CALCULATION OF ELECTROMAGNETIC PROCESSES FOR A TURBOGENERATOR WITH EQUIVALENT STATOR AND ROTOR TOOTH ZONES IN NO-LOAD REGIME

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1 Vol. 2, No. 2, 2012 COMPUTATIONAL PROBLEMS OF ELECTRICAL ENGINEERING CALCULATION OF ELECTROMAGNETIC PROCESSES FOR A TURBOGENERATOR WITH EQUIVALENT STATOR AND ROTOR TOOTH ZONES IN NO-LOAD REGIME Lviv Polytechnic National University, Ukraine ya_kovivchak@yahoo.com Abstract: On the basis o Maxwell s equations with respect to potentials, calculations o electromagnetic processes in movable and immovable areas o a turbogenerator cross-section in transormed and physical coordinate systems or no-load regime have been done. It has been shown that employing the method o equivalentization o the device tooth zones by continuous anisotropic media and ignoring the coordinate systems in which the process is considered in terms o mathematical ield models lead to misrepresentation o real electromagnetic phenomena. Key words: electromagnetic ield, Maxwell s equation, movable media, coordinate systems, tooth zones. 1. Introduction The problem o developing mathematical ield models or electrodynamic devices is o great importance or the theory and practice o designing, developing, manuacturing and maintaining electrical objects o the kind. Here also belongs a turbogenerator being the main source o electric energy in the world. The solution to the task set beore us is impeded by some theoretical and practical problems caused by a need to calculate an electromagnetic ield in movable and immovable, linear and nonlinear, isotropic and anisotropic media [1]. In modern scientiic literature this problem is paid much attention to or the purpose o developing some approaches to determination o electromagnetic processes in electrodynamic devices by exclusively using the theory o electromagnetic ield. The most practically used method o the development o mathematical ield models implies application o equivalent substitution circuits with lumped parameters to the description o an electrodynamic object structure. The parameters o the circuits can be calculated rom spatial distributions o an electromagnetic ield obtained or reciprocal ixed positions o moving and stationary elements o a device under predetermined (given) magnetomotive orces [2, 3]. The models o this kind do not imply taking a vortical component o the ield into account, a possibility o analyzing dynamic electromagnetic processes as well as electromagnetic phenomena occurring in a time slot o the device zones on the basis o Maxwell s equations. This causes a signiicant misrepresentation o real electromagnetic processes in electrodynamic objects. Many scientiic works dedicated to calculations o an electromagnetic ield in electrodynamic structures ignore the coordinate systems in which basic calculation values are ormed in basic vectors or potentials [4, 5]. This results rom the use o timeless techniques to determine unknown variables. Such an approach makes it impossible to analyze electromagnetic phenomena both in movable and immovable elements o a device in a time domain in appropriate coordinate systems. But the electromagnetic process in movable media is known to always depend on a reerence rame in which it is considered. Given the major disadvantages o the above analyzed methods, let us consider some ways o developing mathematical ield models or a turbogenerator with equivalent stator and rotor zones in transormed and physical coordinate systems in the case o a no-load regime. They are based on Maxwell s equations with respect to potentials, and involve both direct time integration o the ield equations and reerence systems the process is described in. 2. Statement o the problem Since the development o a 3-D dynamical model o a turbogenerator on the basis o the theory o electromagnetic ield is a complicated task, to ease the problem let us consider a 2-D model with ield latparallelism along the device axis accepted. The models o the class do not take electromagnetic phenomena in the objects side ace zones into consideration, but as the given model is designed to reproduce a no-load regime o a turbogenerator such an assumption is acceptable. Given the complexity o the object design, and in order to simpliy its mathematical description, we are to replace the stator and rotor tooth zones in the model by the equivalent nonlinear anisotropic media with the electromagnetic characteristics above. Their properties can be determined by making use o the known methods oered by the theory o electromagnetic circuits to calculate parameters o substitution circuits o parallel

