1 INTRODUCTION

AFPM motors have advantages such as compact structure, high ratio of torque to the volume of the machine, high efficiency, and easy maintenance routine, especially for high-torque and low-speed applications. Recently these motors have been employed in many applications, e.g., electromechanical traction drives, electric cars, and marine propulsion systems [1–6]. Great advancements in the field of high-energy permanent magnet (PM) technology, power electronics devices, and modern control algorithms have increased such applications of AFPM motors [7, 8]. AFPM machines have different types and the machine with two rotor/stator discs is the most used AFPM machine [9–12]. The TORUS type Non-Slotted (TORUS NS) AFPM machine presents low torque ripples, vibrations, and acoustic noises due to inherent balanced axial forces applied to both rotor discs and the stator from both sides [2, 12]. The structure of the slotless AFPM machine is shown schematically in Fig. 1 [13]. The machine has a slotless stator core and two rotor discs. The stator part is located between the rotor discs. Generally, for reducing iron losses, the laminated silicon–iron sheet is used to fabricate the stator core. The rotor part rotates with synchronous speed and the iron losses of the rotor part are insignificant compared to the stator core. Therefore, for simplicity of construction, the rotor cores are fabricated from mild steel. The PMs are positioned on the rotor cores. In addition, the machine has a concentrated type of winding [14, 15]. The operation of the AFPM machine is like other common machines. However, in the AFPM machines, the flux travels through the air gap parallel to the machine axis, while in radial flux machines, the flux paths through the air gap along the radius of the machine [16].

Fig. 1.
figure 1

The structure of the slotless AFPM machine [15].

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Commonly for initiating the design process of an AC motor, some data such as input voltage and frequency, output power, and speed are needed. For example, except for the variable speeds, the induction motors are supposed to be connected to the distributed network directly without any interface. Therefore, the voltage and frequency of these motors are fully known in the design stage. Anyway, in synchronous motors, including the PM motors, it is not always the case and requires many careful considerations. First of all, it has to be noted that except for the direct online synchronous motors, a common synchronous motor cannot be connected to the AC network directly due to the lack of starting torque. Therefore, these motors, even for single-speed applications, generally require an inverter-based variable frequency/voltage drive interface [17]. This type of voltage source is much critical for variable speed applications. Therefore, regarding the voltage and frequency, the design objective for variable-speed motors would be much more complicated compared to single-speed motors. Of course, it has not to be assumed an easy task for even a single-speed motor. The reason for this is that, for example, the frequency can be chosen while adjusting the number of machine pole pairs in the design stage to reach the required steady-state speed. In this circumstance, the supply frequency for the rated sustain operating condition is considered an independent parameter and must be evaluated optimally to achieve supreme performance, e.g., maximum efficiency. On the other hand, the existence of such independent parameters may give some feasibility to the designer to design a motor with optimal performances in different previously scheduled operations of a motor [18]. A similar scenario with less importance can be stated regarding the supply voltage. However, usually, the amplitude of the voltage in the terminal of the machine is determined more easily, knowing the available input voltage of the inverter, taking some measures, e.g., avoiding harmonic content of output voltage due to over modulation, etc. In this regard, choosing the optimal value for the frequency is a much bigger issue. Therefore, the present paper is focused on investigating a design procedure for a motor in which the sustain supply frequency is not specified precisely at the beginning and the value can be any in an interval.

So far, numerous papers have been published about the design and construction of AFPM machines [2, 7, 19, 20]. Almost none of them gave a discussion on the supply frequency, pole pair numbers, and output power of the AFPM motor rather they assumed that the customers determine these parameters at the beginning. A relevant matter can be found about the induction motors in [21], in which an experimental formula has been introduced for calculating the efficiency of an induction motor in terms of the pole pair numbers and frequency.

The present paper first investigates the effect of varying frequency, pole pair numbers, and output power on the performance of a slotless AFPM machine. In this consideration, the electromagnetic and thermal phenomena are modeled individually with some details. The electromagnetic model is a magnetic equivalent circuit, which takes the leakage flux and fringing flux paths into account, accurately. It is reminded that these design parameters have a considerable impact on the overall efficiency of the machine.

