Reception WRC for publication in EBS SPbGETI "LETI". Mathematical model of synchronous and asynchronous engines "Maps and schemes in the Foundation of the Presidential Library"

The scope of adjustable alternating current electric drives in our country and abroad is largely expanding. Special position occupies a synchronous electric drive of powerful career excavators, which are used to compensate for reactive power. However, their compensating ability is not used enough due to the lack of clear recommendations on excitation regimes

Solovyov D. B.

The scope of adjustable alternating current electric drives in our country and abroad is largely expanding. Special position occupies a synchronous electric drive of powerful career excavators, which are used to compensate for reactive power. However, their compensating ability is not used sufficiently due to the lack of clear recommendations on excitation modes. In this regard, the task is to determine the highest modes of excitation of synchronous motors in terms of compensation of reactive power, taking into account the ability to regulate the voltage. Efficient use of the compensating ability of a synchronous motor depends on a large number of factors ( technical parameters Engine, load on the shaft, voltages on the clips, the loss of active power on the production of reactive, etc.). Increasing the loading of the synchronous engine by reactive power causes an increase in engine losses, which adversely affects its performance. At the same time, an increase in the reactive power given to the synchronous motor will help reduce the loss of energy and in the career power supply system. Under this criterion, the optimality of the load of the synchronous engine for reactive power is the minimum of the costs of generation and the distribution of reactive power in the career power supply system.

The study of the excitation mode of the synchronous motor is not mediocre on the career, it is not always possible for technical reasons and due to limited funding research. Therefore, it seems necessary description of the synchronous excavator engine with various mathematical methods. Engine like an object automatic control It is a complex dynamic structure described by the system of nonlinear differential equations of high order. Simplified linearized options were used in the management tasks of any synchronous machine. dynamic modelsthat have given only an approximate idea of \u200b\u200bthe behavior of the car. Development of a mathematical description of electromagnetic and electromechanical processes in a synchronous electric drive that take into account the actual nature of nonlinear processes in a synchronous motor, as well as the use of such a structure of a mathematical description when developing adjustable synchronous electric drives, in which the model career excavator It would be comfortable and visual, it seems relevant.

The issue of modeling has always been paid great attention, methods are widely known: analogue of modeling, creating a physical model, digital-analog modeling. However, analog modeling is limited by the accuracy of the calculations and the cost of recruited elements. The physical model most accurately describes the behavior of the real object. But the physical model does not allow changing the model parameters and the creation of the model itself is very expensive.

The most efficient solution is the MATLAB mathematical calculation system, Simulink package. The MATLAB system eliminates all the disadvantages of the above methods. The software implementation of the mathematical model has already been made in this system. synchronous machine.

MATLAB laboratory virtual virtual instruments development medium is an applied graphic programming environment used as a standard tool for objects of objects, analyzing their behavior and subsequent control. Below is an example of equations for modeling a synchronous motor according to the complete equations of the Gorev Park, recorded in streams for the substitution scheme with one damper circuit.

With this software You can simulate all possible processes in the synchronous engine, in full-time situations. In fig. Figure 1 shows a synchronous motor start modes that obtained when solving the equation of the Gorely Park for the synchronous machine.

An example of implementing these equations is presented on a block diagram where variables are initialized, parameters are set and integrating. The results of the start mode are shown on a virtual oscilloscope.

Fig. 1 Example of captured characteristics from a virtual oscilloscope.

As can be seen, at the start of the SD, an impact moment of 4.0 OU and current 6.5 o Е.Е.Е.Е. Starting time is about 0.4 sec. Well visible current oscillations and moments caused by non-symmetry of the rotor.

However, the use of data of ready-made models makes it difficult to study the intermediate parameters of the synchronous machine modes due to the inability to change the parameters of the finished model scheme, the impossibility of changing the structure and parameters of the network and the excitation system other than the received, simultaneously consideration of the generator and motor regime, which is necessary when modeling the start or When resetting the load. In addition, primitive saturation accounting is applied in finished models - the saturation along the "Q" axis is not taken into account. At the same time, due to the expansion of the application of a synchronous motor and an increase in the requirements for their operation, refined models are required. That is, if it is not necessary to get a specific behavior of the model (simulated synchronous engine), depending on the mining and geological and other factors affecting the operation of the excavator, then it is necessary to solve the system of the Park-Growing Park equations in the Matlab package, which allows to eliminate these disadvantages.