2 56 and series connection o electrical and magnetic resistances. Such an approach being more progressive or analyzing electromagnetic processes than the use o equivalent substitution schemes is widely used to develop mathematical ield models o sophisticated electrical devices [6]. Experience o the development o mathematical ield models or electromechanical devices, calculation algorithms as well as analysis o the computer simulation results show that it is practically impossible to create a uniorm mathematical ield model o a turbogenerator intended or calculating all possible operating modes o a real device. Investigations o operating conditions are to be carried out employing several models: a model intended or calculating a no-load regime, a model or computer simulation o a short circuit regime, and a model or the reproduction o operating modes o a turbogenerator. This is due not only to the peculiarity o calculation o an electromagnetic ield in movable media [1] and some restrictions on use o the ield approach to the description o electromagnetic phenomena only in a single link o a complicated electric circuit (o a turbogenerator), but speciically to the necessity o ollowing the laws o switching both in methods o the theory o ield and those o the theory o circuits. As, in the ield consideration, the results o no-load regime calculations are initial data or computer simulation o other modes o the device, let us develop mathematical ield models intended or simulating a noload regime. Besides, there exist some peculiarities o electromagnetic processes in such, at irst sight, a simple no-load regime o a turbogenerator. Let us consider the development o a 2-D mathematical ield model o a no-load regime turbogenerator in transormed and phase coordinate systems. Fig. 1 depicts the location o the available media o the turbogenerator cross-section with equivalent stator and rotor tooth zones, where 1 stands or the massive rotor body; 2 represents the equivalent rotor tooth zone; 3 is the air gap between the stator and rotor; 4 is the equivalent stator tooth zone; 5 stands or the stator body; 6 is the air gap outside the turbogenerator. I we consider a zone scheme given in Fig.1 concerning radii o the device design elements, then R 1 is the external edge radius o the rotor slot; R 2 is the external radius o the rotor; R 3 is the internal radius o the stator; R4 is the external edge radius o the stator slot; R 5 is the external radius o the stator; and R 6 is the nominal radius o the air gap over the turbogenerator in which the electromagnetic process is calculated. The main geometric dimensions o cross-sectional areas o the real turbogenerator ТGV-500 corresponding to Fig. 1 are given in Fig. 2. Fig. 1. Zones o turbogenerator cross section in which ield calculations are carried out. Fig. 2. Main geometrical dimensions o cross-sectional areas o the turbogenerator TGV A turbogenerator mathematical model in the transormed system o coordinates Calculations o electromagnetic processes in electrical devices can be done by Maxwell s equations either in the base vectors H, E, B, D, or in the potentials А,. Choosing the variables depends on the peculiarities o the device design, its symmetry relative to the windings as well as the necessity to minimize the number o equations and the possibility to create both boundary and edge conditions or them. Taking the periodicity o electromagnetic processes on the pole division o a turbogenerator into account, it is acceptable to calculate electromagnetic phenomena only in the sector o a device cross-section that corresponds to the pole division o the object. To simpliy the task o determining boundary conditions and decreasing the number o main equations, let us apply the equation o electromagnetic ield written with respect to potentials. To introduce a single-valued relation between unctions o the vector A and scalar potential, let us choose the known calibration = 0[6]. The only mathematically possible and physically reasonable way o the transer o electromagnetic values rom one reerence rame to another is the Lorentz- Poincare transormation which is written with respect to the vector o electric ield intensity.

3 Calculation o Electromagnetic Processes or a Turbogenerator with Equivalent Stator 57 E E υ B, (1) where E is the vector o electric ield intensity in the moving system o coordinates; E is the vector o electric ield intensity in the stationary system o coordinates; υ is the vector o linear velocity o the moving coordinates system relative to the stationary one; В is the vector o magnetic induction in the immovable reerence rame. As we see rom the equation (1) the dierence between the values o electric ield intensity vectors in the moving and ixed systems o coordinates reaches υ Bwhich deines the character o electromagnetic phenomenon occurring in a moving environment relative to an observer being in an immovable rame o reerence. That is why the key moment when calculating electromagnetic processes in moving elements o electrodynamics devices by means o the theory o