Acceptance of the WRC for publication in the ebs spbgetu "leti". Mathematical model of a two-phase synchronous motor with permanent magnets "Maps and diagrams in the Presidential Library fund"

Design and principle of operation synchronous motor with permanent magnets

Permanent magnet synchronous motor design

Ohm's law is expressed by the following formula:

where is the electric current, A;

Electric voltage, V;

Active resistance of the circuit, Ohm.

Resistance matrix

, (1.2)

where is the resistance of the th circuit, A;

Matrix.

Kirchhoff's law is expressed by the following formula:

The principle of forming a rotating electromagnetic field

Figure 1.1 - Engine design

The engine design (Figure 1.1) consists of two main parts.

Figure 1.2 - The principle of operation of the engine

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

Mathematical description of a permanent magnet synchronous motor

General methods for obtaining a mathematical description of electric motors

General mathematical model of a permanent magnet synchronous motor

Table 1 - Engine parameters

The mode parameters (Table 2) correspond to the motor parameters (Table 1).

The paper outlines the basics of designing such systems.

The works contain programs for the automation of calculations.

Original mathematical description of a two-phase permanent magnet synchronous motor

The detailed design of the engine is given in Appendices A and B.

Mathematical model of a two-phase permanent magnet synchronous motor

4 Mathematical model of a three-phase permanent magnet synchronous motor

4.1 Initial mathematical description of a three-phase permanent magnet synchronous motor

4.2 Mathematical model of a three-phase permanent magnet synchronous motor

List of sources used

1 Computer-aided design of systems automatic control/ Ed. V.V. Solodovnikov. - M .: Mashinostroenie, 1990 .-- 332 p.

2 Mels, J.L. Programs to help students of the theory of linear control systems: trans. from English / J.L. Melsa, 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. Golovenkin, G. D. Kesselman, E. N. Korchagin, B. P. Soustin. - Krasnoyarsk: NII IPU, 2000 .-- 285 p. - ISBN 5-93182-018-3.

4 Bronov, S. A. Precision positional electric drives with dual power motors: Author. dis. ... doc. tech. Sciences: 05.09.03 [Text]. - Krasnoyarsk, 1999 .-- 40 p.

5 A. p. 1524153 USSR, MKI 4 H02P7 / 46. A method for regulating the angular position of the rotor of a dual power engine / S. A. Bronov (USSR). - No. 4230014 / 24-07; Stated 04/14/1987; Publ. 11/23/1989, Bul. No. 43.

6 Mathematical description of synchronous motors with permanent magnets on the basis of their experimental characteristics / S. A. Bronov, E. E. Noskova, E. M. Kurbatov, S. V. Yakunenko // Informatics and control systems: interuniversity. Sat. scientific. tr. - Krasnoyarsk: NII IPU, 2001. - Issue. 6. - S. 51-57.

7 Bronov, S. A. Complex of programs for the study of electric drive systems based on a double-power inductor motor (description of the structure and algorithms) / S. A. Bronov, V. I. Panteleev. - Krasnoyarsk: KrPI, 1985 .-- 61 p. - Manuscript dep. in INFORMELEKTRO 04/28/86, No. 362-et.

To describe AC ​​electric machines, various modifications of systems of differential equations are used, the form of which depends on the choice of the type of variables (phase, transformed), the direction of the vectors of variables, the initial mode (motor, generator) and a number of other factors. In addition, the form of the equations depends on the assumptions made in their derivation.

The art of mathematical modeling consists in choosing from the many methods that can be applied and the factors influencing the course of processes that will provide the required accuracy and ease of performance of the task.

As a rule, when modeling an AC electric machine, the real machine is replaced by an idealized one, which has four main differences from the real one: 1) no saturation of magnetic circuits; 2) absence of losses in steel and displacement of current in windings; 3) sinusoidal distribution in space of curves of magnetizing forces and magnetic inductions; 4) independence of leakage inductive reactances from the position of the rotor and from the current in the windings. These assumptions greatly simplify the mathematical description of electrical machines.

