Optimal Design, Electromagnetic–Thermal Analysis and Application of In-Wheel Permanent Magnet BLDC Motor for E-Mobility

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This study presents the analytical sizing process, multi-objective optimization, finite element method (FEM) analysis, multiphysics using the coupled electromagnetic–thermal analysis, and validation of in-wheel or outer-rotor BLDC motor for the propulsion system of a small-power and low-speed electric vehicle.

Abstract

In this paper, a 96 V, 2.5 kW, 36-slot, and 32-pole brushless direct-current (BLDC) motor is designed, analyzed, and tested in the laboratory and on the prototype vehicle to provide the required output performance for an electric vehicle (EV) according to the rated operating conditions. Applications for in-wheel electric drivetrains have the potential to deliver high efficiency and high torque. Consequently, in-wheel motor topology is proposed for small EVs, and the sizing equations, including primary, stator, and rotor dimensions, are developed step by step for the preliminary design. Then, a multi-goal function is introduced to obtain optimum motor design. This motor has an outer-rotor-type construction. In addition, a concentrated winding arrangement is used, which ensures low-end winding and thus low copper loss. Then, multiphysics using the coupled electromagnetic–thermal analysis is carried out. Elective analysis using the finite element method, a motor prototype, and experimental studies verifies the design effectively.

1. Introduction

Undoubtedly, e-mobility is one of the crucial issues that marked the last quarter of the previous century [1]. These vehicles are electric bicycles, motorcycles, scooters, electric cars, buses, unmanned aerial vehicles, etc., available on many platforms. The foremost cause for using electric motors, especially in-wheel motors, on these platforms is that they do not need mechanical differentials and gearboxes compared to other motors and, therefore, do not have mechanical losses [2,3]. In addition, high power density, wide torque-speed capability, high efficiency, and low maintenance costs are the reasons for their preference [4,5]. Another advantage of EVs over internal combustion engine vehicles is that they allow energy recovery in braking mode. This is another feature that increases the efficiency of EVs. EVs offer comfort when evaluating driving dynamics with high starting torque and smooth acceleration. In addition, developments in battery technology, the increase in range provided, and the reduction in costs are essential reasons that increase the demand and use of electric vehicles [6].

In recent years, due to the intense interest shown by academic and industrial circles in EVs and thus, BLDCs, a great deal of research has introduced new techniques and methods for machine design and analysis [4,7,8,9]. However, the design process for this kind of machine is difficult and time consuming, as many free variables, such as dimensional parameters and electromagnetic values, have to be determined. At the same time, a large number of constraints must be met. Therefore, the dimensioning is performed with analytical calculations in the first step, and then the final dimensions are obtained by optimizing with various software. Designing an efficient, small-volume motor will increase the power and torque values per volume while reducing the cost. Loss minimization must be considered since most losses occur in the stator and rotor. For this, it is necessary to optimize the free parameters in sizing.

The process of optimizing the BLDC motor’s design includes performance improvements, while taking into account different constraints and characteristics. Various optimization techniques, like genetic algorithms, particle swarm optimization, and other metaheuristic algorithms, are performed to improve the performance of BLDC motors [8]. In [10], an Adaptive Network-Based Fuzzy Inference System coupled with a multi-objective optimization algorithm is proposed to track the optimum PM motor design for maximum efficiency. The basic approach to multi-objective optimization is to evaluate all objective functions simultaneously. For example, a Pareto-optimal set yields more than one optimal response, which can be considered a privilege for this multi-objective optimization. This solution set significantly reduces the required computational cost [11]. The decision maker selects a solution from the Pareto set as the compromise that best fits their goals. Differential evolution is an optimization technique that can solve global optimization problems subject to non-linear constraints [12]. In [12], the design of the interior permanent magnet motor with a two- or three-layer barrier in the rotor was carried out using differential evolution, and this method has been proven to be a reliable technique that can handle non-linear constraints, work with integer or discrete variables, and is suitable for multi-objective optimization problems.

The genetic algorithm (GA) can be used to enhance motor performance. Reference [13] discusses an innovative approach to the optimal design of traction motors for electric vehicles utilizing a combination of random forest (RF) and genetic algorithm (GA) methodologies to enhance motor performance metrics such as efficiency and minimize torque ripple.