electromagnetic ield is choosing a system o coordinates in which the main calculated values are written. I we base on the expression (1), we shall obtain the ollowing ratio or converting the values o vector potential unction rom one reerence rame to another [1] A υ A. (2) The ratio (2) cannot be used or direct calculation o electromagnetic processes as it contains two unknown variables A, A - the unctions o vector potential in moving and ixed systems o coordinates. It can be applied only to convert the value o A (according to (1)) rom once rame o reerence to another [1]. It is possible or the electromagnetic quantities to be calculated only in such a coordinate system with respect to which main equations can deinitely be ormed. The equation (2) is not appropriate to be applied in this particular case [1]. From the practical point o view, the calculation o electromagnetic ield can be carried out either in moving or ixed coordinate systems, or simultaneously in moving and ixed rames o reerence, with the boundary conditions being recalculated. As there is a necessity to do calculation o the electromagnetic ield in movable media, the given model implies reducing an immovable coordinate system o a stator to movable coordinates o a rotor. In this way we can dispose o the actual physical movement o the media, but at that the obtained electromagnetic process in the stator tooth zone as well as in its body corresponds to the transormed reerence rame rather than to the physical one In the model considered, the tooth zone o the stator has been replaced by a continuous anisotropic medium, and it is isotropic along the angular coordinate, that is why the use o the transormed system o coordinates is advisable, as this enables the calculations to be considerably simpliied without ignoring the peculiarities o electromagnetic phenomena both in the tooth zone and the body o the stator in the physical system o coordinates. Transition to the physical reerence rame occurs by the certain angular coordinate displacement o the obtained distributions o the electromagnetic quantities that corresponds to the real reciprocal position o the stator and rotor in the given ixed moment o time t [1]. Hence, the calculations o electromagnetic processes in the body and equivalent medium o the rotor tooth zone as well as in the air gap between the rotor and stator will be carried out in the physical rame o reerence related to the rotor while in the equivalent tooth zone o the stator, in its body, and in the air gap beyond the turbogenerator the same procedure will be perormed in the transormed system o coordinates reduced to the coordinates o the movable rotor. The equation or calculating an electromagnetic ield in movable equivalent media (with given external currents) o the rotor tooth zones is represented in the ollowing orm [6]. A, (3) where A is the vector potential o electromagnetic ield in the coordinate system o a moving rotor; Г is the matrix o static electrical conductivities; N is the matrix o static inverse magnetic penetrability o the medium; is the extraneous current density vector. To calculate electromagnetic processes in the rotor body we can use the ollowing expression [6] A, (4) where A is the unction o electromagnetic ield vector potential in the coordinate system o a moving rotor. The electromagnetic process in the air gap zone between the stator and rotor can be calculated by means o the ollowing ratio 0 ν 0 A, (5) where 0 is the inverse magnetic air penetrability. The calculation o an electromagnetic ield both in a tooth zone and body o the stator in the transormed coordinate system or a no-load regime is carried out by the expression below [1] 0 A, (6) where A is the unction o vector potential o the electromagnetic ield in the transormed coordinate system o the stator reduced to the moving rotor. The electromagnetic process in the air gap zone outside the turbogenerator can be calculated by the ollowing equation 0 ν 0 A. (7) The nonlinear electromagnetic characteristics o the stator and rotor materials in the mathematical ield model o a turbogenerator are taken into consideration by means o the ollowing splines

