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 equations of electrical equilibrium increases, and electromagnetic connections become more complex. Therefore, let us reduce the analysis of processes in a three-phase machine to the analysis of the same processes in an equivalent two-phase model of this machine.
In the theory of electrical machines, it has been proven that any polyphase electrical machine with n-phase stator winding and m-phase winding of the rotor, provided that the impedances of the stator (rotor) phases are equal in dynamics, it can be represented by a two-phase model. The possibility of such a replacement creates conditions for obtaining a generalized mathematical description of the processes of electromechanical energy conversion in a rotating electric machine based on the consideration of an idealized two-phase electromechanical converter. Such a converter is called a generalized electrical machine (OEM).
Generalized electrical machine.
OEM allows you to represent the dynamics real engine, both in stationary and rotating coordinate systems. The latter representation makes it possible to significantly simplify the equations of state of the engine and the synthesis of control for it.
Let's introduce variables for OEM. The belonging of a variable to a particular winding is determined by indices that indicate the axes associated with the windings of a generalized machine, indicating the relationship to stator 1 or rotor 2, as shown in Fig. 3.2. In this figure, the coordinate system rigidly connected with the stationary stator is designated,, with a rotating rotor -,, - is the electric angle of rotation.
Rice. 3.2. Generalized two-pole machine diagram
The dynamics of a generalized machine is described by 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 a function of the electrical and mechanical coordinates of the system.
The Kirchhoff equations, expressed in terms of flux linkage, have the form
(3.1)
where and are the active resistance of the stator phase and the reduced active resistance of the rotor phase of the machine, respectively.
The flux linkage of each winding in general view is determined by the resulting action of the currents of all windings of the machine
(3.2)
In the system of equations (3.2), the same designation with a subscript is adopted for the intrinsic and mutual inductances of the windings, the first part of which is 
, indicates in which winding the EMF is induced, and the second 
- by the current of which winding it is created. For example, - self-inductance of the stator phase; - mutual inductance between stator phase and rotor phase, etc.
The notation and indices adopted in system (3.2) ensure the uniformity of all equations, which makes it possible to resort to a generalized form of writing this system, convenient for further presentation,
(3.3)
During the operation of the OEM, the relative position of the stator and rotor windings changes, therefore, the intrinsic and mutual inductances of the windings in the general case are a function of the electric angle of rotation of the rotor. For a symmetrical implicit-pole machine, the intrinsic inductances of the stator and rotor windings do not depend on the rotor position
and the mutual inductances between the stator or rotor windings are zero
since the magnetic axes of these windings are shifted in space relative to each other by an angle. The mutual inductances of the stator and rotor windings pass through full cycle changes when the rotor is turned by an angle, therefore, taking into account those adopted in Fig. 2.1 directions of currents and the sign of the angle of rotation of the rotor can be written
(3.6)
where is the mutual inductance of the stator and rotor windings or when, i.e. when the coordinate systems and coincide. Taking into account (3.3), the equations of electrical equilibrium (3.1) can be represented in the form
, (3.7)
where are defined by relations (3.4) - (3.6). We obtain the differential equation for electromechanical energy conversion using the formula
where is the angle of rotation of the rotor,
where is the number of pole pairs.
Substituting equations (3.4) - (3.6), (3.9) into (3.8), we obtain the expression for the electromagnetic moment of the REM
. (3.10)
Two-phase implicit-pole synchronous machine with permanent magnets.
Consider Electrical engine in EMUR. It is an implicit pole permanent magnet synchronous machine because it has a large number of pole pairs. In this machine, the magnets can be replaced by an equivalent lossless excitation winding (), connected to a current source and creating a magnetomotive force (Figure 3.3.).
Figure 3.3. Scheme of switching on a synchronous motor (a) and its two-phase model in the axes (b)
Such a replacement allows one to represent the equations of equilibrium of stresses by analogy with the equations of the usual synchronous machine, therefore, putting and 
in equations (3.1), (3.2), and (3.10), we have
(3.11)
(3.12)
Let us denote where is the flux linkage on a pair of poles. We make the change (3.9) in equations (3.11) - (3.13), as well as differentiate (3.12) and substitute it into equation (3.11). We get
(3.14)
where is the angular speed of the engine; - the number of turns of the stator winding; - the magnetic flux of one turn.
Thus, equations (3.14), (3.15) form a system of equations for a two-phase implicit-pole synchronous machine with permanent magnets.
Linear transformations of the equations of a generalized electric machine.
The merit obtained in clause 2.2. the mathematical description of the processes of electromechanical energy conversion is that the actual currents of the windings of the generalized machine and the actual voltages of their supply are used as independent variables. Such a description of the dynamics of the system gives a direct idea of the physical processes in the system, but it is difficult to analyze.
When solving many problems, a significant simplification of the mathematical description of the processes of electromechanical energy conversion is achieved by linear transformations of the original system of equations, while real variables are replaced by new variables, provided that the adequacy of the mathematical description to the physical object is maintained. The adequacy condition is usually formulated in the form of the requirement for the power invariance when transforming the equations. Newly introduced variables can be either real or complex quantities associated with real variable transformation formulas, the form of which must ensure that the power invariance condition is satisfied.