LITERATURE

1. Kigel G. A., Trifonov V. D., Chirva V. X. Optimization of excitation modes of synchronous motors on iron ore mining and processing enterprises. - Mining magazine, 1981, NS7, p. 107-110.

2. Nainankov I. P. Automated design. - M.: Nedra, 2000, 188 pp.

Nishovsky Yu.N., Nikolaichuk N.A, Minute E.V., Popov A.N.

Divated hydroda of mineral resources of the Far Eastern shelf

To ensure growing demands in mineral raw materials, as well as in building materials It is required to pay increasingly active exploration and development of mineral resources of the shelf seas.

In addition to the fields of Titano-Magnetitovyk, the sands in the southern part of the Japanese Sea are revealed in passes of gold and construction sands. At the same time, the tapes obtained from the enrichment of gold deposits can also be used as construction sands.

The gold-axis column fields include the placer of a number of bays of Primorsky Krai. The productive reservoir occurs at a depth, ranging from the shore to a depth of 20m, with a capacity of 0.5 to 4.5 m. From above, the reservoir is blocked by sandy-happier sediments with alcohol and clay with a power of 2 to 17 m. In addition to the content of gold in the sands are Ilmenite 73 g / t, Titan-magnetite 8.7 g / t and ruby.

In the coastal shelf of the Far East seas, there are also significant reserves of mineral raw materials, the development of which under the seabed at the present stage requires the creation of new techniques and the use of environmentally friendly technologies. The most explored reserves from the number of minerals are coal layers of previously operating mines, gold-bearing, titanium-magnetite and casstetic sands, as well as deposits of other minerals.

These preliminary geological surveys of the most characteristic deposits in the early years are shown in the table.

Deployed mineral deposits on the shelf seas of the Far East can be divided into: a) airborne-clay and pending sediments (place of metal-containing and building sands, materials and sewers); b) Located on: a significant blowout from the bottom under the thickness breed (coal layers, various ores and minerals).

An analysis of the development of placer deposits shows that none of the technical solutions (both domestic and foreign development) cannot be used without any environmental damage.

The experience of developing non-ferrous metals, diamonds, golden sands and other minerals abroad indicates the overwhelming use of all sorts of drags and dredgers leading to a widespread violation of the seabed and environmental condition of the environment.

According to the Institute of TsNIIsvetmet, economics and information on the development of non-ferrous deposits of metals and diamonds are used abroad more than 170 drags. At the same time, it is mainly used by dummy (75%) with a bucket capacity up to 850 liters and a drop of drawing up to 45 m, less often - suction drags and dredgers.

Dashboards on the seabed are conducted in Thailand, New Zealand, Indonesia, Singapore, England, USA, Australia, Africa and other countries. Metal production technology in this way creates an extremely strong violation of the seabed. The foregoing leads to the need to create new technologies to significantly reduce the impact on the environment or completely eliminate it.

Known technical solutions for the underwater removal of titanium-magnetite sands, based on non-traditional methods of underwater development and removal of bottom sediments based on the use of the energy of pulsating streams and the effect of the magnetic field of permanent magnets.

The proposed development technologies though reduce the harmful effect on the environment, but do not retain the bottom surface from violations.

With the use of other methods of working with cutting and without cutting off the landfill from the sea, the reassembly from harmful impurities of the enrichment of placing the planers in the place of their natural occurrence also does not solve the problem of environmental recovery of biological resources.

Construction and principle of synchronous engine with permanent magnets

Construction of a synchronous engine with permanent magnets

Ohm's law is expressed by the following formula:

where - electric current, and;

Electrical voltage, in;

Active resistance chain, Ohm.

Matrix of resistance

, (1.2)

where is the resistance of the contour, and;

The matrix.

