IMPROVING EFFICIENCY IN DETERMINING THE INDUCTANCE FOR THE ACTIVE PART OF AN ELECTRIC MACHINE'S ARMATURE BY METHODS OF FIELD MODELING

39 and a system of three-phase armature winding the main task of which is formation of rotating magnetic field. Uniform distribution of rotating magnetic field and installed power of the EM armature, starting current and maximum torque depend on the ratio of active and inductive resistances (dissipative and mutual inductances) of phase windings [1]. Well-known engineering procedures take into account design features of phase windings and active part of the EM stator just approximately and do not enable determination of its parameters with high accuracy. Therefore, error in design calculations of the winding data and active part of the EM stator is at least 15 %. In this regard, many issues related to the processes of electromagnetic and electromechanical conversion of energy including features of determining inductive parameters of


Introduction
Further improvement of designs and development of the theory of electrical machines (EM) are directly related to the search for new technical solutions and technologies. This would ensure creation of devices with improved technical characteristics and high indicators of power efficiency of regulated electric drives, autonomous power supply systems for a series of consumers in industry, power engineering, agriculture and special-purpose devices.
The most common types of electric machines are asynchronous and synchronous [1,2], synchronous with permanent magnets [3,4], and synchronous with combined stator windings [5]. They have a fundamentally similar core design aptation of three-dimensional field modeling to the problems of optimal design and calculation of EM parameters.

The aim and objectives of the study
The study objective consists in development of an effective approach to determining self-and mutual inductances of phase armature windings by the method of circuit-field modeling taking into account magnetic and electrophysical properties and design features of multicomponent EM elements.
To achieve this objective, the following tasks were set: -elaborate mathematical description of a circuit-field 3D model of electromagnetic field reflecting features of electromagnetic energy conversion taking into account magnetic properties of materials for active part of the EM armature with multicomponent design elements; -establish laws of dependence of electromagnetic processes on self-and mutual influences of phase currents of the armature causing appearance of effects of self-and mutual induction in presence and absence of magnetic properties of materials of the active part of the EM armature; -determine self-and mutual inductances of dissipation from phase windings in presence and absence of magnetic properties of materials of active part of the EM armature and correction coefficients for inductive parameters of the EM armature phase windings calculated by a classical engineering procedure in order to minimize current errors.

Development and mathematical description of a circuit-field 3D model of electromagnetic processes in the electric machine armature
A 3D calculation range of field modeling of electromagnetic processes occurring in the active part of the tested synchronous machine (SM) armature includes the following subranges: armature core 1; three-phase armature winding 2; insulation system 3 in the groove zone of the armature core. A coil group of the armature winding consists of frontal and groove parts which are geometrically connected and form an solid coil group. Coil groups are geometrically symmetrical. Each coil group of the winding has input 4 and output 5 ( Fig. 1). In mathematical description of electromagnetic processes, assumptions of isotropy of electrophysical and electro-EM armature have not been studied sufficiently [6]. This would make it possible to take more accurately into account magnetic coupling with various electrotechnical complexes and power systems and their mutual influence on features of their work. This determines relevance of high-precision and highly efficient methods for calculating inductive parameters of phase windings of EM armature at the stages of their design and modernization.