Avoiding thermal modeling and so briefly considering the temperature of various portions of the machine as a constant in the design stage has been reported previously [22–30]. Applying such a simple assumption defects the accuracy of the design algorithm. This fact necessitates precise local loss and thermal modeling. Therefore, the thermal model of the present paper built based on the local loss, evaluates precisely the local temperatures of various parts of the machine.

Many practices are accomplished and some simulation results are demonstrated in this paper showing the impact of the frequency and pole pair numbers on machine efficiency. Moreover, these results are described as a reliable formula, which specifies the efficiency versus the output power, frequency, and pole pair numbers. This formulation, which may be interpreted as a meta-model can be simply included in a design algorithm.

2 THERMAL MODEL OF THE CASE STUDY MACHINE

The temperature of the machine parts has a significant influence on the machine’s efficiency. The excessive temperature rise of the machine can cause many problems. Therefore, it is essential to predict the temperatures of all parts or at least some critical points of the machine much watchfully in the design stage. Two familiar methods for the thermal modeling of electrical machines are [2]:

— The finite volume method (FVM);

— The thermal equivalent circuits.

In general, FVM is known as a very accurate and time-consuming method. For this reason, many times the designers prefer to employ a quick alternative method if available. Nevertheless, the thermal equivalent circuit model can offer accuracy of FVM likewise if it is formed properly while requiring a lower computer run-time.

Figure 2 illustrates a thermal equivalent circuit for a slotless TORUS AFPM machine. In this figure, Rdc, Rds, Rdr, and Rdm are the conduction resistances of the windings, stator core, rotor core, and PM, respectively. Likewise, Rcsa and Rcrg are the resistances defined for the heat transferring via convection for the stator and rotor cores, respectively. Convection resistances of the stator windings are described by two resistances i.e., Rcca and Rccg for much accuracy. Similarly, Rcmg and Rcma are two convection resistances defined for the PM, and Rcra and Rcrab are two convection resistances for the rotor core.

Fig. 2.
figure 2

Thermal equivalent circuit of the slotless AFPM machine.

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Heat sources are the losses of the iron core, copper, windage, and bearing friction denoted by Pfe, Pcopper, Pwind, and Pfr, respectively. Ambient temperature is described by Tair.

Each conduction resistance of Fig. 2 can be calculated using the following formula [31]:

$${{R}_{d}} = L{\text{/}}({{K}_{d}}~{{A}_{d}}).$$
(1)

Where Rd is the conduction resistance of the component, Kd is the thermal conductivity of the related material and Ad and L are the cross-section area and length of the flow path, respectively. Similarly, each convection resistance is evaluated by:

$${{R}_{c}} = 1{\text{/}}\left( {Z{{A}_{c}}} \right).$$
(2)

In which, Rc, Z, and Ac are convection resistance, the coefficient for convection heat transfer, and the common surface of the component and surrounding air, respectively.

The stator core loss can be calculated by the following equation [2]:

$${{P}_{{fe}}} = 0.078Wf\left( {100 + f} \right)~B_{{cs}}^{2}~{{G}_{{fe}}} \times {{10}^{{ - 3}}}.$$
(3)

Where W, f, Gfe, and Bcs are the specific core loss, frequency, stator core weight, and flux density in the stator core, respectively.

The copper loss can be evaluated using the formula as [2, 19]:

$${{P}_{{copper}}} = m{{R}_{s}}~I_{{pz}}^{2}.$$
(4)

In which, m, Rs, and Ipz are the number of phases, winding resistance, and RMS value of the armature current, respectively. It should be noted that due to the skin effect, the distribution of the current density inside the coils is nonuniform, which is ignored in (4).

The windage loss is determined by [2]:

$${{P}_{{wind}}} = 0.5{{c}_{f}}\rho {{(2\pi {{n}_{m}}~)}^{3}}~\left( {R_{{out}}^{5} - D_{{sz}}^{5}} \right).$$
(5)

Where:

$${{c}_{f}} = 3.87{\text{/}}\sqrt {{{R}_{e}}} $$
(6)
$${{R}_{e}} = 2\pi {{n}_{m}}\rho R_{{out}}^{2}{\text{/}}\mu .$$
(7)

In which, ρ denotes the specific density of the cooling medium (air), Rout is the outer radius of the rotor core, Dsz is the shaft diameter, nm is the rotor speed in rpm, and μ is the dynamic viscosity of the air.