The Law of Kirchhoff is expressed by the following formula:

Principle of formation of a rotating electromagnetic field

Figure 1.1 - Engine design

Engine design (Figure 1.1) consists of two main parts.

Figure 1.2 - Engine Operation Principle

The principle of operation of the engine (Figure 1.2) is as follows.

Mathematical description of the synchronous engine with permanent magnets

General methods for obtaining a mathematical description of electric motors

Mathematical model of a synchronous engine with permanent magnets in general

Table 1 - Engine Parameters

Mode parameters (Table 2) correspond to the engine parameters (Table 1).

The paper presents the basics of designing such systems.

Works provide programs to automate calculations.

Source mathematical description of a two-phase synchronous engine with permanent magnets

Detailed engine design is shown in applications A and B.

Mathematical model of a two-phase synchronous engine with permanent magnets

4 Mathematical model of a three-phase synchronous engine with permanent magnets

4.1 Source mathematical description of a three-phase synchronous engine with permanent magnets

4.2 Mathematical model of a three-phase synchronous engine with permanent magnets

List of sources used

1 Automated design of automatic control systems / ed. V. V. Solodovnikova. - M.: Mechanical Engineering, 1990. - 332 p.

2 Melsa, J. L. Programs to help learn the theory of linear control systems: per. from English / J. L. Mesa, Art. K. Jones. - M.: Mechanical Engineering, 1981. - 200 p.

3 Problem of safety of autonomous spacecraft: monograph / S. A. Bronov, M. A. Volovik, E. N. Golovovkin, G. D. Kesselman, E. N. Korchagin, B. P. Sustin. - Krasnoyarsk: NII IPU, 2000. - 285 p. - ISBN 5-93182-018-3.

4 Brons, S. A. Precision positional electric drives with dual power engines: author. dis. ... dock. tehn Sciences: 05.09.03 [Text]. - Krasnoyarsk, 1999. - 40 s.

5 A. s. 1524153 USSR, MKA 4 H02P7 / 46. A method for regulating the angular position of the rotor of the dual power engine / S. A. Bronov (USSR). - № 4230014 / 24-07; Declared 14.04.1987; Publ. 11/23/1989, Bul. № 43.

6 Mathematical description of synchronous motors with permanent magnets based on their experimental characteristics / S. A. Bronov, E. E. Noscova, E. M. Kurbatov, S. V. Yakunenko // Informatics and control systems: Interunion. Sat Scientific Tr. - Krasnoyarsk: NII IPU, 2001. - Vol. 6. - P. 51-57.

7 Brons, S. A. A set of programs for the study of an electric drive system based on the inductor dual power engine (Description of the structure and algorithms) / S. A. Bronov, V. I. Panteleev. - Krasnoyarsk: Crapp, 1985. - 61 p. - Manuscript dep. In Informelectro 28.04.86, No. 362-fl.

To describe the AC electrical machines, various modifications of differential equations systems are used, the type of which depends on the choice of the type of variables (phase, transformed), directions of velauses of variables, the source mode (motor, generator) and a number of other factors. In addition, the type of equations depends on the assumptions adopted when it is derived.

The art of mathematical modeling is to make many methods that can be applied and factors affecting processes, choose such that ensure the required accuracy and ease of performing the task.

As a rule, when modeling the AC electric machine, the real machine is replaced by an idealized, having four basic differences from the real: 1) the absence of saturation of magnetic circuits; 2) lack of losses in steel and turning out current in windings; 3) the sinusoidal distribution in the space of the curves of the magnetizing forces and magnetic induction; 4) the independence of inductive scattering resistance from the position of the rotor and on the current in the windings. These assumptions greatly simplify the mathematical description of the electrical machines.

Since the axis of the stator windings and the rotor rotor of the synchronous machine during rotation is moved mutually, the magnetic conductivity for the winding streams becomes a variable. As a result, mutual inductance and inductance of windings change periodically. Therefore, when modeling processes in a simultaneous machine using equations in phase variables, phase variables U., I.,  Prepaid periodic values \u200b\u200bthat significantly makes it difficult to fix and analyze modeling results and complicates the implementation of the model on the computer.