The purpose of the transformation is always one or another simplification of the initial mathematical description of dynamic processes: the elimination of the dependence of the inductances and mutual inductances of the windings on the rotor rotation angle, the ability to operate not with sinusoidally changing variables, but with their amplitudes, etc.
First, consider real transformations that allow us to pass from physical variables defined by coordinate systems rigidly connected with the stator and rotor to the red variable corresponding to the coordinate system u, v rotating in space at an arbitrary speed. For a formal solution of the problem, we represent each real winding variable - voltage, current, flux linkage - in the form of a vector, the direction of which is rigidly connected with the coordinate axis corresponding to the given winding, and the modulus changes in time in accordance with changes in the displayed variable.
Rice. 3.4. Generalized Machine Variables in Different Coordinate Systems
In fig. 3.4 winding variables (currents and voltages) are designated in general form by a letter with a corresponding index, reflecting the belonging of a given variable to a particular coordinate axis, and the relative position at the current time of the axes rigidly connected to the stator of the axes is shown d, q, rigidly connected to the rotor, and an arbitrary system of orthogonal coordinates u, v rotating with respect to the stationary stator at a speed. Real variables in the axes (stator) and d, q(rotor), the corresponding new variables in the coordinate system u, v can be defined as the sum of the projections of real variables on the new axes.
For greater clarity, the graphical constructions necessary to obtain the transformation formulas are shown in Fig. 3.4a and 3.4b for stator and rotor separately. In fig. 3.4a shows the axes associated with the stationary stator windings, and the axes u, v rotated relative to the stator at an angle . Vector components are defined as projections of vectors and on the axis u, the components of the vector - as the projection of the same vectors on the axis v. Summing up the projections along the axes, we obtain the direct transformation formulas for the stator variables in the following form
(3.16)
Similar constructions for rotary variables are shown in Fig. 3.4b. Shown here are the fixed axes rotated relative to them at the angle of the axis d, q, rotor-connected machines rotated with respect to the rotor axes d and q per axis angle and, v, rotating with speed and coinciding at each moment of time with the axes and, v in fig. 3.4a. Comparing Fig. 3.4b with fig. 3.4a, it can be established that the projections of vectors and onto and, v similar to the projections of stator variables, but as a function of the angle. Consequently, for rotor variables, transformation formulas have the form
(3.17)
Rice. 3.5. Variable transformation of a generalized two-phase electrical machine
To clarify the geometric meaning of linear transformations carried out according to formulas (3.16) and (3.17), in Fig. 3.5 additional constructions are performed. They show that the transformation is based on the representation of the variables of a generalized machine in the form of vectors and. Both real variables and, and transformed and are projections on the corresponding axes of the same resulting vector. Similar relationships are valid for rotor variables.
If necessary, transition from transformed variables 
to real variables of a generalized machine 
inverse transformation formulas are used. They can be obtained using the constructions performed in Fig. 3.5a and 3.5 are similar to the constructions in Fig. 3.4a and 3.4b
(3.18)
Formulas of direct (3.16), (3.17) and inverse (3.18) transformations of coordinates of a generalized machine are used in the synthesis of controls for a synchronous motor.
Let us transform equations (3.14) to the new coordinate system. For this, we substitute the expressions for the variables (3.18) into equations (3.14), we obtain
(3.19)
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 physical entity the latter, which for the SD is the moment of resistance Мс 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 from 0, as well as for the basic moment m s= m HOM / cosq> H, which corresponds to the nominal power and the base frequency with 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. With considering this fact, 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 graphically.
The versatility of the developed mathematical model of SD or IM is ensured by automated or automatic controllability M st, as well as 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 excitation current and ARV with compounding of various types occurs basically analogously to 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;
- ARV for inner corner load 0 and its derivatives, which is usually replaced by a less efficient, but simpler ARV in terms of 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:
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Design and principle of operation of a permanent magnet synchronous motor
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
Mathematical model general view 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 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: per. from English / J.L. Melsa, Art. K. Jones. - M .: Mashinostroenie, 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 double 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. A set of programs for the study of electric drive systems based on a dual-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.
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 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. 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 supplied 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.
The study 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 funding research works... 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. In problems of control of any synchronous machine, simplified linearized versions of dynamic models were used, 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 of a synchronous machine has already been made.
MatLAB laboratory virtual instruments development environment is an applied graphical programming environment used as a standard tool for fashion objects, analysis of 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 simulating a start 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 hydro-mining of mineral resources of the Far East 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. From 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 operated 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 mineral deposits 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-clay and pebble deposits (placers of metal-containing and building sands, materials and shell rock); b) located at: significant deepening from the bottom under the stratum 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 environment or exclude it entirely.
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 tailings cleaned from harmful impurities to the place of their natural occurrence also does not solve the problem of ecological restoration of biological resources.