Friction loss of the bearings is calculated as [2]:

$${{P}_{{fr}}} = 0.06{{K}_{{fb}}}~\left( {{{m}_{r}} + {{m}_{{sz}}}~} \right)~{{n}_{m}}.$$
(8)

Where Kfb = 1 to 3 m2/s2, mr is the rotor mass in kg and msz is the shaft mass in kg.

Generally, each heat source is assumed equal to its related loss shown by a current source in the equivalent circuit, as seen in Fig. 2.

3 OPTIMIZATION ALGORITHM OF SLOTLESS AFPM MACHINE

Firstly, for studying the impact of pole pair numbers/output power/frequency on the performance of the slotless AFPM machines, let us proceed with the design practice of a small machine getting help from the formulae presented in the previous publication [19]. Here, some formulae for calculating the main dimensions of the machine are presented.

The average diameter of the machine is calculated as follows:

$${{D}_{g}} = m{{N}_{t}}{{I}_{s}}{\text{/}}\left( {\pi A} \right).$$
(9)

Where Dg, m, Nt, Is, and A are the average diameter of the machine, number of phases, turn number of the stator winding, current of the stator winding, and specific electrical loading respectively.

The outer diameter of the machine is determined as follows:

$${{D}_{o}} = 2{{D}_{g}}{\text{/}}\left( {1 + \lambda } \right).$$
(10)

Where Do, and λ are the machine’s outer diameter and the ratio of inner to outer diameters of the machine respectively.

The machine’s inner diameter is obtained as follows:

$${{D}_{i}} = \lambda {{D}_{o}}.$$
(11)

Where Di is the inner diameter of the machine.

The axial length of the rotor core is calculated as follows:

$${{l}_{{cr}}} = \pi {{B}_{{avg}}}{{D}_{o}}\left( {1 + \lambda } \right){\text{/}}\left( {8{{B}_{{cr}}}p{{K}_{d}}} \right).$$
(12)

In which lcr, Bavg, Bcr, p, and Kd are the axial length of the rotor core, the average air-gap flux density, maximum flux density in the rotor core, the number of pole pairs, and the leakage flux factor respectively.

The axial length of the stator core is determined as follows:

$${{l}_{{cs}}} = \pi {{B}_{{avg}}}{{D}_{o}}\left( {1 + \lambda } \right){\text{/}}\left( {4{{B}_{{cs}}}p} \right).$$
(13)

Where lcs, and Bcs are the axial length of the stator core and the maximum flux density in the stator core.

The thickness of the PM is calculated as follows:

$${{l}_{{PM}}} = 2{{K}_{l}}{{\mu }_{{rPM}}}{{B}_{{avg}}}\left( {g + {{l}_{w}}} \right){\text{/}}\left[ {\left( {1 + {{\alpha }_{i}}} \right)\left( {{{B}_{r}} - {{B}_{u}}} \right)} \right].$$
(14)

In which lPM, Kl, µrPM, g, lw, αi, Br, and Bu are the thickness of the PM, factor for considering MMF drop of the iron cores, the relative permeability of the PM, air gap length, the thickness of the winding, the ratio of pole arc to pole pitch, the remanence flux density of the PM, and flux density at the surface of PM respectively.

The motor is supposed to be operated at a given speed while supplied by a power supply with the frequency assumed unknown previously but fallen within the range of 5 to 275 Hz as common. The range of pole pair numbers is selected between 1 and 20. In addition, the study is performed for the motors with an output power between 0.1 and 20 kW. The design optimization algorithm must specify the precise values of the frequency/pole pair numbers, optimally.

The optimal value of the ratio of the motor’s inner radius to the outer radius is almost constant, as reported by some literature [32, 33]. In addition, the RMS value of the terminal voltage of the machine is nearly known regarding the supply voltage and type of the driver employed. Let us assume 220 volts phase voltage, as is very common.

