The cascade equivalent A-H-circuit of the salient-pole generator on the base of the anisotropic model

Received Feb 17, 2020 Revised Jun 18, 2020 Accepted Jun 30, 2020 In the scientific studies of the electric machines, cylindrical and Cartesian laminated models have long proved their operability when it is necessary to transform the general solutions of the partial differential equations into fourterminal network equations and to create the cascade equivalent circuits of the electric machines. In the case of the salient-pole machines, piecewise continuous Sturm-Liouville eigenfunctions are usually used as the general solution. Unfortunately, we cannot create the cascade equivalent circuit of the salient-pole electric machine when (to ensure both the uniqueness and the accuracy of the solution) the field is modeled with many piecewise continuous Sturm-Liouville eigenfunctions (in the zone of poles) and many sinusoidal functions (in the air gap). Nevertheless, the author developed the approximate method for modeling the magnetic field of the salient-pole electric machines when only one piecewise continuous Sturm-Liouville eigenfunction is used and many sinusoidal functions. In this case, it becomes possible to transform the general solution of the partial differential equation into four-terminal network equations and to create the cascade equivalent circuit of the salient-pole electric machine. In this paper, the cascade equivalent A-H-circuit of the synchronous salient-pole machine is considered (without using piecewise continuous Sturm-Liouville eigenfunctions but also without properties averaging).


INTRODUCTION
In the second half of the twentieth century, many papers were published [1][2][3][4][5][6][7] that examined the problem of modeling the electromagnetic field of electric machines based on the laminated models. Each layer of the laminated model corresponded to some zone of the machine (air gaps, teeth, slots, the yokes of the stator and the rotor). The general solution of the partial differential equation had to be found in each layer of the laminated model and integration constants had to be defined (according to the boundary conditions).
Unfortunately, the calculations were so complex, and the results were so cumbersome that the laminated models did not receive any practical development in the future. However, it became apparent that the solutions of the partial differential equations that were obtained for the laminated models could be reduced into the four-terminal network equations known in the circuit theory. If now the electric intensity is considered equivalent to the voltage and the magnetic intensity is considered equivalent to the current, it is possible to create the equivalent circuit of the four-terminal network equations (the so-called E-H-cascade equivalent circuit). If the vector magnetic potential is considered equivalent to the voltage and the magnetic  [8][9][10][11][12][13][14][15][16][17][18][19][20], as well as for electric technological installations [21][22][23][24][25][26][27].
In the case of the salient-pole machines, in which alternating poles and air spaces between poles cannot be considered as the homogeneous zone, piecewise continuous Sturm-Liouville eigenfunctions are usually used as the general solution of the partial differential equation [28][29][30]. Unfortunately, we cannot create the cascade equivalent circuit of the salient-pole electric machine when (to ensure both the uniqueness and the accuracy of the solution) the field is modeled with many piecewise continuous Sturm-Liouville eigenfunctions (in the zone of alternating poles and air spaces between poles) and many sinusoidal functions (in the air gap). Nevertheless, the author developed the approximate method for modeling the magnetic field of the salient-pole electric machines when only one piecewise continuous Sturm-Liouville eigenfunction is used (in the zone of poles) and many sinusoidal functions (in the air gap) [31,32]. In this case, it becomes possible to transform the general solution of the partial differential equation into four-terminal network equations and to create the cascade equivalent circuit of the salient-pole electric machine [33][34][35][36]. This paper deals with the same subject as [36]. However, as opposed to [36], the A-H-circuit of the electric machine is created without using piecewise continuous Sturm-Liouville eigenfunctions (but also without properties averaging).