Literature review and problem statement
It is known that classical methods applied in calculation of parameters [1] and characteristics [2] of EMs and analysis of their operation in various modes are based on a series of assumptions [6]. They lead to an essential calculation error and cannot be used for optimization [7] and design [8] of modern energy-efficient electric drive systems. Along with classical methods, methods for calculating parameters and characteristics of transformers [9], EM [10] and converters [11] are the most popular today in engineering practice as well as among researchers. They are based on the theory of electric and magnetic circuits. Depending on operating modes of electrotechnical and electromechanical systems, these calculation methods are grounded on commonly known exchange circuits and are built on a series of assumptions and simplifications. Moreover, authors do not take into account influence of surface effects [12], multicomponent spatial structure [13], nonlinear electrophysical [14] and magnetic [15] properties of active materials and winding circuits [16]. All these assumptions cause a decrease in accuracy of determining active and inductive parameters of phase windings of the EM armature.
A more complete account of influence of design features of active part of the EM armature on parameters of their phase windings as well as nonlinear properties of structural materials was made by authors of [17] using field modeling and parametric analysis [18]. However, in multicomponent sections of the active part with nonlinear electrophysical properties, computer implementation of a model based on finite element methods is complicated by large time consumption and requirements to computing resources [19]. A three-dimensional model of electromagnetic processes was proposed in [20] for determining electrical parameters. However, its use is limited and possible only for the systems with materials having linear magnetic properties. For effective numerical implementation of field models for EM [21] and transformers [22], it was proposed to use differentiation of sizes of finite elements and approximations by first-order Lagrange polynomials. Application of this approach would significantly reduce computation and time resources when modeling in a PC. Some authors try to simplify geometric model [23], refine nonlinearity of electrophysical and magnetic [24] properties by empirical dependences or use two-dimensional models [25]. These assumptions and simplifications reduce accuracy of modeling results. To obtain high-precision calculation results, the model of electromagnetic field of armature and the EM in general must reflect in detail multicomponent structure of the active part and take into account nonlinearity of electrophysical and magnetic properties of active materials. Effectiveness of numerical implementation of the three-dimensional EM model must correspond to requirements of the problems of design parameter optimization. This necessitates development of special approaches that will improve accuracy and efficiency of calculations and ensure ad- magnetic properties of materials, absence of bias currents and free charges are made [26]. In this case, non-stationary electromagnetic processes in the SM in a short-circuit mode can be described by a conjugated system of nonlinear partial differential equations [27]: where A is vector magnetic potential; V is electric potential; σ(Q) is electrical conductivity; B is magnetic field induction; μ is relative magnetic permeability; e r is relative dielectric constant; Q is temperature; ω is angular frequency; J e is density of external current source; j indices correspond to subranges of geometric calculation range (Fig. 1).
In accordance with [28], the equation system (1) is supplemented with the Coulomb calibration condition div(A)=0.
The conditions of conjugation for magnetic and electric fields can be formulated as in [30]: , , where H is the magnetic field strength; E is the electric field strength.
Boundary conditions are set at external boundaries of the computation range [30]: Temperature conditions of the SM are considered stationary and estimated in accordance with [31].
Design of a two-layer winding of the SM armature contains four coil groups with three coils in each coil group per phase. The first and third, second and fourth coil groups are interconnected in series. The second and fourth coil groups are connected in parallel to the first and third coil groups. Outputs of coil groups of all phases are Y-system connected (Fig. 2).
Initial conditions correspond to the first law of switching:  Geometric model of the armature winding coil in the tested SM (Fig. 2) is implemented as a single turn consisting of one effective conductor. To take into account the number of turns, coil zones of the SM armature winding are described by the equation from [30]: where n is the number of turns in the winding; I cir is a phase current; A is the cross section of an effective conductor; e coil is the vector variable representing local density of effective conductors in a coil, length and cross section. According to [32,33], electrophysical processes in dielectric materials can be considered similar to the processes of electrical conductivity in materials with a relatively small value of electrical conductivity.
For j-elements of the active part of the SM armature, magnetic field energy, its average values for each j-th zone, active losses of the armature windings and current density are calculated [22]: In addition, phase currents in the armature winding are calculated according to (5).
Computer implementation of the 3D field model (1)-(5) is performed in the structure of COMSOL Multiphysics software tools [28,29]. Calculation subranges of the 3D model are divided into three-dimensional finite elements and have a tetrahedral shape. The tetrahedra faces are approximated by the first-order Lagrange polynomials which, according to [21], is sufficient to ensure high accuracy of calculations.
To improve efficiency of numerical implementation of the model, dimensions of finite 3D elements are differentiated. For the subregion of the anchor core made of cold rolled isotropic electrical steel with a pronounced nonlinearity of magnetic properties, dimensions of finite elements decrease and increase as they approach the outer boundary of the calculation subrange. This makes it possible to implement numerically the model with high accuracy at smaller computational and time resources [21,22].