Different rotor topologies, like outer and inner rotors, are used in BLDC and PMSM design, optimization, and analysis studies [14]. Fatemi et al. [15] studied the design optimization of electric machines for 48 V hybrid vehicles, focusing on rotor technologies and pole–slot combinations. While it provides insights into rotor design performance and efficiency, it may not address the broader impacts on overall vehicle performance or integration into hybrid systems. Exploring these areas could lead to further research in electric vehicle design and analysis. Reference [16] presents a novel modular bearingless permanent magnet synchronous motor (PMSM) designed for ultra-clean applications, highlighting gaps in the existing literature on advanced motor designs. It utilizes 2D and 3D finite element method (FEM) analyses to evaluate motor performance and suspension control via Model Predictive Control (MPC). Additionally, the focus on decoupling characteristics and structural parameters indicates a need for further research into optimizing these factors across various applications, which have not been thoroughly explored in prior studies. Im et al. [17] propose a two-step design process for permanent magnet synchronous motors (PMSMs) to boost efficiency and minimize harmonics in lightweight electric vehicles. This study presents practical methods for improving motor performance but does not consider the broader impact on vehicle efficiency and alternative technologies. It also indicates possible directions for future research to enhance electric vehicle systems. The study [18] presents the design and optimization of an outer-rotor BLDC for medical devices, which will also meet high torque, high efficiency, and small size requirements. Different techniques are used to increase power density and efficiency in applications where size and weight are essential. For example, the Halbach magnet array can improve power density and efficiency by concentrating the magnetic flux [19]. In [20], a rotor design with optimal step-skew angles and optimum step lengths to reduce the cogging moment was proposed to improve the torque. The step-skew method is frequently used as a practical approach to minimize cogging torque in PM machines. Reference [21] explores the adjustments of pole pitch and pole arc in rotor magnets to minimize cogging torque in brushless DC (BLDC) motors. Their analytical findings on cogging torque reduction are consistent with Finite Element Analysis (FEA), establishing a strong basis for assessing the impact of design modifications on cogging torque reduction. Reference [22] presents a design optimization method to reduce cogging torque in PMSMs, contributing to the literature on their performance and reliability issues. However, it may overlook the impact of manufacturing variability on robustness and lacks a comprehensive comparison with other optimization methods, highlighting the need for further research.

Many research studies mainly focus on optimizing motor efficiency, power density, and torque using different multi-objective optimization methods. It is only possible to optimize some of the machine’s sizing parameters to improve all performance values. This is a time-consuming and complex process. For example, many machine performance improvement studies were carried out in this research, but the thermal performance still needs to be studied. In this paper, multiphysics using the coupled electromagnetic–thermal analysis method is proposed. Compared to the literature’s design, analysis, and optimization methods, the techniques used in this paper are different. When the studies are examined, the design, optimization, analysis, and verification studies have yet to be fully and comprehensively discussed.

In this study, the analytical sizing process, multi-objective optimization, finite element method (FEM) analysis and validation of the in-wheel or outer-rotor BLDC motor for the propulsion system of a small-power and low-speed electric vehicle are presented, as shown in Figure 1. In the first step, basic sizing is provided within certain electromagnetic and thermal limits to achieve the specified motor performance. Multi-parameter and parametric optimization methods are used to obtain the most appropriate output values within magnetic and thermal limits. Afterwards, the magnetic and thermal values of the motor are examined by performing coupled electromagnetic and thermal analyses with FEM. Finally, the method is validated with experimental results.

2. Design and Analysis Methods

This section outlines the design and analysis methodology for the in-wheel BLDC motor. It begins by presenting technical specifications and preliminary design considerations (Section 2.1) and defining the motor’s main dimensions, including magnetic and electrical loading, airgap flux density, and rotor geometry (Section 2.2). Subsequent sections detail permanent magnet dimensions and coil MMF calculations for optimal flux linkage and torque (Section 2.3), followed by stator and rotor dimensioning, focusing on efficient power transfer through slot shape, tooth width, yoke height, and magnet thickness (Section 2.4 and Section 2.5). Multi-objective optimization and parametric analysis are then explained, highlighting trade-offs between efficiency, torque, and cogging effects (Section 2.6 and Section 2.7). The design is validated using 2D FEM simulations to analyze electromagnetic performance and compare results with experimental data (Section 2.8). Finally, the coupled electromagnetic–thermal analysis assesses heat generation and cooling performance to ensure thermal stability (Section 2.9). Figure 2 provides a flowchart explaining the design and analysis processes.