THE EQUIVALENT A-H-CIRCUIT OF THE ROTOR
The object of the study is the synchronous generator with the salient-pole rotor (that is also the inductor). Both the rotor yoke and the stator yoke are perfect ferromagnetic, their magnetic permeability is infinite. The magnetic permeability of poles is constant and finite. Poles and air spaces between poles are bounded by coordinate surfaces of the cylindrical coordinate system as shown in Figure 1. The current 0 I creates the exciting field. Let us substitute equivalent current sheets (that will be located on the rotor yoke between poles) for the exciting current. Fourier's expansion turns the current sheet into where 2p is the number of poles. It is known that in case of both the exciting field and the longitudinal armature reaction, field lines are directed generally parallel to the r-axis within the poles and generally perpendicular to the r-axis within the air spaces between poles. Therefore, we can substitute the anisotropic zone for the poles and the air spaces where the components of the magnetic permeability must be where f  is the magnetic permeability of poles and 0  is the magnetic permeability of air. In case of the quadrature-axis armature reaction, field lines are directed generally parallel to the r-axis within the air spaces between poles and generally perpendicular to the r-axis within the poles. Then the components of the magnetic permeability must be 0 , .
In the anisotropic zone, Maxwell's equations determine electromagnetic field: In case of the cylindrical coordinate system, Maxwell's equations (4) are equivalent to the partial differential equation for the vector potential: where A is the z-component of the vector potential.
When the rotor revolves, we can see the traveling waves of the field. Therefore, the complex plane representation of the vector potential can be used. For the each k-th harmonic, the (5) turns into 22 22 1 0 The general solution of the equation (6) is The tangential component of the magnetic intensity (excluding the exciting field source) is 12 1 kk kk k Suppose that at the boundaries of the anisotropic zone, the vector potential and the tangential component of the magnetic intensity vector are known. As in Figure 1 for the first field harmonic, if 1 rr  , If 2 rr  (see Figure 1), Suppose that, for the first field harmonic, the equivalent A-H-circuit (in which vector magnetic potential is considered equivalent to the voltage and the tangential component of the magnetic intensity vector multiplied by a radius is considered equivalent to the current) corresponds to the rotor as shown in Figure 2. Hr corresponds to the first harmonic of the current expression (1): The expressions (9)-(12) describe the operation conditions of the equivalent A-H-circuit. On the other hand, when operation conditions change, the constants of the equivalent A-H-circuit (including the impedances 11 Z , 12 Z , 13 Z ) shall not change. Therefore, we can determine the impedances 11 Z , 12 Z , 13 Z by analyzing the expressions (9)- (12) in the short-circuit conditions and the no-load conditions:

THE CASCADE EQUIVALENT A-H-CIRCUIT OF THE GENERATOR
The cascade equivalent A-H-circuit of the generator has two cells as shown in Figure 3. The first cell ( 11 Z , 12 Z , 13 Z ) corresponds to the rotor and the second cell ( 21 Z , 22 Z , 23 Z ) corresponds to the air gap. Constants of the second cell may be deduced from expressions (14) and (15) The air gap irregularity can be taken into account if the air gap is multiplied by where   is the pole overlap factor and k  is Carter's factor. If the magnetic permeability of stator teeth is infinite then the zone of stator teeth may be removed from the A-H-circuit and the ideal current source 2 I (that corresponds to the armature reaction) will be connected to the output of the second cell. In compliance with Ampere's circuital low, this current source is determined as 12 22 4 bb I h z     (19) where  is the current density in the stator slots (the amplitude of the first harmonic); 2 z is the number of stator teeth; h is the tooth height;   Table 1 contains the initial data of the control calculations. The results of control calculations are offered in Figure 4 (the normal component of the magnetic induction vector on the stator). Both the numerical simulation and the calculations of the A-H-circuit lead to the same result. It is evidence of the correctness of modeling.

CONCLUSION
In this paper, the cascade equivalent A-H-circuit of the synchronous salient-pole generator is considered. In this model, both the poles of the rotor and air spaces between the poles are replaced by the anisotropic zone. It makes it possible to avoid piecewise continuous Sturm-Liouville eigenfunctions. The constants of the A-H-circuit change depending on the field configuration (the exciting field or the longitudinal armature reaction or the quadrature-axis armature reaction). The air gap irregularity can be taken into account if the air gap is increased. Control calculations of the cascade equivalent A-H-circuit and the numerical simulation lead to the same result. It is evidence of the correctness of modeling.