On the other hand, for the case study slotless machine, the effective air gap is inevitably too large. Therefore, the synchronous reactance of this machine is smaller than conventional synchronous machines. In this regard, for the case study structure with proper design, amplitudes of the Back-EMF and terminal voltage would be much closer. In addition, the machine has good stability and operates at small power angles almost for all load values within the permissible rated value. In this case, the Back-EMF amplitude is nearly identified at the primary step of the design, i.e., 220 volts for our case study machine.

Now, a design algorithm needs to be developed to perform intended frequency/pole pair numbers studies getting help from above mentioned constant or initial guess values. The suggested design algorithm of the present paper is shown in Fig. 3. In this figure, the analysis for pole pair numbers is shown. The process analysis of two other parameters is identical. In this algorithm, the design variables are classified into three groups, i.e., “rated,” “selective” and “calculable” design parameters. The nominal parameters include the nominal values and the constraints such as the number of phases, terminal voltage, the maximum value of outer diameter, the maximum value of the axial length of the machine, etc. Since for a design optimization problem, the number of unidentified parameters is more than the number of available relationships, the values of some variables are regarded as the independent variables called here selective. The other variables are named the calculable parameters. These parameters are written as equations versus the rated and selective parameters. For example, the electrical loading factor, and current density of the conductors may belong to the group of selective parameters. These parameters are typically determined according to knowledge, experience, manufacturing constraints, or optimization works. The values of calculable parameters (e.g., power factor, efficiency, leakage flux factor, machine inner and outer diameters, etc.) can be determined by knowing the rated and selective parameters.

Fig. 3.
figure 3

The suggested algorithm to investigate the influence of pole pair numbers.

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In the flowchart of Fig. 3 at the preliminary step, the values of rated parameters are introduced for the computer. Moreover, a group of possible pole pair numbers/frequencies/output power in a specific interval are assumed for these parameters instead of a fixed value. Each frequency in the group has to present an integer value for the pole pair numbers regarding the required speed. The algorithm commences with the smallest value of the pole pair numbers/frequency/output power in the given group. After that, some initial guess values are assigned for the selective parameters. In the next step, the equations related to the calculable parameters must be solved simultaneously. Even though these equations inherently have only one set solution due to the presence of some complications, they cannot be solved by applying familiar analytical methods. As mentioned above, in the design process of an electrical machine, the number of unknown parameters is more than the number of existing equations. Therefore, for some unknowns, an initial value is considered by the designer. However, this initial value may not be very accurate. The machine designers determine the initial values based on their experiences and specifications of the previously fabricated machines. It should be noted that the values of different unknown parameters are dependent on each other. Therefore, the imprecise selection of the initial values of the unknown parameters causes even the values obtained from the existing relationships to be imprecise. Hence, the unknown parameters cannot be determined by applying familiar analytical methods. Therefore, the iterative technique is employed to solve the relationships and determine the correct value of the parameters. In this iterative method, some well-estimated values are assigned to a few calculable parameters, such as efficiency, power factor, and leakage flux factor, first. Let us name these parameters as “re-calculable.” Then other parameters are evaluated simply using some reorganized versions of the previously defined equations.

When the machine dimensions are determined for the specific pole pair numbers/frequency/output power, the magnetic equivalent circuit is built to calculate the flux density distribution, fringing, and leakage flux factors. This magnetic circuit for the slotless TORUS-NS AFPM machine with all details has been reported, previously [34].

In the next step, a thermal equivalent circuit is employed to determine the temperature in various portions of the machine. Now, the efficiency and power factor of the machine can be calculated more precisely. At the end of this part of the algorithm, the new values of the leakage factor, efficiency, and power factor obtained using the magnetic equivalent circuit and thermal models are substituted for the first estimated values of re-calculable parameters, and the whole calculations of this section are repeated until the differences between two consequence values of the parameters become negligible. Just the convergence criteria are realized, and the calculations of the dimensions and other design parameters of the machine corresponding to the first set of values assigned to the selective parameters are completed. Many designs can be suggested assigning various values for the group of selective parameters. A comparison between the efficiencies of various designs is then performed, and the design with the maximum efficiency is chosen as the optimal one for the given pole pair numbers/frequency/output power. Then the value of the pole pair numbers/frequency/output power increases. The whole design process is completed for each possible pole pair number/frequency/output power and the associated optimal machine is specified.