4 58 3 k a B B m B, k = 1, 2,..., n, (8) i m k where n is the number o segmentations along the В axis, with the module value o a magnetic induction vector in the spatial grid nodes o the cylindrical coordinate system being obtained rom the ollowing ratios B r ; B ; B Br B, (9) r r where B r, B, B are the radial tangential components and module o a magnetic induction vector in the grid nodes o the physical and transormed coordinate system. The recalculation o anisotropic electromagnetic characteristics o the equivalent stator and rotor tooth zones as well as the zone o laminated stator body is carried out by means o the known ratios or the determination o series and parallel connection o electric and magnetic resistances [6] d ν0 d0 ν d d0 νr ν ; νr ν ; d d d ν d ν where material; 0 γ d γcu d0 γ, (10) d d 0 γ is the electrical conductivity o the rotor γcu is the copper conductivity; d, d 0 are either the wih o the tooth and slot, or the erromagnetic sheet and isolation o the laminated stator. Since the main electromagnetic ield equations written with reerence to the vector potential contain the second spatial derivative with respect to A, and the calculation o electromagnetic quantities is perormed only on the pole turbogenerator division, it is reasonable that the boundary condition given below be used so that the boundary conditions on a level o the second spatial derivative o the vector potential can more ully be taken into consideration. H r H r. (11) 0 80 This ratio is theoretically grounded and physically valid as the electromagnetic process on the pole turbogenerator division is periodical not only on the level o the irst spatial derivative o the electromagnetic ield vector potential, but also on the level o the higher derivatives o this unction. The boundary conditions along the radii o the rotor body zone, equivalent rotor tooth zone and air gap between the stator and rotor on the turbogenerator pole division in the physical coordinate system o the rotor, with the equations (9) and (11) being considered, can be represented as [7] A 2 A A A A A ; A k k 2 2 k n k 3 k n2 k n Ak 2 Ak 4 Ak n 2Ak k n 1 2A, (12) k n whereas along the radii o the equivalent stator tooth zone, the stator body and the air gap outside the turbogenerator correspondingly as [7] k Ak 2 2Ak n Ak 3 Ak n2 k n Ak 2 Ak 4 Ak n 2Ak 3 A 2 Ak n ; A 1 2Ak n, (13) where k is the index corresponding to certain nodes o the spatial grid along the angular coordinate. The boundary conditions or the equations o vector potential outside the turbogenerator are determined rom the ratio below A A A, (14) im 2 im im where і is the index corresponding to the nodes o the spatial grid in the cylindrical system o coordinates along the radius. On the internal boundaries o the tooth zone o the rotor and its body, the body o the rotor and the air gap between the stator and rotor, the tooth zone o the rotor and the air gap between the stator and rotor, the boundary conditions are ound rom the expression νi Ai νi Ai A i, (15) νi νi and between the tooth zone o the stator and its body, the body o the stator and the air zone by means o ν 1 ν i Ai i Ai A i, (16) νi νi in which the indexes і point to the numbers o the nodes that correspond to the media distribution boundary. The value o the rotor s winding current is determined by the equation di d u r i L, (17) where d w k l n r i Ri, (18) with w being the number o the rotor s windings; l r being the axis winding length; A Ri being the value o the unction o vector potential in the coordinate system o the rotor in the grid nodes being located in the winding; k being the coeicient involving the number o the nodes along the angular coordinate which get into the rotor s winding zone. The determination o the voltage in the stator winding phases is perormed on the basis o the ratios given below n di Si ui wikil, i = A, B, C, (19) m where u i is the voltage in the stator s windings; w i is the number o stator armature windings in each phase; l is the axial winding length; A Si is the value o the vector potential unction in the nodes o a dimensional discretization grid connected to the transerred stator coordinate system within the windings zone.

5 Calculation o Electromagnetic Processes or a Turbogenerator with Equivalent Stator Mathematical model o a turbogenerator in physical system o coordinates Let us consider the development o a mathematical ield model o a turbogenerator in a physical system o coordinates. The speciic eature o the model is that electromagnetic processes in a movable rotor are considered in the coordinate system o a moving rotor, while in a stator this is done in the coordinate system o an immovable stator. The given model implies the realization o physical movement o the rotor relative to the stator along the angular coordinate. The calculation o electromagnetic phenomena in the equivalent tooth zone and the massive rotor body is perormed by the expressions (1) and (2) respectively. In these equations the unction o vector potential in the electromagnetic ield А is connected with the rotor coordinates system. In all the other zones o the turbogenerator cross-section, А-values belong to the coordinate system o an immovable stator. The electromagnetic processes in the tooth zone and in the stator body can be ound rom the ratio [1] 0 A, (20) whereas the electromagnetic phenomena in the air gaps zones between the stator and rotor, and outside the turbogenerator can be calculated by the expression 0 ν 0 A, (21) where A is the unction o vector potential o the electromagnetic ield