More simple and convenient for modeling are the so-called transformed equations of the mountain park, which are obtained from equations in phase values \u200b\u200bby special linear transformations. The essence of these transformations can be understood when considering Figure 1.

Figure 1. Picture vector I. and his projections on the axis a., b., c. and axis d., q.

In this figure, two coordinate axes are depicted: one symmetrical three-line fixed ( a., b., c.) And the other ( d., q., 0 ) - orthogonal, rotating at the angular speed of the rotor . Also in Figure 1 shows the instantaneous values \u200b\u200bof phase currents in the form of vectors I. a. , I. b. , I. c. . If you geometrically add the instantaneous values \u200b\u200bof phase currents, then the vector will be I.which will rotate with the orthogonal axis system d., q.. This vector is called the current current vector. Similar depicting vectors can be obtained for variables U.,  .

If we design the depicting vectors on the axis d., q.The corresponding longitudinal and transverse components of the depicting vectors are new variables that are replaced by phase variables, voltages and streams.

While phase values \u200b\u200bin the steady mode periodically change, depicting vectors will be permanent and fixed relative to the axes d., q. And, therefore, they will be constant and their components I. d. and I. q. , U. d. and U. q. ,  d. and  q. .

Thus, as a result of linear transformations, the AC electric machine is represented as a two-phase with perpendicularly located windows over the axes d., q.that eliminates mutually induction between them.

The negative factor in the transformed equations is that they describe the processes in the machine through fictitious, and not through actual values. However, if you return to the above Figure 1, you can establish that the reverse transformation from fictitious values \u200b\u200bto phase does not represent a special complexity: sufficiently according to the components, for example, current I. d. and I. q. Calculate the value of the image vector

and design it on any fixed phase axis, taking into account the angular velocity of rotation of the orthogonal system of the axes d., q. relatively fixed (Figure 1). We get:

,

where  0 is the value of the initial phase of the phase current at T \u003d 0.

System of the synchronous generator equations (Park-Gorev), recorded in relative units in the axes d.- q., rigidly related to its rotor, has the following form:

;

;

;

;

;

;(1)

;

;

;

;

;

,

where  d,  q,  d,  q - the streaming of stator and sedative windings along the longitudinal and transverse axes (D and Q);  f, i f, u f - streaming, current and excitation winding voltage; i d, i q, i d, i q - states of stator and sedative windings along axes d and q; R is the active resistance of the stator; x d, x q, x d, x q - reactive resistance of stator and sedative windings along axes D and Q; x F - reactive resistance of the excitation winding; X AD, X AQ - resistance of the immigration of the stator along the axes D and Q; u d, u q - voltage over the axes D and Q; T DO - the time constant of the excitation winding; T d, t q - constant time of sedative windings along the axes d and q; T j - inertial time constant diesel generator; S is a relative change in the rotor of the generator rotor (sliding); M kr, m SG - torque of the drive motor and the electromagnetic moment of the generator.

In equations (1), all essential electromagnetic and mechanical processes in a simultaneous machine are taken into account, both sedative windings, so they can be called complete equations. However, in accordance with the previously admitted assumption, the angular speed of rotation of the rotor of the SG in the study of electromagnetic (rapid) processes is accepted unchanged. It is also permissible to take into account the sedative winding only on the longitudinal axis "D". Taking into account these assumptions, the system of equations (1) will take the following form:

;

;

;

; (2)

;

;

;

;

.

As can be seen from the system (2), the number of variables in the system of equations is greater than the number of equations, which does not allow for simulating to use this system in direct form.

More convenient and efficient is the transformed system of equations (2), which has the following form:

;

;

;

;

;

; (3)

;

;

;

;

.

The synchronous motor is a three-phase electrical machine. This circumstance complicates the mathematical description of dynamic processes, since with an increase in the number of phases, the number of electrical equilibrium equations increases, and electromagnetic connections are complicated. Therefore, we will reduce the analysis of the processes in a three-phase machine to analyze the same processes in the equivalent two-phase model of this machine.