Since the axes of the stator and rotor windings of a synchronous machine mutually move during rotation, the magnetic conductivity for the winding fluxes becomes variable. As a result, the mutual inductances and inductances of the windings change periodically. Therefore, when modeling processes in a synchronous machine using equations in phase variables, the phase variables U, I,  are represented by periodic quantities, which greatly complicates the fixation and analysis of the simulation results and complicates the implementation of the model on a computer.

The so-called transformed Park-Gorev equations, which are obtained from equations in phase quantities by means of special linear transformations, are simpler and more convenient for modeling. The essence of these transformations can be understood by looking at Figure 1.

Figure 1. Representing vector I and its projection on the axis a, b, c and axes d, q

This figure shows two systems of coordinate axes: one symmetric three-line fixed ( a, b, c) and another ( d, q, 0 ) - orthogonal, rotating with the rotor angular velocity . Figure 1 also shows the instantaneous values ​​of the phase currents in the form of vectors I a , I b , I c... If we geometrically add up the instantaneous values ​​of the phase currents, we get the vector I which will rotate along with the orthogonal axis system d, q... This vector is usually called the imaging current vector. Similar depicting vectors can be obtained for the variables U,  .

If projecting imaging vectors on the axis d, q, then the corresponding longitudinal and transverse components of the imaging vectors will be obtained - new variables, which, as a result of transformations, replace the phase alternating currents, voltages and flux linkages.

While the phase quantities in the steady state change periodically, the imaging vectors will be constant and motionless relative to the axes d, q and, therefore, 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, an AC electric machine is represented as a two-phase one with perpendicular windings along the axes d, q, which excludes mutual induction between them.

The negative factor of the transformed equations is that they describe the processes in the machine in terms of fictitious, and not in terms of actual values. However, if we return to Figure 1, discussed above, we can establish that the reverse conversion from fictitious values ​​to phase values ​​is not particularly difficult: it is enough in terms of components, for example, current I d and I q compute the value of the imaging vector

and project it onto some fixed phase axis, taking into account the angular speed of rotation of the orthogonal system of axes d, q relatively motionless (Figure 1). We get:

,

where  0 is the value of the initial phase of the phase current at t = 0.

The system of equations of a synchronous generator (Park-Gorev), written in relative units in the axes d- q, rigidly connected to its rotor, has the following form:

;

;

;

;

;

;(1)

;

;

;

;

;

,

where  d,  q,  D,  Q are the flux linkages of the stator and damping windings along the longitudinal and transverse axes (d and q);  f, i f, u f - flux linkage, current and voltage of the excitation winding; i d, i q, i D, i Q - currents of stator and damping windings along the d and q axes; r is the active resistance of the stator; x d, x q, x D, x Q are the reactances of the stator and damping windings along the d and q axes; x f is the reactance of the excitation winding; x ad, x aq - stator mutual induction resistance along the d and q axes; u d, u q - stresses along the d and q axes; T do is the time constant of the excitation winding; T D, T Q - time constants of the damping windings along the d and q axes; T j is the inertial time constant of the diesel generator; s is the relative change in the generator rotor speed (slip); m cr, m g - the torque of the drive motor and the electromagnetic moment of the generator.

Equations (1) take into account all essential electromagnetic and mechanical processes in a synchronous machine, both damping windings, so they can be called complete equations. However, in accordance with the previously accepted assumption, the angular speed of rotation of the SG rotor in the study of electromagnetic (fast) processes is assumed to be unchanged. It is also permissible to consider the damping winding only along the longitudinal axis "d". Taking these assumptions into account, the system of equations (1) will take the following form:

;

;

;

; (2)

;

;

;

;

.

As can be seen from system (2), the number of variables in the system of equations is greater than the number of equations, which does not allow using this system in a direct form in modeling.

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

;

;

;

;

;

; (3)

;

;

;

;

.

