The traditional system of designing electrical machines is based on the analysis of a previous experience of their creation, as shown in [1
]. With this approach, new samples inherit the advantages and disadvantages of the previous models. Electric machines of the traditional design have a big statistical base accumulated over the previous century of design, allowing for the design of efficient machines of any power. However, when the task is the design of an unusual electrical machine appears, this approach is not possible because there are no statistics available. There are various methods for designing such machines to produce the most effective results.
There are different indicators for evaluating the design quality of an electrical machine: power density [2
], the volume torque density and weight torque density [3
], the volume and weight of permanent magnets per Newton-meter (for permanent magnet synchronous motor, PMSM) [4
], machine price per Newton-meter (for motors [5
]), machine price per rated power (for generators [6
]), and others [7
]. When designing any electrical machine, one of the main tasks is obtaining a maximum or minimum one or several of these parameters. When solving problems of maximization of various parameters, different results can be obtained, as shown in [5
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 [1
]. Various soft computing tools such as neural networks [3
] and metaheuristics are used to solve such problems. The most effective methods for optimizing electrical machines are various matavistic methods. These are computational methods that provide a fairly good approximate solution to the problem in polynomial time.
Metaheuristics includes several methods, such as Ant Colony, evolutionary optimization, and genetic algorithms [5
] as well as some combinations of them.
For example, L.S. Batista et al. in [13
] use the Ant Colony metaheuristic method to solve the problems of optimization of the Interior Permanent Magnet Machine, namely, the task of minimizing the volume of permanent magnets per Newton-meter and the problem of improving the shape of the MMF curve in the air gap. The result of this study is two different topological forms of rotor design, providing solutions to two set tasks. Both structures are technologically feasible. However, a great disadvantage of the developed technique is that it is aimed only at optimizing the shape of the rotor and shape of stator slots but the stator core is not optimized.
A.C.F. Mamede and J.R. Camacho in [14
] use an evolutionary optimization algorithm to maximize electromagnetic torque per total volume of a single-phase switched recovery machine and to decrease electrical losses in copper. The main parameter of motors of similar design is the angle of rotor and stator arc. The result of this study is the design of an engine having the maximum possible torque in a given volume. However, as with any other metaheuristic method, the result was achieved through repeated effort. In this case, the result was achieved in 50 iterations, but it should be noted that the work was carried out through the modernization of the already existing engine, i.e., the initial data for starting the evolutionary algorithm were known at the beginning. It would take many more iterations to calculate an engine without initial data.
P. Virtic and M. Vrazic in [5
] apply genetic algorithms to the optimum design of the axial flux permanent magnet synchronous motor with two stators. Several optimization tasks were solved in this work: (1) maximized weight torque density, (2) maximized volume torque density, (3) minimized mass of PMs per Newton-meter, (4) minimized volume of PMs per Newton-meter, and (5) minimized machine price per Newton-meter. The result of the work was five completely different electric motor designs, solving different tasks. Each of the tasks required about 4000 iterations. With the use of computer equipment available to the authors, the calculation of each iteration took 145 s, and the complete calculation of the engine design took about 13.5 h.
T. Raminosoa and B. Blunier in [11
] also use genetic algorithms to upgrade a high-speed switched reluctation machine in an air compressor.
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 [9
] use a combination of the metaheuristic method “particle swarm optimization” and four neural networks to optimize a permanent magnet flux switching generator for a low-power wind turbine, namely, to minimize the weight and price of the generator, as well as to improve the harmonic composition of the induced voltage. Neural networks are designed to model complex relationships between weight, volume, cost, and the harmonic distortion factor. The metaheuristic algorithm serves to find the best solution in the entire search space. FEM calculations were used to train neural networks. Such combined methods are very effective, however, training neural networks, as well as working out all iterations required a metaheuristic algorithm which takes a long time and computational power.
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 short-circuited rotor will be designed in operation. However, this technique can be adapted to any type of a motor.
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 [16
], optimize the machine for the inner diameter of the stator without controlling its outer diameter. A distinctive feature of our technique is taking into account the external dimensions of the machine and optimizing the dimensions of the stator and rotor already inside this given volume.
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 low-voltage 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.