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 along 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.
Streaming of each winding in general Determined by the resulting current of the currents of all windings 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. Mutual inductance of stator and rotor windings pass full cycle Changes when rotating the rotor at the angle, therefore, taking into account the adopted 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 the usual synchronous machine, therefore, 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 transform equations (3.14) to a new coordinate system. To do this, we substitute the expressions of the variables (3.18) in equations (3.14), we get
(3.19)
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