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"Maps and diagrams in the Presidential Library collection"

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The field of application of variable-voltage AC drives in our country and abroad is expanding to a large extent. A special position is occupied by the synchronous electric drive of powerful mining excavators, which are used to compensate for reactive power. However, their compensating ability is underutilized due to the lack of clear recommendations for arousal modes.

D. B. Soloviev

The field of application of adjustable AC drives in our country and abroad is expanding to a large extent. A special position is occupied by the synchronous electric drive of powerful mining excavators, which are used to compensate for reactive power. However, their compensating ability is underutilized due to the lack of clear recommendations for arousal modes. In this regard, the task is to determine the most advantageous excitation modes for synchronous motors from the point of view of reactive power compensation, taking into account the possibility of voltage regulation. The effective use of the compensating ability of a synchronous motor depends on a large number of factors ( technical parameters motor, shaft load, terminal voltage, active power losses for reactive power generation, etc.). An increase in the load of a synchronous motor in terms of reactive power causes an increase in losses in the motor, which negatively affects its performance. At the same time, an increase in reactive power delivered by a synchronous motor will help to reduce energy losses in the power supply system of the open pit. According to this criterion of optimality of the load of a synchronous motor in terms of reactive power is the minimum of the reduced costs for the generation and distribution of reactive power in the power supply system of the open pit.

Investigation of the excitation mode of a synchronous motor directly in the open pit is not always possible due to technical reasons and due to limited research funding. Therefore, it seems necessary to describe the synchronous motor of the excavator by various mathematical methods. The engine, as an object of automatic control, is a complex dynamic structure described by a system of high-order nonlinear differential equations. Simplified linearized variants were used in control problems for any synchronous machine. dynamic models, which gave only an approximate idea of ​​the behavior of the machine. Development of a mathematical description of electromagnetic and electromechanical processes in a synchronous electric drive, taking into account the real nature of nonlinear processes in a synchronous electric motor, as well as the use of such a structure of mathematical description in the development of controlled synchronous electric drives, in which the study of the model mining excavator it would be convenient and clear, it seems relevant.

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

The most effective solution is the MatLAB system of mathematical calculations, SimuLink package. The MatLAB system eliminates all the disadvantages of the above methods. In this system, a software implementation of the mathematical model has already been made. synchronous machine.

The development environment for laboratory virtual instruments MatLAB is an applied graphical programming environment used as a standard tool for fashioning objects, analyzing their behavior and subsequent control. Below is an example of equations for a simulated synchronous motor using the full Park-Gorev equations written in flux linkages for an equivalent circuit with one damper circuit.

With this software it is possible to simulate all possible processes in a synchronous motor in standard situations. In fig. 1 shows the modes of starting a synchronous motor, obtained by solving the Park-Gorev equation for a synchronous machine.

An example of the implementation of these equations is shown in the block diagram, where variables are initialized, parameters are set, and integration is performed. Trigger mode results are shown on the virtual oscilloscope.

Rice. 1 An example of the characteristics taken from a virtual oscilloscope.

As you can see, when the SM is started, a shock torque of 4.0 pu and a current of 6.5 pu arise. The start-up time is about 0.4 sec. Oscillations of current and torque caused by non-symmetry of the rotor are clearly visible.

However, the use of these ready-made models makes it difficult to study the intermediate parameters of the modes of a synchronous machine due to the impossibility of changing the parameters of the circuit of the finished model, the impossibility of changing the structure and parameters of the network and the excitation system, different from the accepted ones, simultaneous consideration of the generator and motor modes, which is necessary when modeling start-up or during load shedding. In addition, in the finished models, a primitive account of saturation is applied - saturation along the "q" axis is not taken into account. At the same time, due to the expansion of the field of application of the synchronous motor and the increased requirements for their operation, refined models are required. That is, if it is necessary to obtain the specific behavior of the model (simulated synchronous motor), depending on mining and geological and other factors affecting the operation of the excavator, then it is necessary to give a solution to the Park-Gorev system of equations in the MatLAB package, which allows eliminating the indicated disadvantages.