In the theory of electrical machines, it is proved that any multiphase electric machine with n.phase stator winding and m.-Fased rotor winding under the condition of the equal impedance of the phases of the stator (rotor) in the dynamics can be represented by a two-phase model. The possibility of such a replacement creates the conditions for obtaining a generalized mathematical description of the processes of electromechanical energy transformation in a rotating electrical machine based on the consideration of an idealized two-phase electromechanical converter. Such a converter was called a generalized electric machine (OEM).

Generalized electric machine.

OEM allows you to present a dynamics real Engine, both in fixed and in rotating coordinate systems. The last idea makes it possible to significantly simplify the equation of the status of the engine and the synthesis of control for it.

We introduce variables for OEM. An affiliation of a variable of one or another winding is determined by the indices that are indicated by the axis associated with the windings of the generalized machine, indicating the ratio to Stator 1 or Rothor 2, as shown in Fig. 3.2. In this figure, the coordinate system is rigidly associated with a fixed stator, designated, with a rotating rotor -, - an electrical angle of rotation.

Fig. 3.2. Scheme of a generalized bipolar machine

The dynamics of the generalized machine describe four equations of electrical equilibrium in the circuits of its windings and one equation of electromechanical energy conversion, which expresses the electromagnetic moment of the machine as the function of the electrical and mechanical coordinates of the system.

Kirchhoff equations, expressed through streaming, have

(3.1)

where and is the active resistance of the phase of the stator and the active impedance of the phase of the rotor of the machine, respectively.

The streaming of each winding in general is determined by the resulting effect of all windows of the machine

(3.2)

In the system of equations (3.2) for its own and mutual inductors, the windings adopted the same designation with a substitution index, the first part of which , indicates which winding makes EMF, and the second - What kind of winding it is created. For example, the own inductance of the phase of the stator; - mutual inductance between the phase of the stator and the phase of the rotor, etc.

The designations and indices adopted in the system (3.2) provide the same type of all equations, which makes it possible to resort to a generalized form of recording this system convenient for further

(3.3)

When operating the OEM, the mutual position of the stator and rotor windings changes, so the own and mutual inductance of the windings in the general case are the function of the electrical angle of rotation of the rotor. For a symmetric non-operating machine, the own inductance of stator and rotor windings does not depend on the position of the rotor

and the mutual inductance between the stator or rotor windings is zero

since the magnetic axes of these windings are shifted in space relative to each other at an angle. The mutual inductance of the stator and rotor windings passes a full cycle of changes when rotating the rotor at an angle, therefore, taking into account the figures taken in Fig. 2.1 directions of currents and angle of rotor rotation can be recorded

(3.6)

where is the mutual inductance of stator and rotor windings or when, i.e. With coordinate systems coincide and. Taking into account (3.3), the equation of electrical equilibrium (3.1) can be represented as

, (3.7)

where relations are determined by relations (3.4) - (3.6). The differential equation of electromechanical transformation of energy will be obtained by using the formula

where is the rotor rotation angle,

where is the number of pairs of poles.

Substituting equations (3.4) - (3.6), (3.9) in (3.8), we obtain an expression for the electromagnetic moment of OEM

. (3.10)

Two-phase immovable synchronous machine with permanent magnets.

Consider electrical engine In Emur. It is an innovable synchronous machine with permanent magnets, as it has a large number of pairs of poles. In this machine, the magnets can be replaced by an equivalent winding of excitation without loss () connected to the current source and creating magnetorevizable force (Fig. 3.3.).

Fig.3.3. Scheme for switching on the synchronous motor (s) and its two-phase model In the axes (b)

Such a replacement allows you to represent the equilibrium equations by analogy with the equations of a conventional synchronous machine, so, putting and in equations (3.1), (3.2) and (3.10), we have

(3.11)

(3.12)

Denote where - the streaming to a couple of poles. We will replace (3.9) in equations (3.11) - (3.13), as well as subjectedly (3.12) and substitute to equation (3.11). Receive

(3.14)

where - the angular speed of the engine; - the number of turns of the stator winding; - Magnetic stream of one turn.

Thus, equations (3.14), (3.15) form a system of equations of a two-phase immocent synchronous machine with permanent magnets.