4 VALIDATIONS OF THE DESIGN ALGORITHM

To validate the design algorithm a sample three-phase machine with a rated power of 3.7 kW and a speed of 1400 rpm has been chosen for design practice. The phase voltage is assumed 220 volts while the design algorithm determines the frequency. A few machines are designed for some frequencies to discover the optimal machine and frequency. Various electromagnetic and thermal results of the design algorithm are investigated to confirm the precision of the design algorithm. In addition, the operation of the machine is evaluated more carefully, getting the help of various time-consuming Finite Element (FE) simulations for double-check purposes. For example, FE simulation can calculate the temperature distribution of various portions of the machine by applying the local losses as thermal sources. Here the presented formulae and the FE analysis have already calculated the locally distributed stator core losses and copper losses. Then they were applied in a 3-D FE thermal analysis simulation as the heat sources. The heat transfer coefficients for various parts of the machine are given in Table 1. Using these coefficients, the temperatures of different parts of the machine have been calculated. It should be noted that the motor housing is fully enclosed and there is no fan for cooling the motor. Therefore, this is a Totally Enclosed Naturally Ventilated (TENV) motor. The thermal analysis results of FE simulation for the stator core, stator windings, and rotor core are depicted in Figs. 4 and 5, correspondingly. Similar results obtained from the thermal equivalent circuit are given in Table 2 for comparison. There are slight differences between the results of the two models, which approve the validity of the thermal equivalent circuit model of the design algorithm.

Table 1.   The heat transfer coefficients for various parts of the machine
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Fig. 4.
figure 4

The distribution of the temperature in the stator core and windings.

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Fig. 5.
figure 5

The distribution of the temperature in the rotor core and PMs.

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Table 2.   The temperatures calculated by FE simulation and thermal equivalent circuit
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5 THE SUGGESTED FORMULA FOR DETERMINATION OF MACHINE EFFICIENCY

At the beginning of the design algorithm, the machine speed and output power are identified. In addition, the speed of the machine is influenced by two parameters the feeding frequency and the pole pair numbers. In the following, the influence of these parameters on machine efficiency is explored by designing machines with different powers, pole pair numbers, and frequencies. For this purpose, the algorithm of Fig. 3 is employed. To fulfill this goal, by keeping constant two variables of these three variables and changing the third variable in a given range, different machines are designed. First, the pole pair numbers are changed between 1 to 20 while the feeding frequency and output power remain constant, and different designs are performed. In Fig. 6, the variations of the efficiency due to the pole pair number changes for the designed machines are shown. There are many variables in the design process of an electrical machine. In addition, all of these design variables are dependent on each other. In Fig. 6, it is assumed that the other design parameters are constant, and only the effect of changes in the number of pole pairs on the efficiency of the machine is investigated. The machine designer must consider various factors in the design of an electrical machine. For example, if the machine structure studied in this article is designed with a small number of poles, the dimensions of the permanent magnets will increase. This issue is especially important about high-power and low-speed machines, where the dimensions of the machine are also large. The possibility of preparing large magnets is difficult and, in some cases, impossible. It is also very difficult to install large magnets on the rotor. Therefore, the machine designer in such cases may not be able to design a machine with a low number of poles. In general, existing limitations and important factors should be considered in the design process and the number of machine pole pairs should be chosen accordingly.

Fig. 6.
figure 6

The efficiency variations against the number of pole pair variations.