LITERATURE

1. Kigel GA, Trifonov VD, Chirva V. X. Optimization of excitation modes of synchronous motors at iron ore mining and processing enterprises.- Mining journal, 1981, Ns7, p. 107-110.

2. Norenkov IP Computer-aided design. - M .: Nedra, 2000, 188 p.

Niskovsky Yu.N., Nikolaychuk N.A., Minuta E.V., Popov A.N.

Well-bored hydraulic mining of mineral resources of the Far Eastern shelf

To meet the growing demand for mineral raw materials, as well as building materials it is required to pay more and more attention to the exploration and development of mineral resources of the shelf of the seas.

In addition to deposits of titanium-magnetite sands in the southern part of the Sea of ​​Japan, reserves of gold-bearing and building sands have been identified. At the same time, the tailings of gold-bearing deposits obtained from beneficiation can also be used as building sands.

Placer deposits in a number of bays in Primorsky Krai belong to gold-bearing placer deposits. The productive stratum lies at a depth, starting from the coast and up to a depth of 20 m, with a thickness of 0.5 to 4.5 m. At the top, the stratum is overlain by sandy-hapey deposits with silts and clay with a thickness of 2 to 17 m. In addition to the gold content, the sands contain ilmenite 73 g / t, titanium-magnetite 8.7 g / t and ruby.

The coastal shelf of the seas of the Far East also contains significant reserves of mineral raw materials, the development of which under the seabed at the present stage requires the creation of new technology and the application of environmentally friendly technologies. The most explored reserves of minerals are coal seams of previously operating mines, gold-bearing, titanium-magnetite and kasritic sands, as well as deposits of other minerals.

The data of preliminary geological study of the most characteristic deposits in the early years are shown in the table.

Explored deposits of minerals on the shelf of the seas of the Far East can be divided into: a) lying on the surface of the sea bottom, covered with sandy-clayey and pebble deposits (placers of metal-containing and building sands, materials and shell rock); b) located at: significant deepening from the bottom under the strata of rocks (coal seams, various ores and minerals).

Analysis of the development of alluvial deposits shows that none of the technical solutions (both domestic and foreign development) can be used without any environmental damage.

The experience of developing non-ferrous metals, diamonds, gold-bearing sands and other minerals abroad indicates the overwhelming use of all kinds of dredges and dredgers, leading to widespread disruption of the seabed and the ecological state of the environment.

According to the TsNIItsvetmet Institute of Economics and Information, more than 170 dredges are used in the development of non-ferrous deposits of metals and diamonds abroad. In this case, mainly new dredges (75%) with a bucket capacity of up to 850 liters and a digging depth of up to 45 m are used, less often - suction dredges and dredgers.

Dredging works on the seabed are carried out in Thailand, New Zealand, Indonesia, Singapore, England, USA, Australia, Africa and other countries. The technology of mining metals in this way creates an extremely strong disruption of the seabed. The foregoing leads to the need to create new technologies that can significantly reduce the impact on the environment or completely eliminate it.

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

The proposed development technologies, although they reduce the harmful effect on the environment, do not preserve the bottom surface from disturbances.

When using other mining methods with and without fencing the landfill from the sea, the return of placer enrichment tailings cleaned from harmful impurities to their natural occurrence also does not solve the problem of ecological restoration of biological resources.

The fundamental differences between a synchronous motor (SM) and SG are in the opposite direction of the electromagnetic and electromechanical moments, as well as in the physical essence of the latter, which for the SM is the moment of resistance Mc of the driven mechanism (PM). In addition, there are some differences and the corresponding specificity in CB. Thus, in the considered universal mathematical model of the SG, the mathematical model of the SG is replaced by the mathematical model of the PM, the mathematical model of the SV for the SG is replaced by the corresponding mathematical model of the SV for the SD, and the specified formation of moments in the equation of motion of the rotor is provided, then the universal mathematical model of the SG is converted into a universal mathematical model of SD.