5.
Studying the electromagnetic processes occurring in the active part of the synchronous machine armature 5. 1. Results obtained in the study of magnetic field in the active part of the armature of a synchronous machine Field 3D modeling of the electromagnetic field in the time-dependent statement of the problem in subregions of the active armature part was carried out on a sample of experimental SM. The experimental SM was built on the basis of a 2.6 kW MTF-111 induction motor with a non-factory four-pole two-layer armature winding. Design and diagram of connection of coil groups and phase windings of the armature fully complied with Fig. 1, 2. Inputs of the first and second coil groups of the three-phase winding were connected to an AC voltage source of 50 Hz industrial frequency. Angle of interphase shift was 120 el. grades. Phase voltage was corresponding to short-circuit voltage of 0.5 U nom .
Results of 3D modeling are presented by distribution of the z-component of magnetic potential vector Az along entire active part of the armature (Fig. 3, a) and as XY-planes in the zones of S (plane 1) and j (plane 2) of the armature core length and in the zone of its face part (plane 3) (Fig. 3, b).
According to the data obtained from 3D field modeling, it was found that when testing the active part of the SM armature in a short-circuit (SC) mode, magnetic field is localized in the system of its phase windings. Surfaces of magnetic field localization close around bores of the armature core grooves and extend along entire zone of the grooves. Concentration of magnetic field decreases in the region of the face part of the core and beyond it, in the region of frontal parts of the winding. Moreover, values of Az j and magnetic field energy W j were no more than 12 % of their values in the central region of the active armature part (Fig. 3, a). Magnetic field had a plane-parallel character (projections 1, 2, Fig. 3, b) in the central zone of the active core part measuring up to 75-80 % of its length. Prevailing effects of self-induction and mutual surface induction and a series of other end effects were observed closer to the core face zones and in the region of frontal parts of the armature winding [2]. They were caused by peculiarities of distribution of scattering fields depending on the change in spatial orientation of frontal parts of the SM armature phase windings.
In order to study the effect of self-induction and mutual induction on electromagnetic parameters of the armature phase windings, cases of individual turn-on of each A, B, C phase and a group turn-on of A-B, B-C and C-A phases of the armature winding were considered. This approach has made it possible to apply the principle of superposition to the electromagnetic fields according to the Bio-Savart law [34] for the calculation range of the active part of the SM armature.

2. Establishing the laws of electromagnetic processes caused by self-and mutual influences of phase currents in the synchronous machine armature
According to the data of numerical modeling, oscillograms of phase currents were obtained for corresponding cases of alternating turn-ons of phase windings of the SM armature (Fig. 4). When A-B phase windings are turned on, self-induction EMF is induced in the A phase with its vector aligned in concert with the phase voltage vector. In vector (Fig. 4, a). When A-C phase windings are turned on, self-induction EMF is generated in the A phase with its vector directed opposite to the phase voltage vector. In this case, a resulting vector of  (Fig. 4, a). When all phase windings are turned on, resulting current , ABC A I  vector is formed which can also be determined from the following expression:

3. Determining self-and mutual inductances of armature phase windings of a synchronous machine
In accordance with [22,23] and based on the data obtained in calculating the values of magnetic field energy (6), self-dissipation inductances and mutual inductances between phases of the SM armature were determined:    and mutual inductances between the winding phases are given in Table 1. They were found with and without taking into account magnetic properties of the SM armature core. According to the EM theory [1,2], mutual inductances of the armature phase windings were М AB =М ВА =М АС = =М СА =М ВС =М СВ =1/2·L AA (L AA =L BB =L СС ) which is confirmed by calculations according to (8) with μ=1 based on the modeling data with an error of ( ) 1 0,4 %. M µ= δ ≤ When magnetic properties of the active part of the SM armature were taken into account, only equality of self-dissipation inductances of the phase windings was preserved with values 3.5 higher than self-values at 1.

µ =
In presence of magnetizing and demagnetizing effects from currents of neighboring phases, symmetry of the mutual inductance system ; was violated. Despite this, symmetry of total inductances of the armature winding phases (9) was preserved, and according to the data in Table 1, it is  Table 1 Calculated values of dissipation inductance and mutual inductance between phases of the SM armature winding according to numerical modeling 6. Discussion of the results obtained in modeling electromagnetic processes in the active armature part of the tested electric machine Procedures of calculating inductive parameters of the SM armature winding based on classical EM theory are valid if magnetic properties ( ) 1 µ = of materials of the active part of the SM armature are not taken into account. This was confirmed by verification of inductive parameters according to the field modeling data ( Table 2). Based on the results of field modeling, description of the phenomena of self-induction in phases of armature winding and formation of components of induced currents in a phase from action of phase currents in neighboring phases were obtained and their magnetizing and demagnetizing properties were considered. These processes cause asymmetry of mutual inductance systems between winding phases but do not violate symmetry of total inductances of the armature winding phases. Electromagnetic parameters and ohmic resistances of the armature winding of the tested SM were validated by comparing them with the results of prototype tests conducted in accordance with GOST 11828-86 at a laboratory of Zaporizhzhia Polytechnic National University, Ukraine (Fig. 6).
Electromagnetic parameters of the armature winding were measured using OWON XDS3202E oscilloscope. According to the results of numerical modeling and actual measurements of the experimental sample, error in ohmic resistance of the armature phase windings was 0.00694 %. R δ = Relative values of current errors for numerical modeling shown in Table 2 were found from the following expression [22]: where I t , I s are the values of currents according to the test data and numerical modeling, respectively. Measurements were carried out both for turning on individual phases of the armature winding, A-B, A-C, B-C phase groups and a three-phase turning on.
The proposed approach to determination of self-and mutual inductances between phases of the SM armature winding ensures reliability and accuracy of the data obtained in 3D modeling of magnetic fields. This approach is based on decomposition of electromagnetic processes through combinations of turning in phase windings of the armature to the network using a combination of electrical circuits of phase windings and geometric 3D region of the active part of the SM armature. To ensure adequacy of the widely used well-known three-and two-phase EM models based on the systems of differential equations of the first order [6,10], the calculated inductive parameters of their windings should be refined using correction coefficients k L ( Table 2). Correction coefficients obtained in field modeling can be applied to inductive parameters of the armature winding а b Fig. 6. General view of prototype anchor assembly of a synchronous machine: a -front view; b -rear view of the tested SM and the armature winding of EM of other series with similar designs. For other designs and parameters of windings and the core of EM armature, coefficients k L should be determined according to the proposed approach. In the future, the approach proposed in this paper will be extended to determination of self-and mutual inductances of the rotor winding as well as the main inductances between phases of the armature windings and the SM rotor.
The results of this work can be applied to the problems of optimal design and calculation of EM parameters and in determining optimal magnetic compatibility of EM with elements of the electrotechnical complex and power systems. This will improve energy efficiency indicators of these elements depending on features of their operation.