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Here, (3)–(8) are used to calculate the machine losses. In addition, the rotor core losses are neglected. In this case, the machine input power Pin is obtained as:

$${{P}_{{in}}} = {{P}_{{out}}} + {{P}_{{loss}}}.$$
(15)

Where, Pout denotes the output power, and Ploss is the total losses of the machine, respectively. In this case, the efficiency is calculated as follows (Darabi, Baghayipour and Mirzahosseini, 2017):

$$\eta = {{P}_{{out}}}{\text{/}}{{P}_{{in}}}.$$
(16)

According to Fig. 6, we can consider the efficiency variations versus pole pair numbers as a Gaussian function. By changing the frequency, and keeping constant the pole pair numbers and output power, different machines have been designed. The machine efficiency versus the frequency variations is shown in Fig. 7. Based on Fig. 7, the machine efficiency changes versus the frequency as an exponential function. Finally, by keeping constant the frequency and pole pair numbers, different designs have been done for different output powers. In this case, the machine efficiency versus output power variations will be as Fig. 8. According to Fig. 8, the efficiency variations versus output power are considered as an exponential function, too. According to the relationships presented in published references such as [2, 19], it is clear that all design parameters and relationships are dependent on each other, so finding an analytical relationship for machine efficiency versus pole pair numbers, feeding frequency, and output power of the machine is a very complicated task. Whereas the intended analytical relationship is obtained easily using the MATLAB curve fitting technique. In this case, according to Figs. 6, 7, and 8 and with the help of MATLAB tools (curve fitting); a relationship is proposed to determine the machine efficiency η versus the frequency, the pole pair numbers, and output power of the machine as follows:

$$\eta = {{K}_{1}}\left( {{{f}^{{{{K}_{2}}}}}} \right)\left( {{{e}^{{ - {{K}_{3}}~{{f}^{{{{K}_{2}}}}}}}}} \right)\left( {{{e}^{{\left( { - {{{\left( {\left( {p - {{K}_{4}}} \right)/{{K}_{5}}} \right)}}^{2}}} \right)}}}} \right)\left( {P_{{out}}^{{{{K}_{6}}}}~} \right)\left( {{{e}^{{ - {{K}_{7}}P_{{out}}^{{{{K}_{6}}}}}}}} \right),$$
(17)

where the values of K1 … K7 are given in Table 3.

Fig. 7.
figure 7

The efficiency variations against frequency variations.

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Fig. 8.
figure 8

The efficiency variations against output power variations.

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Table 3.   The values of equation (17) coefficients
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6 EVALUATING THE SUGGESTED FORMULA

To validate the precision of the suggested relationship, numerous constructed machines with various specifications have been investigated. Three types of these machines are revealed in Fig. 9. Table 4 presents the main parameters of these machines. These machines have been tested in full-load operating conditions. Here, for more clarity, the process of evaluating the machine with a power of 3.7 kW is presented. For this purpose, by connecting the machine to a voltage source, both in the FE software and in the laboratory, machine performance has been studied. For this study, the phase amplitude and frequency of the voltage source are set at 91 V and 70 Hz, respectively.

Fig. 9.
figure 9

The constructed slotless AFPM machines (a) Machine 1, (b) Machine 2, (c) Machine 3.

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Table 4.   Specifications of the Constructed machines
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A separately excited DC generator has been employed as the mechanical load. These generator specifications have been presented in Table 5. This generator generates 186.7 V at 1400 rpm. Therefore, a current with an amplitude of 19.8 A must flow at the generator’s terminal to obtain a power of 3.7 kW. These parameters are shown in Figs. 10 and 11, respectively. The induced voltage in the armature winding of a DC generator is obtained as:

$${{E}_{a}} = K\phi {{\omega }_{m}}.$$
(18)

Where Ea, K, ϕ, and ωm are the internal induction voltage, constant factor, field flux, and rotational velocity of the generator, respectively. The armature winding voltage drop of the generator is not significant. Therefore, by neglecting this voltage drop, the terminal voltage VT can be used instead of Ea. In this case, the term Kφ is calculated as:

$$\begin{gathered} K\phi = {{E}_{a}}{\text{/}}{{\omega }_{m}} \approx {{V}_{T}}{\text{/}}{{\omega }_{m}}~ = {{V}_{T}}{\text{/}}\left( {2\pi {{n}_{m}}{\text{/}}60} \right) \\ = 200{\text{/}}\left( {2\pi \times 1500{\text{/}}60} \right) = 1.273. \\ \end{gathered} $$
(19)

