curve of electrical or structural steel [\5].
A description of the mathematical model used to build an algorithm for analyzing circuits with a controlled source F® is shown below. The input flow is directed from the node g to the node h, the input magnetic voltage is zero, the output magnetic voltage is directed from the node r to node n against the source, the output stream and the source of the magnetomotive force is directed from node n to node r.
The component equations of the controlled source of the magnetomotive force:
«2(0 = kR ■ 0l(t) , Vr(t) - Vn(t) = kR ■ 0\(t),
Vn (t) - V (t) + kR ■ 0(t) = 0, Vh(t ) - Vg(t) = o.
(3)
The system of extended nodal equations for a controlled source of the magnetomotive force F® is made up taking into account the control and controlled branches (3).
g h r n 01 02
g " 0 0 0 0 1 0 " \Vg(t ) 1 "0
h 0 0 0 0 -1 0 Vh(t ) 0
r 0 0 0 0 0 -1 Vr(t ) 0
n 0 0 0 0 0 1 Vn(t ) 0
0l -1 1 0 0 0 0 0(t) 0
02 0 0 -1 1 kR 0 02 (t )_ 0
(4)
The presented expression has no features [8, 9] and is automatically embedded in the general description of the circuit model (Fig. 5) generated before performing the computational procedures. It can be noted that behind each controlled source in a multiphase chain model there is a matrix stamp (4) of the sixth order and higher, therefore, the order of the resulting system of equations is quite high [\0]. However, for machine methods and analysis tools, the dimension of systems of equations is not decisive.
According to the results of numerical iterative calculations in tabular form, the integral values of the amplitudes of the magnetic fluxes in the teeth and the yoke
are obtained (Fig. 4, a). For a graphical representation, it is convenient to apply a vector mapping, when magnetic flux is combined with inductive magnetomotive forces in a single field, as shown in fig. 7
The distribution of the integral magnetic fluxes of the dentate zone turned out to be relatively uniform, although the vectors are different. It is characterized by the magnetomotive forces of the three-phase windings (Fig. 5) in the initial state kiFi, k2F2, k3F3. The vector diagram shows the possibility of regulating each magnetomotive force due to the magnitude of AFk.
In this way, it is relatively easy to manage the distribution of the magnetic flux of the dentate zone, changing not only the amplitudes, but also the values of 912, 923, 934. The target redistribution control function is formed at the optimization stage, according to a set of requirements for the distribution of the traction forces of the linear machine in the melt. In addition to the typical three-phase configuration of a three-zone induction machine, there is an additional constructive option to improve traction characteristics by expanding
the raster of the distribution of magnetic fluxes in the melt.
A means of improving performance, in addition to the main windings 1, are additional windings 4, located along the edges of the MI with an extended magnetic core 2 (Fig. 8). An example of a modified design of the linear induction MHD machine of the longitudinal magnetic field considered above is shown in Fig. 8.
Iu£u Ivg"v wjlw
Fig. 8. Sketch of the design of a modified three-zone linear IM
The main difference between the design in the presence of additional power windings w4 and w5 on the edges of the magnetic circuit. This solution slightly increases the longitudinal dimensions of the inductor and is only suitable in the absence of design limitations
of the furnace. The order and phasing of the winding connection shown in the figure determine the new spatial distribution of the magnetomobile forces of the windings w4, w5 according to the vector diagram shown in Fig. 9.
Fig. 9. The distribution of the magnetizing forces of the windings of a modified linear IM
To calculate the magnetic fluxes of the tooth zone of IM of the shown configuration in fig. 8, a new circuit model of a nonlinear magnetic circuit should be compiled by analogy with Fig. 5, perform iterative calculations, as described in (14, 16) and carry out parametric optimization of the force field, according to the specified requirements.
When developing equipment for the technology of electromagnetic mixing of melts [5, 6, 17], one has to solve a whole range of problems, from creating efficient inductors to building low-frequency inverters and coordinating their modes. Practical measures to develop power supplies for linear induction machines of different configurations should take into account not only the number of phases, but also the actual operating characteristics of the inductors [20, p.228].
And first of all it is necessary to take into account the extremely high asymmetry of currents in phases. It should be noted that the results presented here should be considered as a first approximation and formulation of one of the tasks, in the format of developing an induction MHD machine of the above configuration.
Conclusion. When building energy-efficient induction MHD machines, several interrelated problems should be solved. Evaluation of the effectiveness of the effect of inductors on the molten metal when changing the operating characteristics is the essence of the mag-netohydrodynamic problem. The study of the characteristics and features of the electromagnetic field of an induction machine, as well as the methods of controlling the redistribution of magnetic flux, relates to the field of mathematical modeling and optimization of the inductor magnetic system. Creating an effective winding switching circuit, controlling the number of poles and the speed of a traveling magnetic field should also be considered as a task in the field of research into flat induction machines of a longitudinal magnetic field. In addition, it should be understood that standard three-phase inverters of a rotating asynchronous electric drive are unsuitable for powering metallurgical equipment, the modes of which are sharply asymmetric and extreme. Therefore, when constructing complexes of various dimensions intended for electromagnetic stirring of the melt, it is necessary to create a series of economical and reliable power sources for induction machines, with a different number of phases and various circuitry for switching windings. Each of the designated tasks for the whole variety of designs of induction machines should be devoted to a separate study.
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