2.1. Specifications and Preliminary Design Considerations

There are so many factors, such as economic and material limitations, technical features, and particular elements in the design process of electrical machines, that it is impossible to work on design within strict limits. In any design, a compromise must be made between numerous conflicting requirements to achieve the desired performance. Therefore, a design problem has a variety of solutions. The designs to meet the exact technical specifications will differ as the designers emphasize each requirement differently [23]. A new brushless motor design includes a range of performances such as power, speed, torque, efficiency, and physical or structural properties, as seen in Figure 3. There is no explicit and unlimited sizing to satisfy the performance requirement. In most cases, the initial dimensions are calculated in the first step; then, the optimum design is presented by considering the physical constraints, material limitations, cost, and desired performance characteristics [24].

2.2. Main Dimensions

Design specifications, including machine type, construction type (axial flux, radial flux, external rotor, and internal rotor), number of phases, rated power, rated rotational speed, power factor, and rated voltage, are the first step in the design process [25,26]. In [25,26], the design procedure of electrical machines is presented in detail.

Table 1. Reference data of the BLDC motor.

Parameter	Value
Rated power	2.5 kW
Maximum output power	3.8 kW
Rated speed	750–900 rpm
Rated torque	>26 Nm
Targeted efficiency	>85%
DC voltage Vdc (automobile battery)	96 V
Stator material	M330-50A
Rotor material	AISI/SAE 4340 Alloy Steel

presents reference parameters for the present machine. Although the design of electrical machines is computer aided today, obtaining the basic size parameters for all electrical machines, from micromotors to the largest motors [24], is essential. The fundamental question in electric motor design is what volume a motor must be to produce the required torque. This can be expressed using Equations (1)–(9) for radial flux motors [23,24,27,28].

Current density and airgap magnetic flux density, also expressed as specific electrical and magnetic loading, are the most critical factors affecting the size and shape of the machine. Some key parameters that are independent variables in the design of electrical machines are given in

Table 2. Independent variables in the design.

Parameter	Value
Magnetic loading (T)	0.8
Electrical loading (kA/m)	15
Stator windings current density (A/mm2)	4.5
No-load EMF to phase voltage ratio	0.89
The ratio of stack length to pole pitch	1.5
Slot fill factor (%)	45
Maximum flux density in stator tooth (T)	1.8
Maximum flux density in stator yoke (T)	1.5
Maximum flux density in rotor yoke (T)	1

.

In radial flux motors, the flux crosses the airgap radially, and the magnetic loading is determined by the average radial flux density in the airgap [24]. The first stage is to choose the specific magnetic and electric loadings on electric machines’ general or optimum design. These loadings are described, respectively, as follows [28]:

$$B_{mg}={\displaystyle \frac{{\Phi }_g}{{\alpha }_i{\tau }_p{L}_e}}={\displaystyle \frac{2p{\Phi }_g}{\pi {\alpha }_i{L}_e{D}_g}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{A}_m={\displaystyle \frac{2m\sqrt{2}{N}_1{I}_a}{\pi D_g}}$$

(1)

where Φ_g is the airgap flux with the armature reaction taken into account. D_g, L_e, N₁, and 2p are the airgap diameter of the machine or the stator outer diameter (for an external rotor motor), the axial length of the machine, the number of turns per phase, and the number of poles, respectively. In Equation (2), b_p is the pole shoe width. The effective pole arc coefficient α_i (pole arc/pole pitch ratio) and pole pitch τ_p are expressed as follows.

$${\alpha }_i={\displaystyle \frac{b_p}{{\tau }_p}};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.55<{\alpha }_i<0.75,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tau }_p={\displaystyle \frac{\pi D_g}{2p}}$$

(2)

In PMSMs, magnets are usually placed on the rotor surface, and the mean flux can be defined by the relative magnet width α_PM = w_PM/τ_p. Thus, αPM is used instead of α_i in Equation (2). The machine’s angular speed, frequency, and synchronous speed can be related as in Equation (3).