Linear transformations of the equations of the generalized electrical machine.

The advantage of obtained in paragraph 2.2. The mathematical description of the processes of electromechanical energy transformation is that as independent variables, the actual currents of the summary of the generalized machine and the actual voltages of their power are used. Such a description of the dynamics of the system gives a direct idea of \u200b\u200bphysical processes in the system, however, is difficult to analyze.

When solving many problems, a significant simplification of the mathematical description of the processes of electromechanical energy transformation is achieved by linear transformations of the original system of equations, while replacing real variables with new variables, provided that the adequacy of the mathematical description is preserved by the physical object. The condition of adequacy is usually formulated as a requirement of power invariance when converting equations. The newly administered variables can be either valid or complex values \u200b\u200bassociated with real variables of conversion formulas, the type of which should ensure the condition of the power invariance.

The purpose of the transformation is always one or another simplification of the original mathematical description of dynamic processes: elimination of the dependence of inductors and mutual inductance of windings from the rotor rotation angle, the ability to operate in non-sinusoidally changing variables, but their amplitudes, etc.

First, consider valid transformations that allow you to move from physical variables defined by coordinate systems that are rigidly associated with the stator and with a rotor with a good variable corresponding to the coordinate system u., v.rotating in space with arbitrary speed. For a formal solution of the problem, we will present every real winding variable - voltage, current, streaming - in the form of a vector, the direction of which is rigidly associated with the coordinate axis corresponding to this winding, and the module varies in time in accordance with the changes in the variable depicted.

Fig. 3.4. Variable generalized machine in various coordinate systems

In fig. 3.4 Winding variables (currents and voltages) are indicated in a general form of a letter with the corresponding index reflecting the affiliation of a given variable to a certain axis of coordinates, and the mutual position is currently in the current time of the axes, rigidly related to the stator, axes d, Q,rigidly related to the rotor, and an arbitrary system of orthogonal coordinates u, V.Rotating relatively fixed stator at speeds. Reminted as defined real variables in the axes (stator) and d, Q. (rotor) corresponding to them new variables in the coordinate system u, V. You can determine as the amount of projections of real variables on new axes.

For greater clarity, the graphic constructions necessary to obtain the transformation formulas are presented in Fig. 3.4A and 3.4B for the stator and the rotor separately. In fig. 3.4A are the axes associated with the windings of a fixed stator, and the axis u, V.rotated relative to the stator at the angle . The components of the vector are defined as projections of vectors and on the axis u., components - as the projections of the same vectors on the axis v.Having summarizing the projections on the axes, we obtain a direct conversion formula for stator variables in the following form

(3.16)

Similar constructions for rotary variables are presented in Fig. 3.4b. Shows fixed axes, rotated relative to them to the angle of the axis. d, Q,machines related to rotor rotated relative to rotary axes d.and q.at the angle of axis and, V,rotating at speed and coinciding at every moment of time with axes and, V.in fig. 3.4A. Comparing fig. 3.4B Fig. 3.4A, you can establish that the projections of the vectors and on and, V.similar to the projections of stator variables, but in the function of the angle. Therefore, for rotary variables, the conversion formulas are

(3.17)

Fig. 3.5. Transformation of variable generalized two-phase electrical machine

To explain the geometrical meaning of linear transformations carried out by formulas (3.16) and (3.17), in Fig. 3.5 Additional construction. They show that the conversion is based on the representation of the variable generalized machine in the form of vectors and. Both actual variables and, and converted and are projections on the appropriate axes of the same result vectors. Similar ratios are valid for rotary variables.

If you need to go from transformed variables to the actual variable of the generalized machine Reverse conversion formulas are used. They can be obtained by constructions made in Fig. 3.5A and 3.5Banalogic constructions in Fig. 3.4A and 3.4B

(3.18)

Formulas Direct (3.16), (3.17) and reverse (3.18) conversion coordinates of the generalized machine are used in the synthesis of controls for a synchronous motor.

We convert equations (3.14) to new system coordinates. To do this, we substitute the expressions of the variables (3.18) in equations (3.14), we get

(3.19)

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