Where nm is the speed of the machine in rpm. Hence, instantaneous speed and torque Tm(t) can be attained from the terminal voltage VT(t) and the current ia(t) as follows:

$$\begin{gathered} {{n}_{m}}~\left( t \right) = {{V}_{T}}~\left( t \right){\text{/}}\left( {2\pi \times K\phi {\text{/}}60} \right) \\ = {{V}_{T}}~\left( t \right){\text{/}}\left( {2\pi \times 1.273{\text{/}}60} \right) = 7.5{{V}_{T}}~\left( t \right), \\ \end{gathered} $$
(20)
$${{T}_{m}}~\left( t \right) = K\phi {{i}_{a}}~\left( t \right) = 1.273{{i}_{a}}~\left( t \right).$$
(21)
Table 5.   The Parameters of the DC generator
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Fig. 10.
figure 10

The measured output voltage of the DC generator (oscilloscope probe: a ×10).

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Fig. 11.
figure 11

The measured current of the DC generator (attenuation of current transducer: 1/20, and oscilloscope probe: a ×1).

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According to (20), with the multiplication of Fig. 10 by 7.5, the mechanical speed of the machine is obtained. In addition, according to (21), with the multiplication of Fig. 11 by 1.273, the electromagnetic torque is attained. Based on these figures, the average value of rotor speed in rpm is 1400. In addition, the average electromagnetic torque of the motor is 25.3 N.m. In this case, with the multiplication of these two values, the output power would be equal to 3.7 kW. For better clarity, the phase current of the fabricated machine has been calculated by 2-D FE simulation and by real test in our laboratory. The obtained results from the two approaches are illustrated in Figs. 12 and 13, respectively. As can be seen, there is a minor difference between the results of the two procedures. It is possible to measure the input power of the machine Pin in the laboratory. Therefore, after determining the Pin, the efficiency of the machine has been calculated with the help of (16). An analogous technique has been employed to evaluate the performances of the other constructed motors. The efficiency of constructed machines has been evaluated by experiment. In addition, the efficiency has been calculated by (17). The gained results of the two approaches are presented in Table 6. By comparing the results, it can be concluded that (17) has satisfactory accuracy and the error between two procedures is about 4.6%. Therefore, at the primary stage of designing a slotless AFPM machine, the machine designers can employ the suggested formula to choose the optimum values of frequency and the pole pair numbers to achieve maximum efficiency. In practice, when the machine designer wants to design an AFPM machine with a specific power and speed, the (17) is a good relationship to determine the optimum values of numbers of pole pairs and frequency. The machine designer can test various numbers of pole pairs and frequencies and calculate machine efficiency using (17). Then, the pole pair number and frequency that the machine efficiency is maximum is chosen as the best choice. After that, the AFPM machine is designed for the chosen number of pole pairs and frequency. Based on these explanations, the (17) could be used in the optimization design process of the machine to maximize machine efficiency.

Fig. 12.
figure 12

The current of one phase of the motor attained by FE.

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Fig. 13.
figure 13

The current of one phase of the motor measured in the laboratory.

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Table 6.   The efficiencies attained from measurement and equation (17)
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7 CONCLUSIONS

The influence of important machine parameters on the efficiency of the slotless AFPM machine has been studied. Due to the direct effect of the temperature on the losses and consequently the machine’s efficiency, first, the thermal equivalent circuit of the machine has been presented. By using this thermal equivalent circuit, magnetic equivalent circuit, and sizing equations, an algorithm for studying the influence of frequency, pole pair numbers, and output power on the efficiency of the slotless AFPM machine has been proposed. With the help of this algorithm, different machines with different frequencies, pole pair numbers, and output powers have been designed. Based on the obtained results, a formula for determining the efficiency of the machine in terms of frequency, pole pair numbers, and output power has been proposed. To approve the precision of the suggested formula, three constructed slotless AFPM machines have been chosen as tested machines. The efficiency of selected machines has been obtained by the proposed relationship and from the experimental test. The difference among the results of the two approaches was lower than 5%, which approves the precision of the suggested formula. Therefore, this formula for calculating efficiency can be used by designers of slotless AFPM machines.