$${\omega }_e=2\pi f_e\to n_s={\displaystyle \frac{120f_e}{2p}}$$

(3)

where n_s, ω_e, and f_e are synchronous speed (rpm), angular speed, frequency, and pole number, respectively. Thus, the airgap apparent electromagnetic power can also be written as [28,29,30]

$$\begin{array}{l}{S}_{elm}=m_1{E}_f{I}_a=\pi \sqrt{2}f{N}_1{k}_{\omega 1}{\varphi }_f{\displaystyle \frac{m_1\pi D_g{A}_m}{2m_1\sqrt{2}{N}_1}}\\ =\left({\displaystyle \frac{{\pi }^2}{120}}\right)\left(n_{s}2p\right)k_{\omega 1}{\displaystyle \frac{D_g^2{L}_e}{2p}}{B}_{mg}{A}_m\\ =\left({\displaystyle \frac{{\pi }^2}{120}}\right)k_{\omega 1}{D}_g^2{L}_e{n}_s{B}_{mg}{A}_m\end{array}$$

(4)

where k_ω1 is the winding factor. If Equation (4) is rearranged by power coefficients cosφ and cosθ by including efficiency, i.e., rotor iron and copper loss and rotational losses, electromagnetic torque and output (shaft) power are obtained as given in Equations (5) and (6), respectively.

$$T_d={\displaystyle \frac{S_{elm}\cos \phi }{2\pi n_s}}={\displaystyle \frac{\pi }{4}}{k}_{\omega 1}{D}_g^2{L}_e{B}_{mg}{A}_m\eta \cos \phi$$

(5)

$$P_o=P_{in}\eta =m_1{V}_1{I}_a\eta \cos \theta ={\displaystyle \frac{1}{\epsilon }}{S}_{elm}\eta \cos \theta$$

(6)

where the back emf and phase voltage ratio for the excited motor ε = E_pm/V₁ = 0.6…0.95 [28], E_pm ≈ 0.9…1.1 V₁ [30], or ε ~ 0.98–0.005 p [31]. The output coefficient or motor constant σ_p given in Equation (7) is the ratio of output power to rotor volume and speed [28].

$$\begin{array}{l}{\sigma }_p={\displaystyle \frac{P_o\epsilon }{D_g^2{L}_e{n}_s}}=\left({\displaystyle \frac{{\pi }^2}{120}}\right)k_{\omega 1}{B}_{mg}{A}_m\eta \cos \theta \\ \to D_g^2{L}_e={\displaystyle \frac{P_o\epsilon }{{\sigma }_p{n}_s}}\end{array}$$

(7)

Equations (4) and (7) show that the power developed in a PMSM is mainly proportional to the speed and volume of the rotor; that is, the output coefficient depends on both the rotor geometry and the speed [32]. In the sizing procedure, NdFeB is initially evaluated as B_mg ≈ (0.6…0.8)B_r for magnet motors, while the stator current density A_m’s amplitude varies depending on the size of the motor, from 10 kA/m for small motors to 55 kA/m for medium-power motors [28]. Another critical step in sizing is the separation of D_g and L_e. These values cannot be chosen arbitrarily, as they are essential for the losses and cooling of the machine. These values can be determined with the help of the ratio between the effective core length and the pole pitch, called the aspect ratio. This ratio can be selected in the value range given in Equation (8) [23,31,32]. Thus, the airgap diameter of the machine is shown in Equation (9).

$$\lambda ={\displaystyle \frac{L_e}{{\tau }_p}}={\displaystyle \frac{L_{e}2p}{\pi D_g}};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.15<\lambda <3$$

(8)

$$D_g=\sqrt[3]{{\displaystyle \frac{2p}{\pi \lambda }}{\displaystyle \frac{P_o\epsilon }{{\sigma }_p{n}_s}}}$$

(9)

In the next step, the slot and pole number must be selected. Q_s = 36 slot and 2p = 32 pole configurations are preferred for low-speed and high winding factors. Airgap g is in the range of 0.2–2.5 mm for small- and medium-power motors, and larger values for MW are suggested to limit losses due to field harmonics caused by windings and slots [32]. In [28], it is meant to be 0.3–1.0 mm for small-power PMSMs. In another reference [30], it is given as g = (0.18 + 0.006 × P_o^0.4)/1000 (m). Also, the following equations can be used to estimate the airgap as a function of pole pitch and number of poles [23]:

$$g=3·{10}^{-3}\left(\sqrt{{\displaystyle \frac{2p}{2}}}\right)·{\tau }_p$$

(10)

2.3. Pm Dimension and Coil Mmf

Maximum flux linkage ψ_pm, airgap flux density B_mg, and magnet thickness h_m can be expressed as follows [32].

$${\psi }_{pm}=B_{mg}{b}_{pm}{L}_e{n}_c{N}_{ct}$$

(11)

$$B_{mg}\cong {\displaystyle \frac{B_r}{1+k_{fr}}}·{\displaystyle \frac{h_m}{h_m+g}}\Rightarrow {\displaystyle \frac{h_m}{h_m+g}}\cong {\displaystyle \frac{B_{mg}}{B_r}}\left(1+k_{fr}\right)$$

(12)

where k_fr is the fringing factor in the range of 0.1…0.2, b_pm is the pole arc or magnet width, n_c is the coil number per phase, and N_ct is the turn number per coil. B_r = 1.25 T and H_c = 959 kA/m for NeFeB N40UH PMs. Electromagnetic torque and coil mmf can be expressed as Equations (13) and (14).

$$T_e=3p·{\displaystyle \frac{{\mathsf{\Psi }}_{pm}}{\sqrt{2}}}·I_{ph}=3p·{\displaystyle \frac{B_{mg}{b}_{pm}{L}_e{n}_c{N}_{ct}}{\sqrt{2}}}·I_{ph}$$

(13)

$$N_{ct}{I}_{ph}(rms)={\displaystyle \frac{T_e\sqrt{2}}{3p{B}_{mg}{b}_{pm}{L}_e{n}_c}}$$

(14)

To increase the winding factor and torque and reduce cogging torque, a 36-stator slot and 32-rotor pole configuration are used. So, the coil number per phase n_c is 12 for the 3-phase machine. The machine functions similarly to a DC machine if the voltage waveforms are applied to the rotor position by a PWM and with phase commutation every 60° (electrical). The 120° broad, smooth current blocks cause the machine to operate in two phases in a steady state, ignoring phase commutation. The sum of two-phase EMFs in series and emf per phase can be expressed as Equations (15) and (16) [32].

$$E_{LL}=V_{dc}-2R_s{i}_a-2L_s{\left({\displaystyle \frac{d{i}_a}{dt}}\right)}_{=0}$$

(15)

$$e_1={\displaystyle \frac{V_{dc}-2R_s{i}_a}{2}}$$

(16)

where Ia is the flat-topped value of the squarewave current equal to the inverter input current, V_dc is the DC input voltage that supplies the inverter, R_s is the stator phase resistance, and Ls is the synchronous inductance. When a PM brushless motor is powered by an inverter, and the DC bus voltage is known, the fundamental harmonic of the six-step, three-phase inverter’s output AC line-to-line voltage is [28]

$$V_{1L}={\displaystyle \frac{\sqrt{6}}{\pi }}{V}_{dc}\approx 0.78V_{dc}$$

(17)

The ratios of the first harmonic amplitudes to the maximum values of the normal components of the armature reaction magnetic flux densities in the d-axis and q-axis, respectively, are known as the form factors of the armature reaction (k_fd = B_ad1/B_ad and k_fq = B_aq1/B_aq). Additionally, the armature reaction form factors for the surface PM rotor are k_fd = k_fq = 1 [28]. For a square-wave excitation, the EMF induced in a phase turn and turn number per phase, respectively

$$e_1=\pi \sqrt{2}f{N}_1{k}_w{\varphi }_g=\pi \sqrt{2}f{N}_1{k}_w{\alpha }_i{B}_{mg}{\tau }_p{L}_e$$

(18)

$$N_1={\displaystyle \frac{e_1}{\pi \sqrt{2}f{k}_w{\alpha }_i{B}_{mg}{\tau }_p{L}_e}}\cong {\displaystyle \frac{\epsilon V_{1L}/\sqrt{3}}{\pi \sqrt{2}f{k}_w{\alpha }_i{B}_{mg}{\tau }_p{L}_e}}$$

(19)

The phase current can be calculated from Equations (14) and (19). So, the copper area S_Cu and copper diameter d_cu can be expressed as follows:

$$S_{cu}={\displaystyle \frac{I_{ph}}{j_{con}}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{d}_{cu}=2\sqrt{{\displaystyle \frac{S_{cu}}{\pi }}}\hspace{0.17em}$$

(20)

where current density j_con = 4…6.5 A/mm².

2.4. Stator Dimension

For the slot shape given in Figure 4, some variables such as the slot opening width b_so = 2–3 mm, slot opening height h_so = 0.5–1 mm, and wedge height h_s1 = 1–4 mm are given in [31].