Conclusions
1. It was established that mathematical description of the 3D circuit-field model of magnetic field of the active part of the EM armature with multicomponent elements of its structure reflect features of electromagnetic energy conversion. Reliability and accuracy of the results obtained in numerical modeling as well as the results of calculation of inductive parameters of the SM armature winding were substantiated by validating the calculation data of numerical modeling and the results of physical tests on a prototype EM. When taking into account magnetic properties of materials of the active part of the SM armature, relative current error did not exceed 2.68-2.91 % and when magnetic properties were not taken into account, the error was 103.09-106.32 %.
2. The phenomena of self-and mutual induction in phases of the armature winding, formation of components of induced currents in the phase under the action of phase currents in neighboring phases as well as their magnetizing and demagnetizing properties were considered. It was established that currents with a leading phase possess magnetizing property in a three-phase system of currents of the EM armature winding with respect to the phase under consideration. In this case, an induced current is formed in this phase. It is purely reactive and has an inductive character. Currents with a lagging phase have demagnetizing property. In this case, an induced current is formed which is purely reactive and has a capacitive character. In the case when magnetic properties of materials are not taken into account (μ=1), their modular values are respectively 0.1-0.15 of the current module of a separately turned in phase. The slight predominance of the capacitive component of the induced current module is due to the fact that concentration of magnetic field decreases in the region of frontal parts of the winding. At the same time, value of the magnetic potential vector and the magnetic field energy make up no more than 12 % of their values in the central region of the active part of the armature which is a consequence of action of end effects from frontal parts of the phase windings of the SM armature. When taking into account magnetic properties of materials (μ=B(H)), their module values get 2 times higher which brings about a significant non-symmetry in terms of the modulus of inductanceand capacitance-induced currents in the phase. These laws of electromagnetic processes are also valid for other phases.
3. It has been found that self-dissipation inductances of the SM armature phase windings without taking into account magnetic properties of materials (μ=1) were equal by their values (L AA =L BB =L СС ) and mutual inductances М AB =М ВА =М АС =М СА =М ВС =М СВ were equal to 1/2 L AA . Discrepancy in mutual inductances between winding phases was δМ| (μ=1) ≤0.4 %. When magnetic properties of the active part of the SM armature are taken into account, only equality of self-inductances of dissipation of phase windings is preserved. They were 3.5 times higher than self-values at μ=1. Under action of magnetizing and demagnetizing effects from currents of neighboring phases, symmetry of the mutual inductance system М AB ≠М ВА ; М АС ≠М СА ; М ВС ≠М СВ is broken. Despite this, symmetry of the complete inductances of the phases of the SM armature winding is preserved. The discrepancy error in terms of symmetry of total inductances of the armature winding phases was δL| (μ=B(H)) =0.224÷0.817 %. For accurate determination of inductive parameters of the SM armature winding by a classical method, correction coefficients for values of dissipative inductances and mutual inductances of the armature phase windings were determined taking into account magnetic properties of materials of the active part of the SM armature. This will make it possible to minimize current errors and ensure adequacy of widely used well-known threeand two-phase EM models based on the systems of differential equations of the first order.