in the coordinate system o an immovable stator. The boundary condition along the radii o the turbogenerator cross-section zones on the pole division o the turbogenerator in physical coordinate systems takes the ollowing orm [7] Ak 2 Ak 2 2Ak n Ak 3 Ak n2 Ak n ; Ak n Ak 2 Ak 4 Ak n 2Ak 3 2Ak n, (22) And outside the turbogenerator it (this)can be expressed as A im 2Ai m Ai m. (23) On all the internal boundaries o the turbogenerator s cross-section zones, the boundary conditions are to be ound on the basis o the equations (15). The voltage values o the stator s winding phase are ound rom the expression n di Si ui wikil, i = A, B, C, (24) m where A Si is the vector potential o the electromagnetic ield in the stator coordinate system. All the other auxiliary mathematical ratios (6) (8), (14), (15) are o the same orm as in the previous model. The considered mathematical ield model o a turbogenerator in the phase coordinates implies reciprocal displacement o movable media in physical reerence rames relative to the nodes o spatial discretization grids provided the ratio (22) is ollowed, (25) where is the angular step o the discretization grid; t is the step o time integration o the dierential equations system; is the angular rotary speed. 5. Computation results On the basis o the developed models there was carried out computer simulation o the transition process o a no-load regime o the real turbogenerator ТGV-500 when the voltage o its excitation winding was equal to u = 141 V. Fig. 3 represents the time dependence o excitation winding current i o the turbogenertor s rotor in the transition process o the no-load regime. Fig. 3. Time values o an excitation winding current o the turbogenertor rotor. Fig. 4 9 demonstrate the calculated spatial divisions o the magnetic induction vector module in the turbogenerator cross-section zones in the transormed and physical coordinate systems. On the given spatial radius distribution, the coordinate grid nodes correspond to the ollowing zones o the turbogenerator cross-section: 054 is the erromagnetic rotor body zone; 5475 is the equivalent rotor tooth zone; 7587 is the zone o an air gap between the stator and rotor; 8717 is the equivalent stator tooth zone; is the stator body zone; is the air gap outside the turbogenerator. The number o the discretization grid nodes along an angular coordinate on the pole division o the turbogenerator is equal to 192. Fig. 4 depicts the spatial distribution o magnetic induction vector module in a coordinate system on the pole division o a turbogenerator when t = 1 sec o the transition process in a no-load regime obtained on the basis o the mathematical ield model in the transormed system o coordinates.

6 60 time when t = 650, sec o the transition process in a no-load regime obtained on the basis o the mathematical ield model in the phase coordinates. Fig. 4. Spatial distribution o magnetic induction vector module in transormed coordinate systems on the pole divisi on o a turbogenerator when t = 1 sec o transition process in a no-load regime. Fig. 5 shows the spatial distribution o magnetic induction vector module in transormed coordinate systems on the turbogenerator pole division when t = 100 sec o transition process in a no-load regime obtained on the basis o the mathematical ield model in the transormed system o coordinates. Fig. 6. Spatial distribution o the magnetic induction vector module in transormed coordinate systems on turbogenerator pole division when t = 650 sec o transition process in a noload regime. Fig. 5. Spatial distribution o magnetic induction vector module in transormed coordinate systems on turbogenerator pole division when t = 100 sec o transition process in a no-load regime. Fig. 6 illustrates the spatial distribution o the module o a magnetic induction vector in the rotor coordinate system on the pole division o the turbogenerator when t = 650 sec o the transition process in a no-load regime, obtained on the basis o the ield mathematical model in the transormed coordinate system. Fig. 7 represents the spatial distribution o magnetic induction vector module on the pole division o the turbogenerator in the phase coordinate systems when t = 650,01543 sec o the transition process in a no-load regime obtained on the basis o the mathematical ield model in the phase coordinates. Fig. 8 shows the spatial distribution o the magnetic induction vector module on the pole division o the turbogenerator in the phase coordinate systems at the Fig. 7. Spatial distribution o the magnetic induction vector module on turbogenerator pole division in phase coordinate systems when t = 650,01543 sec o transition process in a no-load regime. Fig. 8. Spatial distribution o magnetic induction vector module on turbogenerator pole division in phase coordinate systems when t = 650, sec o transition process in a no-load regime.

