MATHEMATICAL MODEL OF INDUCTION MOTOR WITH SERIES-CONNECTED STATOR AND ROTOR WINDINGS

2017. Т. 17, No 1. С. 77–87 77 Introduction There are diverse types of industrial mechanisms having technical process that requires relatively longstanding speed reduction with a decrease in static loads and restrained requirements for the accuracy of speed regulation. There are industrial fans, street conveyors in cold times, pumps of hydraulic presses, etc. [1, 2]. A complete stop of the drive is unacceptable for these systems, while decreasing the speed to preserve energy and to use resources effectively is considered more appropriate. In most cases, squirrel-cage induction motors and wound rotor induction motors are used as driving engines in the mentioned mechanisms. The motor speed remains unregulated or to the use of added resistances in circuits of rotor or stator is required for regulation. Having no regulation or deploying relay contact switching circuits causes high power consumption. High prices for energy and other recourses make it important to enhance the mentioned electric drives. The implementation of two-unit frequency converter based on key elements is not appropriate for the mentioned mechanisms due to high price, complicated maintenance, and the high level of the operating staff’s skills required [3, 4]. Impulse-vector control system for alternating current drives is used to create the current of the same frequency in the rotor and the stator, i.e. to exclude motor slip. Impulse-vector control is suitable for copious amounts of mechanisms because of its low price, low power consumption in low velocity mode and the absence of abundant control possibilities. There are some scheme solutions featuring the impulse-vector control system with wound rotor induction motor [5]: – impulse-vector control system with thyristor switches; – impulse-vector control system with thyristor switches and damping circuit; – impulse-vector control system with diodes and transistor; – impulse-vector control system with damping circuit by symistors. The control in these systems is performed depending on the angular location of a rotor, and, therefore, it is necessary to set a shaft encoder or to use sensorless control methods. Using shaft encoders for the mechanisms mentioned above is inappropriate, as it increases the cost and complicates the system. Furthermore, sensorless control methods have some advantages: the absence of the shaft encoder; simple installation, replacement, and operation; a possibility to use sensorless systems in cases when the measurement unit installation on the shaft is impossible (exposure to vibrations, radiation, high temperatures, etc.); a possibility of applying equipment for electric drives diagnostics; a possibility of decreasing of sustaining speed error, and augmenting the regulation range in contrast to open loop systems. Using the known sensorless control methods [6, 7] or the development of unique algorithms of shaft position determination [8] for pulse-vector control system [5, 9] in most cases are complicated. It is due to the lack of appropriate mathematical description of electromagnetic processes in the schemes with DOI: 10.14529/power170111


Introduction
There are diverse types of industrial mechanisms having technical process that requires relatively longstanding speed reduction with a decrease in static loads and restrained requirements for the accuracy of speed regulation. There are industrial fans, street conveyors in cold times, pumps of hydraulic presses, etc. [1,2]. A complete stop of the drive is unacceptable for these systems, while decreasing the speed to preserve energy and to use resources effectively is considered more appropriate.
In most cases, squirrel-cage induction motors and wound rotor induction motors are used as driving engines in the mentioned mechanisms. The motor speed remains unregulated or to the use of added resistances in circuits of rotor or stator is required for regulation. Having no regulation or deploying relay contact switching circuits causes high power consumption. High prices for energy and other recourses make it important to enhance the mentioned electric drives.
The implementation of two-unit frequency converter based on key elements is not appropriate for the mentioned mechanisms due to high price, complicated maintenance, and the high level of the operating staff's skills required [3,4]. Impulse-vector control system for alternating current drives is used to create the current of the same frequency in the rotor and the stator, i.e. to exclude motor slip. Impulse-vector control is suitable for copious amounts of mechanisms because of its low price, low power consumption in low velocity mode and the absence of abundant control possibilities.
There are some scheme solutions featuring the impulse-vector control system with wound rotor induction motor [5]: -impulse-vector control system with thyristor switches; -impulse-vector control system with thyristor switches and damping circuit; -impulse-vector control system with diodes and transistor; -impulse-vector control system with damping circuit by symistors.
The control in these systems is performed depending on the angular location of a rotor, and, therefore, it is necessary to set a shaft encoder or to use sensorless control methods. Using shaft encoders for the mechanisms mentioned above is inappropriate, as it increases the cost and complicates the system. Furthermore, sensorless control methods have some advantages: the absence of the shaft encoder; simple installation, replacement, and operation; a possibility to use sensorless systems in cases when the measurement unit installation on the shaft is impossible (exposure to vibrations, radiation, high temperatures, etc.); a possibility of applying equipment for electric drives diagnostics; a possibility of decreasing of sustaining speed error, and augmenting the regulation range in contrast to open loop systems.
Using the known sensorless control methods [6,7] or the development of unique algorithms of shaft position determination [8] for pulse-vector control system [5,9] in most cases are complicated. It is due to the lack of appropriate mathematical description of electromagnetic processes in the schemes with To provide for the cost-effective use of resources and energy conservation it is vital to enhance unregulated electric drives of copious working mechanisms. The technical process of these mechanisms requires relatively longstanding speed reduction under low static loads. Moreover, another relevant issue is choosing control systems of electric drives in the mentioned systems in accordance with the economic and maintenance aspects. The authors suggest using the systems of impulse-vector control system with wound rotor induction motor that have one essential drawback -the shaft encoder installation. The replacement of the shaft encoder with sensorless impulse-vector control system is complicated due to the lack of proper mathematical description of the electromagnetic processes in schemes with non-traditional windings stator and rotor connection . To solve this problem the authors have developed a mathematical description of impulse-vector control system with wound rotor induction motor supposed to be multiphase and asymmetrical. Angular dependencies of inductances, flux linkages, voltages of engine windings, circuit current and electromagnetic torque relatively to rotor location are derived. In addition to that, the mathematical modeling and research of induction motor configuration with a series connected windings fed by AC voltage source are presented. Equating rotor position in impulse-vector control system with wound rotor induction motor is considered possible through angular dependencies of drop voltages on stator and rotor windings.
Keywords: industrial mechanisms, electric drive, asynchronous motor, phase rotor, impulse-vector control system, sensorless control, mathematical model. non-traditional windings connection in the wound rotor induction motor.
Sensorless impulse-vector con wound rotor induction motor are mu trical, nonlinear, impulse systems w connection of stator and rotor, and, velopment of mathematical descript is complicated and requires a comp solution. So, the paper offers a desc magnetic processes in asymme schemes with series windings conne rotor fed by AC voltage source for tation in sensorless impulse-vector c

Theoretical part
Operating principles of impulse system with wound rotor induct In impulse-vector control syste tor induction motor [5,9] the elect is created with the series connect phases through valve elements the two windings of the rotor. Th force (MMF) vector of stator windin bore discretely with the step of 60 (el. degrees). Switching is effect the shaft position, providing the or vectors of the stator and rotor windi the motive torque. One of the stator de-energized while two others are w rotor winding is connected in pa de-energized during the operating cy Consider the operation princip rotor induction motor with the ser nection of the stator and rotor fed by source. Assuming that the stator wi are fed by source voltage, the stat de-energized at the considering (Fig n of stator and rotor ntrol system with ultiphase, asymmewith series windings , therefore, the detion of such system plex stage-by-stage cription of electroetrical multiphase ection of stator and further implemencontrol systems. e-vector control tion motor em with wound rotromagnetic torque tion of two stator consequently to he magneto motive ngs moves in stator 0 electrical degrees ted depending on rientation of MMF ngs respectively to r windings remains working. The third arallel or remains ycle. ples of the wound ries windings cony harmonic voltage indings AX and BY tor winding CZ is g. 1).

Idealization of the mach
The magnetic field crea in alternating current machi due to the complex configura cores boundaries, the peculi with currents and the nonlin characteristic in the magnet tions, strict determination of and, therefore, studying the e in the researched machine an ferential equations employ a the determination of magneti ing simple electromagnetic the main electromagnetic p The idealization is based on absence of the magnetic circ losses in steel; absence of copper windings; harmoniou MMF and magnetic induction reactive resistances of win the rotor location; complete uniformity of air gap [10,11] Abstract designations an Consider: i -instantane rent; u, ψ -instantaneous winding flux linkage; r -acti L -inductance of winding; values of voltages and curren phasize which winding is windings -AX, BY, CZ; phase c0; equivalent rotor winding replaced with one equivalen [12]; voltages on rotor ringsscripts are applied to desig between the first and the seco inductance of stator winding winding R is marked as L AX windings connection Fig. 2 ine ated by windings currents ines is very non-uniform ation of the ferromagnetic iar layout of conductors nearities of the magnetic core. Under such condif the field is complicated, electromagnetic processes nd the compilation of difa known idealization for ic field. It allows obtainconnections representing process in the machine. the following principles: cuit saturation, hysteresis, current displacement in us space distribution of n curves; independence of ndings dispersion from e symmetry of windings; ]. nd equivalent circuit eous value of circuit curvalues of voltage and ive resistance of winding; U, I -root-mean-square nt. We use indexes to emdesignated: phase stator e rotor windings -a0, b0, g -R (rotor winding is nt winding according to -ab, bc, ca. Double subgnate mutual inductances ond windings, e.g. mutual g AX and equivalent rotor XR . In circuit AB ( Where L AXBY -mutual inductance of the stator windings; L AXR -mutual inductance of the stator winding AX and equivalent rotor winding R; L 1σ -leakage inductance of the stator winding.
As stator windings have identical features, the magnetic flux created by the second winding current linking to the turns of the first winding is identical to the magnetic flux created by the first winding current linking to the turns of the second winding based on the assumed windings axes similarity and currents values equality. It is obvious that the pattern of the magnetic field will be similar and will not dependent on which winding current flows in under these conditions. Consequently, the inductance of the main magnetic flux will be equal to the stator windings mutual inductance in case the windings axes are the same.
The shift of the windings axes in space will cause the alteration of its mutual inductance proportional to the angle cosine of the shift. For the considered scheme, mutual inductance between stator windings L AXBY is always positive. It is true due to two facts: these windings spaced at the angle of -120 el. degrees and current flows into the beginning of one winding outflowing from the beginning of another: L AXBY = = L 0 cos(-120 + 180) = 0,5L M , where L 0 = L M -mutual inductance in case the windings axes are the same.
Current i flows in the equivalent rotor winding R. As a result, the magnetic flux created by equivalent rotor winding is linked to the stator windings leading to the emergence of an additional constituent in the equation of stator winding flux linkage L AXR ·i. As a rule, the amount of rotor winding turns w 2 for the typical induction motors is not equal to the number of the stator windings turns w 1 . Consequently, the current flows in stator and rotor windings create different total magnetic fluxes. Assume L AXR ≠ L M to consider that fact. Then, mutual inductance of stator winding AX and equivalent rotor winding is equal to: where E 20 -rotor rings voltage; U 1N -rated stator voltage.
Considering the mentioned dependencies of inductance, the equation of flux linkage for the phase A may take the following form: Stator windings BY and AX have the same properties. Therefore, considering the location of stator winding BY relating to AX it is possible to represent mutual inductance between BY and equivalent rotor winding as follows: Stator winding CZ is de-energized (Fig. 2), and, therefore, it produces no leakage flux and part of the main magnetic flux. The magnetic fluxes created by windings AX and BY and linked to CZ, are equal in value and oppositely directed. Hence, the flux linkage of de-energized winding CZ is equal to the flux linkage created by the equivalent rotor winding relying on the relative position: Mutual inductance between equivalent rotor winding and stator windings AX and BY: cos( ) , cos( 60) .
Flux linkage value of equivalent rotor winding is equal to: To evaluate the rotor windings flux linkages a0, b0, c0 it is crucial to consider the scheme with three-phase rotor winding (Fig. 1). Symmetric three-phase rotor winding is wye-connected; windings are shifted by 120 el. degrees in relation to each other. Windings a0 and b0 are connected with stator windings in series, winding c0 is de-energized.
The leakage inductance of rotor winding is L 1σ . Making the same assumptions as those mentioned above for the flux linkages of rotor winding and considering the interaction of stator windings AX and BY separately, we obtain the following equations: Using the mentioned idealization to calculate the magnetic flux leakage inductances, own inductances illustrating the interaction of winding with its own magnetic flux and mutual inductances between stator windings does not dependent on the angle of rotor rotation.
Voltages, circuit current and torque of the studied machine Instantaneous values of source voltages in three-phase network may be written as follows: sin( ); sin( 120); where u А , u В , u С -instantaneous values of three-phase network; U m -amplitude of source voltage instantaneous value;  = 2 f -circular frequency (f -power frequency); t -time.
Equations of instantaneous values representing the electrical condition of the system according to the second Kirchhoff law and Faraday law for the considered equivalent circuit may be written in the following way: where r 1 , r 2 -active resistances of stator and rotor.
Applying the values of inductances and flux linkages to instantaneous values equations, we calculate the derivative value of the circuit current:   where n d dt   -rotor speed. For locked rotor n = 0. According to the second Kirchhoff law and Faraday's law for the considered equivalent circuit (Fig. 2) equations for the instantaneous values of voltages on stator windings and rotor rings take the following form: . 2 The voltage drop on the engine windings are equal to the sum of self-induced EMF and mutual inductance EMF. Self-induced EMF of stator windings and equivalent rotor winding depends on the rotor angle. There is no current in a free stator winding CZ. Moreover, CZ induces only self-induced EMF from the equivalent rotor winding (U C = E C ).
The EMF inductance value may be split into two components. The first component dψ / dt is related to flux linkage variation in time due to currents variations in time is called transformation EMF similarly to the processes of motor field in the machine. The second component is ωψ, which comes from flux linkage variation in time due to the rotor rotation, and it is called rotational EMF.
It is appropriate to calculate the electromagnetic torque of the engine via the instantaneous values of induction motor internal coordinates. Furthermore, we use the equation of electromagnetic torque coming from the common assumption that the electromagnetic torque of the electric machine is equal to partial differential coefficient in geometric angle α from the total value of the electromagnetic energy: where L j -own winding inductance (in considered case L AХ , L BY , L R ); i j -current in the winding; L ij -mutual inductances (L AXR , L BYR , L AXBY ). Consider: 0 j dL d  and 0 AXBY dL d  , therefore, no torque created by these values.
In addition to that, the one current i flows in the machine windings, electromagnetic torque is created only by two components, and then equation may be written as follows: where T АX , T ВY -torques created by currents of stator windings АX and ВY.

Mathematical modeling
It is implied that in case of rotor rotating by the value of d current succeeds steady-state value, i.e. 0 di d  . It is true for low rotation speed. Computed dependencies of root-mean-square values of currents on the engine windings, circuit current and torque from angle of rotor rotation are obtained with the mentioned differential equations of instantaneous values using a model developed in MATLAB.
Instantaneous values of source voltage u A , u B , u C , u A -u B are shown in Fig. 3a, instantaneous value of circuit current i - Fig. 3b, instantaneous values of voltages on stator and rotor windings - Fig. 3 c, d (α = 0, U m = 117 V). Root-mean-square values of voltages on the engine windings, root-mean-square value of circuit current and electromagnetic torque variation α from 0° to 360° are illustrated in Fig. 4, 5.
The circuit current value is the lowest at the rootmean-square value of source voltage. 2. The voltage on the de-energized winging U CZ is equal to -(U BY + U AX ) representing gap field strength depending on the torque. It is zero at α = 150 and α = 330 el. degrees. In these positions the equivalent rotor winding is located orthogonally to the deenergized stator winding CZ and does not induce EMF (Fig. 9).
The first position at α = 150 el. degrees is a position of instable equilibrium: having minimal alteration of angle the rotor turns clockwise or anticlockwise under the influence of the highest value of torque (Fig. 7). The vectors of stator MMF F S and rotor MMF F R are coincided (Fig. 9a). The second position (α = 330 el. degrees) is a position of stable equilibrium, changing angle in the range from 300 to 360 el. degrees the rotor is fixed or returned to position at α = 330 el. degrees, i.e. the position where the electromagnetic torque value is equal to zero (Fig. 7). The vectors of sta-tor MMF F S and rotor MMF F R coincide (Fig. 9b). The type of the electromagnetic torque dependence complies with equation (17).
3. At the positions where the voltage on the deenergized winding is equal to zero, the voltages on the stator windings switched "forward" and "backward" are equal. If the voltage on working "backward" winding is bigger than the voltage on working "forward" winding (in the range from 150 to 360 el. degrees not including) the engine leads to rotation clockwise. In addition to that, if the voltage on working "forward" winding is bigger than the voltage on working "backward" winding (in the range from 0 to 150 el. degrees) the engine leads to anticlockwise rotation.
4. Experimental dependencies U ab , U bc , U ca on the angle of rotor rotation shown in Fig. 8

Conclusion
1. The interaction analysis of the rotor and stator windings changing angular position of the rotor allowed finding out that the impulse-vector control system with wound rotor induction motor has unambiguous dependencies from the angular position of the rotor. The obtained angular dependencies vividly explain the physics of the stator and rotor windings interaction when the windings connected in series. 2. The obtained dependencies of the windings voltages, current and electromagnetic torque in the scheme with stator and rotor windings connected in series and fed by harmonic voltage source allowed the conclusion that the voltages values on the engine values may be used as input variables of angular rotor position estimator.
The work was supported by Act 211 Government of the Russian Federation, contract № 02.A03.21.0011.