To convert a universal mathematical model of SD into a similar model asynchronous motor(IM) provides for the possibility of zeroing the excitation voltage in the equation of the rotor circuit of the motor, used to simulate the excitation winding. In addition, if there is no asymmetry of the rotor contours, then their parameters are set symmetrically for the equations of rotor contours along the axes d and q. Thus, when modeling AM, the excitation winding is excluded from the universal mathematical model of the SD, and otherwise their universal mathematical models are identical.

As a result, in order to create a universal mathematical model of SD and, accordingly, HELL, it is necessary to synthesize a universal mathematical model of PM and SV for SD.

According to the most common and proven mathematical model of many different PMs, there is an equation of the moment-speed characteristic of the form:

where t start- initial statistical moment of PM resistance; / and nom - the nominal moment of resistance developed by the PM at the nominal torque of the electric motor corresponding to its nominal active power and synchronous nominal frequency with 0 = 314 s 1; o) d - the actual speed of the rotor of the electric motor; with di - the nominal rotational speed of the rotor of the electric motor, at which the moment of resistance of the PM is equal to the memorial one, obtained at the synchronous nominal rotational speed of the electromagnetic zero of the stator from 0; R - exponent, depending on the type of PM, taken most often equal p = 2 or R - 1.

For an arbitrary load of the PM SD or HELL, determined by the load factors k. t = R / R noi and arbitrary network frequency © s F with 0, as well as for the basic moment m s= m HOM / cosq> H, which corresponds to rated power and the basic frequency ω 0, the above equation in relative units has the form

m m co „co ™

where M c - -; m CT =-; co = ^ -; co H = - ^ -.

m s"" Yom “o“ o

After the introduction of the notation and the corresponding transformations, the equation takes the form

where M CJ = m CT -k 3 - coscp H - static (frequency-independent) part

(l-m CT)? -coscp

the moment of resistance of the PM; t w =--so "- dynamic

a certain (frequency-independent) part of the moment of resistance of the PM, in which

It is usually believed that for most PMs the frequency-dependent component has a linear or quadratic dependence on ω. However, in accordance with the power-law approximation with a fractional exponent is more reliable for this dependence. Taking this fact into account, the approximating expression for A / ω -co p has the form

where a is a coefficient determined based on the required power-law dependence by calculation or graphical means.

The versatility of the developed mathematical model of SD or IM is ensured by automated or automatic controllability M st, and M w and R by means of the coefficient a.

The used SV SD have a lot in common with SV SG, and the main differences are:

• in the presence of a dead zone of the ARV channel according to the deviation of the stator voltage of the LED;
• ARV for the excitation current and ARV with compounding of various types is basically the same as for similar SV SG.

Since the operating modes of the SD have their own specifics, special laws are required for ARV SD:

• ensuring the constancy of the ratio of the reactive and active powers of the SD, called ARV, for the constancy of the given power factor cos (p = const (or cp = const);
• ARV, providing a given constancy of reactive power Q = const SD;
• ARD by the internal load angle 0 and its derivatives, which is usually replaced by a less efficient, but simpler ARD by the active power of the SM.

Thus, the previously considered universal mathematical model of SV SG can serve as the basis for constructing a universal mathematical model of SV SD after making the necessary changes in accordance with the indicated differences.

To implement the dead zone of the ARV channel according to the deviation of the stator voltage of the LED, it is sufficient at the output of the adder (see Fig. 1.1), on which d U, enable the link of controlled nonlinearity of the form of the dead zone and limitation. Replacement of variables in the universal mathematical model of SV SG variables with the corresponding regulation variables of the named special laws of ARV SD completely ensures their adequate reproduction, and among the mentioned variables Q, f, R, 0, the calculation of active and reactive power is carried out by the equations provided in the universal mathematical model of the SG: P = U K m? i q? + U d? K m? i d,

Q = U q - K m? I d - + U d? K m? i q. To calculate the variables φ and 0, also

necessary for modeling the indicated laws of ARV SD, the equations are applied: