This article proposes a mathematical model of an axial flux induction motor (AFIM) with one stator and one rotor. The model is based on the expression for the electromagnetic torque, which presents a function of two independent variables: the axial length of the stator core and the flux density in the air gap. This allows calculating the main dimensions of the motor with the highest possible torque density. Thus, developed model is suitable for designing the motor of specified volume with maximum torque, and solving the inverse problem of minimizing the machine volume with the specified torque. The detailed output of the model and the results of the calculations for the lowpower engine powered by voltage of 7.35 V (RMS) are given. The results are validated using FEM in ANSYS software: with the outer motor diameter of 0.11 m, the flux density in it reaches 1.2 T.
The traditional system of designing electrical machines is based on the analysis of a previous experience of their creation, as shown in [
There are different indicators for evaluating the design quality of an electrical machine: power density [
There is still no single mathematical model of an electrical machine, which solves the problem of maximization for at least one of the quality indicators, as stated in [
Metaheuristics includes several methods, such as Ant Colony, evolutionary optimization, and genetic algorithms [
For example, L.S. Batista et al. in [
A.C.F. Mamede and J.R. Camacho in [
P. Virtic and M. Vrazic in [
T. Raminosoa and B. Blunier in [
In addition, there are combined optimization techniques that use neural networks and various metaheuristic algorithms for design. For instance, S. Meo and A. Zohoori in [
The main contribution of an article is that we propose a motor design procedure aimed for torque density optimization. Such a conception was not proposed before in any of the publications we know. Currently, the most common optimization approaches are based on metaheuristics or the application of neural networks. They provide an optimal solution, with a high degree of accuracy, but they require multiple iterations to be processed. This requires a lot of processing power and a lot of time. This article proposes a simplified mathematical model of an electric machine, allowing to design electric motors in one iteration with the maximum possible torque density. With this model, the Axial Flux Induction Machine (AFIM) with a shortcircuited rotor will be designed in operation. However, this technique can be adapted to any type of a motor.
In the paper [
The principle of this optimization can be illustrated by a simple thought experiment. Suppose that a machine occupies a fixed volume, i.e., its outer diameter
Further, we assume that if we increase the length of the stator core
Based on Rolle’s theorem, the electromagnetic torque as a function of the length of the stator core must have at least one extremum in the considered interval. Thereby, the basis for the mathematical model is the expression of the electromagnetic torque of the induction motor:
Equation (
The phase voltage
In turn,
If the active length of the stator conductors is taken to be equal to
Of greatest interest in Equation (
The phase resistance can be defined as the ratio of the doubled number of turns of the winding
In the process of generating an expression for the average halfturn of the stator winding, it must be taken into account that the stator has two coil spans of different lengths along the outer diameter of the stator
We introduce the notation
Next, we transform Equation (
According to the work in [
The length of the bar is defined as
In the shortcircuited winding of the rotor of the end induction motor there must also be two shortcircuited rings: the outer ring with diameter
Equation (
The crosssectional areas of the bar
The decrease in current density in the shortcircuited rotor rings relative to the current density in the rods (to improve the cooling conditions of the rods) can be taken into account using the coefficient
Substitute this in Equation (
For the final form of expression for
The expression for the stator slot area
A similar expression can be written for the rotor, taking into account the fact that with a squirrel cage rotor, only one bar with a cross section
Now Equation (
The last two coefficients of Equation (
The first factor of (
The remaining part of (
Moreover, the equation of the model can be written as
For simplicity, rectangular open slots on the stator are considered. The width of the slot
In its turn, the stator slot area can be found as the difference between the area of the tooth pitch and the tooth
If flux density in the air gap
The dependence of the area of tooth pitch and tooth can be found as
Then, the tooth pitch and stator slot area will be written as
Now it is possible to determine the width of the stator tooth.
To find the height of the stator core
After the magnetic flux conversion, it is possible to find the stator core thickness as well as the height of the stator slots.
The height of the stator tooth can be determined as the difference between the thickness of the stator steel stack and the thickness of the stator core.
In the simplest case, it can be assumed that the rotor slots are also rectangular, and the outer and inner diameters of the rotor and stator are equal (
In order to obtain a general solution to this problem, we will introduce a system of relative dimensions, taking as the basic total axial length of the engine L. Under this condition, the main relative dimensions of the engine can be written as
In such a case, the geometric dimensions of the stator slots can be converted as
The total area of all stator slots can be written as:
The axial length of the rotor core can be written as
Given (
This conversion allows us to rewrite the expression for the electromagnetic torque in relative units:
As mentioned in
The results below will be for solving the problem of designing a lowvoltage AFIM with four pole pairs. Such motors can be widely used in small electric transport [
The most significant variables defining all the characteristics of the machine are the axial length of the stator, which determines the relationship between the stator and the rotor (see
It should be noted that the values of the parametric factor (
After substituting the initial data in the model expression, the following dependencies were obtained (
From
Rotor steel pack thickness can be found as the difference of total engine thickness, stator steel pack thickness, and air gap values.
Substituting in Equations (
The calculated engine was validated by using FEM, the analytical method based on Maxwell’s equations, and the measured torque. The external view of the designed engine is shown in
Electromagnetic torque is presented in
This article shows the possibility of optimizing an electric machine in a closed volume. Classic electric machine models, such as those based on the output equation [
In our opinion, another important scientific result of this work is obtaining a mathematical model of the motor as a design object. This makes it possible to conduct research on the influence of various design parameters on the characteristics of an electric machine. Note that for a more reliable description of the electric machine, the model should be improved. The model must be supplemented with characteristics of steel magnetization, heat transfer and heat removal processes, the ability to set an efficiency class, and other additional functions.
The article proposes an analytical model of AFIM, with which it is possible to design an electric motor with the maximum possible electromagnetic torque density. By analyzing the expression of the electromagnetic torque, the model allows us to obtain an optimal value of magnetic flux density in the air gap, as well as geometric parameters of the motor, namely, axial lengths of the stator and rotor steel packs and the height and width of the stator and rotor slots.
In the articles considered in the introduction, similar results are achieved through various soft computing techniques, namely, neural networks and metaheuristic algorithms. A common disadvantage of these methods is that they require working out multiple iterations to achieve the goal. It takes a lot of time and computing power for processing these iterations. The model presented in this article allows us to solve the problem in a single iteration.
To illustrate the operability of the model, we calculated a small lowvoltage AFIM with four pairs of poles. In addition, the designed engine was validated using the FEM.
This model not only provides an algorithm for calculating the engine for a single iteration, but is also a valuable research tool. The influence of various parameters on the characteristics of an electric machine can be obtained in the context of its analysis. Therefore, in the following publications, the analysis of the influence of the number of poles and other parameters on the characteristics of the machine will be checked.
Conceptualization, G.B. and T.K.; data curation, V.O.; formal analysis, G.B. and A.V.; investigation, A.Z. and A.V.; methodology, G.B.; project administration, G.B. and T.K.; resources, A.Z., T.K., and A.V.; software, A.Z. and T.K.; supervision, G.B.; validation, V.O. and A.V.; visualization, A.Z. and V.O.; writing—original draft, G.B.; writing—review and editing, V.O. and T.K. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
The authors declare no conflict of interest.
The following abbreviations are used in this manuscript:
Illustration of a thought experiment. (
Stator tooth zone in the plane of the air gap.
Stator teeth in isometric.
Constraints determining electromagnetic torque.
Engine model in isometry.
Electromagnetic torque.
Magnetic flux density in air gap.
Initial data of design process.
Designation  Name  Value  Unit 

U  Line voltage, RMS  7.35 


Maximum line voltage  36 

f  Rated frequency  50 


Maximum frequency  100 


Outer diameter  0.11 


Inner diameter  0.05 


Total thickness  0.03 


The number of phases  3   

The number of pole pairs  4   

The number of slots per pole and phase  1   

Step of span in slot pitches  3   

The number of rotor slots  32   

Space factor for copper  0.5   

Space factor for iron  0.98   

Space factor for aluminum  0.97   

Thermal factor  1.32   

Thickness of air gap  1 


Overload capability  2   

Inductance of stator core  1.4 


Inductance of rotor core  1.3 


Inductance in stator teeth  1.8 


Crossection of 1 conductor 