7 Calculation o Electromagnetic Processes or a Turbogenerator with Equivalent Stator 61 Fig. 9 demonstrates the spatial distribution o the magnetic induction vector module on the pole division o the turbogenerator with an conductive tooth zone o the stator in the phase coordinate systems at the time when t = 650, sec o the transition process in a no-load regime obtained on the basis o the mathematical ield model in the phase coordinates. Fig. 9. Spatial distribution o magnetic induction vector module on turbogenerator pole division with an conductive stator tooth zoneз in phase coordinate systems when t = 650, sec o transition process in a no-load regime. The obtained character o changing the turbogenerator s rotor excitation winding current i (Fig. 3) in the transition process o a no-load regime shows that when t = 1 sec the transition process in the winding is practically completed. While in Figure 4 we see that the transition electromagnetic process in the turbogenerator in a noload regime when t = 1 sec has just started, and is still ar to its end. Even i t = 100 с (Fig. 5) it is still going on. This conirms the importance o considering electromagnetic phenomena in dierent elements o the device construction (not only in the windings) when modeling even such a simple operating mode o the turbogenerator i.e. a no-load regime. The behaviour o electrodynamics devices in dierent modes is deined by electromagnetic processes in all their elements and not in windings only. I we compare the results given in Fig. 6 and 7, then we can see that the spatial distribution o the magnetic induction vector module in the equivalent stator tooth zone and in the stator body (Fig. 6) is displaced relative to the angular coordinate or a value that corresponds to the real reciprocal location o the rotor and stator at the given ixed time (Fig. 7 8). The data given in Fig. 6 are obtained by using the mathematical ield model o a turbogenerator in the transormed coordinate system. It is easier to develop the model o this kind than that in the phase coordinates because o the absence o any physical medium movement in it. That is why, it is reasonable that the mathematical ield model in a transormed coordinate system be used to calculate a no-load regime o a turbogenerator. Transition to a physical system o coordinates is realized by displacing electromagnetic quantities obtained or a certain angular coordinate [1]. This is veriied by the results given in Fig. 7 8 obtained on the basis o the mathematical ield model o a turbogenerator in the physical coordinate system. The relative erro or the dierence o the results o spatial distributions o electromagnetic quantities on the pole division o the turbogenerator cross-section zones obtained on the basis o the two considered models or a random ixed time point o the stator and rotor reciprocal placement can achieve less than 1 %. Special attention should be paid to the spatial distribution o the module o the magnetic induction vector in the turbogenerator cross-section zones (Fig. 9) calculated by means o the mathematical ield model in the phase coordinates. Fig. 9 shows a large increase o the magnetic induction vector module in the equivalent stator tooth zones as well as a considerable magnetic skin-eect on both the surace o the massive rotor and the surace o its equivalent tooth zone. This is because the electromagnetic process in an equivalent stator tooth zone in the mathematical model o a turbogenerator in a physical coordinate system was determined by the expression given below A, (26) as it is the very ratio to be used or describing electromagnetic phenomena in conductive zones o electrical devices [6]. In the turbogenerator no-load regime, the stator winding is open, i.e. there are no conductive currents in it. Since the mathematical model in phase coordinates involves physical displacement o a rotor (a source o turbogenerator magnetic ield in the given mode) relative to a stator, as it occurs in a real object, this leads to changing values o the unction o vector potential o the electromagnetic ield in the stator tooth zone, i.e. 0. (27) This phenomenon, i the medium is conductive (Г 0), results in inducing an eddy component o the ield (an eddy current). Thus, by using the expression (26) we can present a regime o symmetric short circuit o the stator winding rather than the no-load one o the turbogenerator. These are the results that demonstrate the spatial distribution shown in Fig. 9. The results we obtained just conirm the adequacy o the developed model, as in any conductive environment moving in a magnetic ield, eddy currents will be induced.

8 62 To reproduce a no-load regime o a turbogenerator, the electromagnetic process in an equivalent tooth zone o a stator should be calculated by the expression (20) rather than (26), as the equation (20) implies the absence o conductive currents in the tooth zone o the stator unctioning in the no-load regime. It was the very way o obtaining results presented in Fig The availability o Г 0 in the equation (26) o the mathematical ield model o a turbogenarator in a noload regime in phase coordinate systems while describing electromagnetic processes in the tooth zone o the stator leads to reproduction o the winding shortcircuit mode. Fig. 10 shows time dependences o A-phase voltage o stator winding ater completing the transition process or the linear (1) and nonlinear (2) cases o electromagnetic media characteristics achieved on the basis o the mathematical models in both the transerred and physical coordinate systems. Fig. 10. Time values o A-phase