COMPUTER SIMULATION OF ELECTROMAGNETIC FIELD WITH APPLICATION THE FREQUENCY ADAPTATION METHOD

1Dr. Sc., Associate Professor, Head of the Electrical Machines Department, Zaporozhye National Technical University, Ukraine 2Ph. D., Associate Professor, Associate Professor of the Electrical Machines Department, Zaporozhye National Technical University, Ukraine 3Ph. D., Associate Professor, Associate Professor of the Electrical Machines Department, Zaporozhye National Technical University, Ukraine 4Senior lecturer of the Electrical Machines Department, Zaporozhye National Technical University, Ukraine


INTRODUCTION
A present stage of powerful radio-electronic and electrotechnical systems development, with a capacity more than 1 MW, imposes increased requirements to their energy equipment, uninterrupted operation and power supply reliability in operating modes [1][2][3]. High level of competition in the international markets (in civil and defensive spheres) and also essential restrictions on mass-dimensional indicators and prices increase for electrotechnical materials both conductive, and ferromagnetic, cause relevance of creation a new domestic production at the level of the best world samples. Therefore, at the production design preparation stage of powerful radio engineering systems, it is necessary to pay great attention to auxiliary systems and structural elements, in particular, to the systems of special current-conducting wires which are carrying out a reliable power supply the basic modules and units.
Despite the large volume of the researches connected with similar systems design of current-conducting wires for other industries [4][5][6][7][8][9][10][11][12][13][14][15][16] to provide adaptation of their results for powerful radio engineering systems very difficult. This is due to high density of their configuration and installation, the electric power transmission with AC high harmonics and, as a consequence, a significant influence of proximity effects, self-and mutual induction, which are usually neglected in common industrial equipment design. The known calculation procedures and powerful systems design of special current-conducting wires, which are based on circuit simulation methods and root-mean-geometric distances [4,5,[17][18][19], do not satisfy the precision requirements and reliability. Respectively, it should be compensated for, by creating and testing prototypes, and as a result, additional material expenses and financial resources, labor costs and time for production preparation, by increasing the final product cost. Now mathematical simulation methods of electrodynamic processes in high-frequency systems on the basis of Helmholtz [20][21][22] equations were widely adopted, which are implemented by expensive specialized software packages (ANSYS, Comsol Multiphysics, etc.). Their use leads to the essential growth of financial costs on hardware and software production design preparation, which becomes very difficult at the existing crisis factors. In addition, the use of software with a free license, for example, FEMM, for modeling the electrophysical processes of high-frequency alternating current is limited due to the absence of specialized software modules in them. Besides, the software application with a free license, for example FEMM, for AC high frequency electrophysical processes simulation is limited due to the absence in them specialized program modules. Therefore, the elaboration problem of perspective approaches for field models numerical realization on the basis of Maxwell's equations in frequency statement should be consider relevant. These approaches expand the area of software use with a free license and adapted to production design preparation requirements for special systems of currentconducting wires of powerful power supply modules in radio engineering systems.
The purpose of the work is the elaboration a new numerical realization methods of field models taking into account AC electrophysical processes with high frequency on the basis of Helmholtz equations frequency formulations, adapted to software packages use with a free license.
To achieve this purpose, the following tasks are formulated: to elaborate new numerically-field calculations methods of interrelated electromagnetic processes during energy transmission, with AC high frequency, for special systems of current-conducting wires; to verify these methods on test mathematical models experimentally and to elaborate on this base the recommendations about their practical use.

PROBLEM STATEMENT
The electromagnetic field in geometric domains of current-conducting components and their circumambients is described by conjugate system of equations for vector magnetic potential and electric potential complex amplitudes [7][8][9]11]: In the system of equations (1), the projections of A and V are parametric functions of AC angular frequency ω : The system of equations (1) is supplemented by magnetic and electric fields conjugation conditions at environment sections boundaries with different electrophysical properties for magnetic field strength vectors and current density in current-conducting wires [7][8][9]: under boundary conditions on the external shields surfaces [7][8][9]: and on the symmetry planes [7][8][9]:

REVIEW OF THE LITERATURE
The known engineering techniques of calculation and current-carrying wires design [4][5][6][17][18][19], based on the electrical circuit's theory, methods of generalized expressions and mean geometric distances, have essential assumptions and don't answer modern requirements of precision. In recent years, at high-frequency electromagnetic processes description, field simulation has become widespread [7][8][9][10][11][12][13][14][15][16]. The analytical and numerical methods [23,24] realized on the computer can be applied to the problem formulations of electromagnetic fields simulation given above. Numerical methods are conventionally divided into three main groups [24]: integral (a boundary elements method, a secondary sources method, an integration method on field sources and a inductively connected circuits method) [23,25,26], differential (finite differences method and finite volumes method) [27,28] and variation (Ritz's method, Galerkin's method, finite elements method) [29,30,31]. The MBE belong to the most widespread integrated methods [26,27,32]. Their feature is that the integrated equation of rather unknown quantities is formulated only on the computational domain boundary. They are successfully applied to the linear partial equations solution in private derivatives of elliptic type. The FEM belongs to the variation methods based on the use of variation statements equivalent to an initial boundary-value problem [30,33,34]. FEM is based on the fact that for the elements into which the investigated domain is divided, an approximation of the required function is described [34,35]. And this approximation has to provide this function's continuity throughout the definition range. FEM allow to realize rather large number of elements, when using special algorithms for systems numerical solution of algebraic equations with band matrix coefficients. The elaboration of this method became possible thanks to computer technology development and specialized software with user-friendly interfaces. They are successfully applied to the field problems solution in irregular geometrical shape domains. FEM disadvantage is the stability lowering of computational process for nonlinear frequency field models and requirements for the use of specialized expensive software [11][12][13][14] and significant computing resources [10]. Because of the essential differences in the elements geometric dimensions of current-conducting systems, in electromagnetic processes simulation, the authors [12][13][14][15][16] significantly simplify computational models or consider them in a plane-parallel statement [13,14]. Also, the community absence of approaches in describing and simulating low-and high-frequency electromagnetic processes leads to significant increase in computing and time resources that complicate the use of field calculation methods at production design preparation stages. It causes need of elaboration new approaches based both on

MATERIALS AND METHODS
For synthesis a new method for system equations (1)-(3) solution, it is necessary to differentiate system equations of (1) with respect to the parameter ω, neglecting magnetic and electrical properties dependences of current-conducting wires materials on AC angular frequency ω: Replacing continuous range domain of frequency ω , by the discrete sequence , transform this system of the differential equations to an integrated form: For the left parts of system ratios (8) the next designations are accepted: (9) and the right rectangles formula [36] is applied analytical and numerical methods combination and on the frequency formulations reduction of Helmholtz problems to other types of problems.
Perform the averaging of system ratios (7) for frequency parameter change range ( ) (12) and apply the right rectangles formulas [36] to the left parts in the systems (11), (12) and the left rectangles [36] for their right parts. By reducing them to ( ) which allow solutions independent of each other. If to perform summands regrouping between the left and right parts of system (13), (14) integrated equations, then it is possible to obtain the modified systems of equations in the form: which differ from the recurrent equations (13), (14) by frequency parameter 1 + ω k .
In the systems of equations (13)-(16) initial approaches for vector magnetic and electric potentials are defined from independent solutions of homogeneous differential equations (17) and can be considered as approximations to direct current.
Thus, by transformations (8)- (14), the system of equations in frequency formulation (1) adapts to the mathematical description and approaches for solutions in DC analogies (15)- (18). The main advantage of adapted approach should be considered possibility of independent solution the magnetic and electric fields equations with respect to frequency derivatives of vector magnetic and electric potentials on each step of frequency parameter discrete change. Initial approximations of the vector magnetic potential and electric potential are defined according to direct current. Convergence conditions of iterative process are defined by convergence conditions for Euler's method [37].

EXPERIMENTS
In data absence on the application new frequency adaptation method (7)-(18), for realization electromagnetic field problems, it is expedient to estimate its efficiency on (10) the basis of a model problem, which has an exact analytical solution. For this purpose there is considered a skin-effect mathematical description in current-conducting components of waveguide system [20]. There is investigated the phenomenon of electromagnetic field localization and electric current in the domain of electrically conductive plate, ∞wide and ∞ -extended, with end z ( fig. 1). It is accepted that current density vector will be directed along the 0х axis and there is considered its projection x J , and ∞ extended system along the 0y axis.
Since the electric current is a harmonic function of time (with ω ), then, taking into account the ratio , it suffices to consider Helmholtz equation with respect to electric field strength amplitude [20]: with boundary conditions in the form: General solution of equation (19) Taking into account the expression for skin-effect layer thickness , 2πσμω = δ c (22) expression (21) can be transformed to the next form:  (20), we obtain an analytical solution that describes distribution of the intensity complex amplitude Ẽ in electrically conductive domain ( Fig. 1):   (24). A typical feature of frequency adaptation method is the even number of iterations. On odd iterations, the imaginary components of the solution are specified, and on even ones, the real components. At the same time the number of iterations for studied model doesn't exceed 4.

RESULTS
Results of numerical experiments for a research of approximate solutions precision, obtained by an analytical method and a new frequency adaptation method are presented in Fig. 2. According to the exact solution, the nonlinear character of the field strength distribution is determined by the depth of its penetration into electroconductive environment, mainly by AC frequency. At electroconductive layer thickness the distribution will have a parabolic form. Comparison of exact solutions the system of equations (1) and calculations by a new frequency adaptation method (25), taking into account ω = ω Δ , confirms that the error of this method is lower than 0,6% for parabolic form of field strength distribution in current-carrying components domain of current-conducting systems (Fig. 2a).
, the parabolic form of field strength distribution is transformed into a flattened form (Fig. 2), and the error of frequency adaptation method increases to 5.1% due to the effect of the "false" local convexity on the symmetry axis of current-conducting wires. This negative effect is easily compensated by a step reduction of frequency adaptation to ω ⋅ = ω Δ 5 . 0 , and by increasing the number of iterations up to 4, at the same time the error decreases to 1.2% in comparison with exact solution (Fig.  2b). It confirms high precision of the offered frequency adaptation method, which can be also realized in structure of application programs packages for stationary field simulation, as a rule, having a free license (for example, FEMM).
For current-conducting wires with a rectangular plan configuration, field calculations were performed in Comsol Multiphysics software structure for frequency model (1)-(6) and in the FEMM software structure for frequency adaptation model (18). Field calculations in Comsol Multiphysics and FEMM structures used triangular finite elements, the number of which was 55264. When comparing the arrays of local values for magnetic field strength module, the relative discrepancy deviation for these methods did not exceed 1.24% for ω = ω Δ and 0.75% ω ⋅ = ω Δ 5 , 0 . As As

DISCUSSION
Thus, the frequency adaptation method has a simple numerical implementation and doesn't require the use of specialized commercial software. This method can be used in conjunction with final elements method and can be realized in the FEMM structure. At the joint application the frequency adaptation methods and final elements method, the numerical realization efficiency is caused by reduction the problem dimension on each step in frequency twice. It's provided by step-by-step realization the recurrent equations (18)  range. Thus, the frequency adaptation method provides good computational accuracy for engineering calculations. This method can be used in problems of calculation and analysis of electromagnetic fields in one-dimensional, twodimensional and three-dimensional formulations. Numerical realization data, in Comsol Multiphysics and FEMM structures for frequency adaptation to stationary field models, correspond with high precision to numerical realization data for frequency field models, but the time consumption are reduced from 1.12 to 2.8.

CONCLUSIONS
A new method of frequency adaptation is elaborated, which provides systems of Helmholtz equations reduction in frequency domain to the recurrent Maxwell's equations in DC analogies, high precision and field simulation efficiency. At the joint realization the frequency adaptation methods and final elements method the number of freedom degree decreases twice, which is provided by step-by-step realization the recurrent modified Maxwell's equations in DC analogies, both for the real, and for imaginary components of the electrical and vector magnetic potentials. The frequency adaptation method's error for problem solution of AC electric field in a one-dimensional statement doesn't exceed 0.6% for range. At calculating the plane-parallel electromagnetic field for current-conducting components of current carrying systems the frequency adaptation method's error doesn't exceed 1.24%, and time consumption are reduced from 1.12 to 2.8.