voltage in the stator winding ater completing the transition process or linear (1) and nonlinear (2) characteristics o electromagnetic media obtained on the basis o the models developed. Some attention should also be paid to voltage curves shown in Fig. 10. According to speciications o the turbogenerator ТGV-500, the phase voltage peak value equals V. The voltage peak value in the linear model is equal to V, while in the nonlinear one is V (Fig.10). Thus, the most accurate ield model o a turbogenerator is the cause o a 35% error in a linear variant and a 23% error in a nonlinear one. No assumptions introduced by the theory o circuits can explain this. For a long time the results obtained could not be explained logically. Seeking the causes o such an error in the theory and in the calculation algorithm was a ailure. Only ater developing more detailed models accounting real tooth structures o a stator and rotor the identiied problem was solved. Thereore, it must be emphasized that only the ield mathematical model o a turbogenerator more ully taking into account a real structure o a device ensues calculations o high accuracy. Using the method o replacement o complex structures o electrical devices by equivalent media actually changes the structure o the device, and in this case it is a mistake to expect real electromagnetic processes in changed structures o the objects to be reproduced by such mathematical models. But in integral values o such class, the models can demonstrate high accuracy. This is a speciic task o inding appropriate parameters or the developed models. 5. Conclusions When developing mathematical ield models o electrodynamic devices on the basis o Maxwell s equations it is necessary that the systems o coordinates the process is considered in be taken into account. With this actor being ignored, electromagnetic phenomena in the models developed can be misrepresented. Most o the works dedicated to the analysis o electromagnetic processes in electrodynamic devices on the basis o the theory o electromagnetic ield do not ocus on coordinate systems, as they apply timeless calculation methods and consider real dynamic o media movement as a series o reciprocal ixed positions o movable and immovable structures. This oversimpliies the physical nature o real electromagnetic processes in devices as they ensure observing the law o total current and not objective regularities (laws) o electromagnetic phenomena in general. As the results obtained by means o the developed models show, calculations o electromagnetic processes in electrodynamic devices with a constructive periodicity o structures on their pole division should be done by the mathematical ield models in the transormed coordinate system. Absence o mechanical movement o media in such models greatly simpliies the model itsel, its realization program, and the calculation process. The method o replacement o complex structure elements by continuous anisotropic media possessing the above mentioned characteristics can be used in mathematical ield models o electrostatic and electrodynamic devices. This results in actual changing the device design and in obtaining wrong electromagnetic processes. Only the mathematical ield models which most ully take into account design peculiarities o the devices can adequately reproduce electromagnetic phenomena both in the case o dimensional distributions o electromagnetic values and in the case o integral parameters o the devices.

9 Calculation o Electromagnetic Processes or a Turbogenerator with Equivalent Stator 63 Reerences 1. Y. Kovivchak, Features o the Calculation o the Electromagnetic Field in Moving Media // Elektrichestvo. Moscow, Russia: Znak P (Russіan) 2. L. Wu, Z. Zhu, D. Staton, M. Popescu, D. Hawkins, An Improved Subdomain Model or Predicting Magnetic Field o Surace Mounted Permanent Magnet Machines Accounting or Tooth-Tips // IEEE Transactions on Magnetics. Vol. 47, P T. Lubin, S. Mezani, A. Rezzoug, 2-D Exact Analytical Model or Surace-Mounted Permanent-Magnet Motors With Semi-Closed Slots // IEEE Transactions on Magnetics. Vol. 47, P P. Pister, Y. Perriard, Slot Less Permanent-Magnet Machines: General Analytical Magnetic Field Calculation // IEEE Transactions on Magnetics. Vol. 47, P E. Rosseel, H. Gersen, S. Vandewalle, Spectral Stochastic Simulation o a Ferromagnetic Cylinder Rotation at High Speed // IEEE Transactions on Magnetics. Vol. 47, P V. Tchaban, Mathematical Modeling o Electromechanical processes. Lviv, Ukraine: Publishing house o Lviv Polytechnic National University (Ukrainian) 7. Y. Kovivchak, Deinition o Boundary Conditions in the Spatial Problems o Electrodynamics // Elektrichestvo. Moscow, Russia: Znak P (Russіan) РОЗРАХУНОК ЕЛЕКТРОМАГНІТНИХ ПРОЦЕСІВ ТУРБОГЕНЕРАТОРА З ЕКВІВАЛЕНТНИМИ ЗУБЦЕВИМИ ЗОНАМИ СТАТОРА І РОТОРА В РЕЖИМІ НЕРОБОЧОГО ХОДУ Ярослав Ковівчак На основі рівнянь Максвелла в потенціалах проведено розрахунок електромагнітних процесів у рухомих і нерухомих зонах поперечного перерізу турбогенератора у перетвореній і фізичній системах координат для режиму неробочого ходу. Показано, що використання методу еквівалентування зубцевих зон пристрою суцільними анізотропними середовищами та нехтування систем координат, у яких розглядається процес у польових математичних моделях, приводить до спотворення реальних електромагнітних явищ. Ph.D., Associate Proessor, works at the Department o Automated Control Systems at the Institute o Computer Sciences and Inormation Technologies, Lviv Polytechnic National University, Ukraine. Research interests: theory o electromagnetic ield, mathematical modelling.