Slot pitch τ_s, stator tooth width b_ts, stator slot height h_s2, stator yoke height h_sy, the upper b_s1 and lower b_s2 parts of the slot width, slot area A_u0, the active slot area A_u, stator inner diameter D_si, and rotor outer diameter D_ro can be expressed, respectively, as follows [31,32]:

$${\tau }_s={\displaystyle \frac{\pi D_g}{Q_s}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{b}_{ts}={\displaystyle \frac{{\tau }_s}{k_{fill}}}·{\displaystyle \frac{B_{mg}}{B_{sat}}}$$

(21)

$$h_{s2}={\displaystyle \frac{6N_{s}I}{\frac{B_{mg}}{B_{sat}}{j}_{con}{k}_{fill}}}{\displaystyle \frac{1}{\pi D_g}}={\displaystyle \frac{A_m}{\frac{B_{mg}}{B_{sat}}{j}_{con}{k}_{fill}}}$$

(22)

$$h_{sy}={\displaystyle \frac{B_{mg}}{B_{sy}}}·{\displaystyle \frac{{\tau }_p}{\pi }}={\displaystyle \frac{B_{mg}}{B_{sy}{k}_{sf}}}·{\displaystyle \frac{D_g}{2p}}$$

(23)

$$b_{s1}=\pi {\displaystyle \frac{(D_g-2·h_{s0}-2·h_{s1})}{Q_s}}-b_{ts}$$

(24)

$$b_{s2}=\pi {\displaystyle \frac{(D_g-2·h_{s0}-2·h_{s1}-2·h_{s2})}{Q_s}}-b_{ts}$$

(25)

$$\begin{array}{l}{A}_{u0}={\displaystyle \frac{2N_{ct}{I}_{ph}}{k_{fill}·j_{con}}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\\ A_u={\displaystyle \frac{(b_{s1}+b_{s2})}{2}}·(h_{s2}+h_{s1}+h_{s0})\end{array}$$

(26)

$$D_{si}=D_g-2(h_{s0}+h_{s1}+h_{s2}+h_{sy})$$

(27)

where saturation flux density B_sat = 1.8 T, yoke flux density B_sy = 1.4…1.7 T, airgap flux density B_mg = 0.85 T, and slot fill factor k_fill = 0.5 [23,30,31,32]. For efficiency, the diameter ratio D_si/D_so should be close to 0.5 [31] or 0.6 [29].

2.5. Rotor Dimension

The magnet thickness is supposed to produce the desired electromagnetic power and torque in the airgap and also to avoid demagnetization in the case of limited overload. Therefore, the magnet thickness can be calculated as given in Equation (12). The rotor yoke height here is calculated from Equation (28), and then the rotor outer diameter D_ro can be expressed as in Equation (29).

$$h_{ry}={\displaystyle \frac{B_{mg}}{B_{ry}}}·{\displaystyle \frac{{\tau }_p}{\pi }}={\displaystyle \frac{B_{mg}}{B_{ry}}}·{\displaystyle \frac{D_g}{2p}}$$

(28)

$$D_{ro}=D_g+2(g+h_m+h_{ry})$$

(29)

2.6. Optimization and Analysis

Depending on the cooling method and output power of the BLDC, the current density ranges from 3.5 to 10 A/mm². In the meantime, the machine’s electrical and magnetic load ability and the flux per pole must be carefully defined when determining the machine’s dimensions according to the operating conditions and load types.

Many factors, such as efficiency, dimensional limitations, cogging torque, cooling method, demagnetization, output performance, cost, and many other variables, limit the motor design. Optimization studies were carried out considering these factors.

2.7. Multi-Objective Parametric Analysis

Torque, speed, power, and efficiency curves are critical when selecting or designing an electric motor. These curves need to be analyzed under different operating conditions of the motor to understand its performance, determine if it is suitable for the application, and determine the optimum parameters for the desired performance. The optimum design in electrical machines is to obtain an acceptable design that fulfills all the requirements. Since electrical machines are composed of many parameters, the minimization or maximization of a set of objective functions is required for multi-objective parametric optimization [33,34]. One or more variable sweep definitions, each defining a range of variable values, are defined in a parametric analysis. You can parameterize component values, for instance. At every variation, the design is solved using optimetrics. The performance of each design variant can then be ascertained by comparing the outcomes. Because parametric analyses aid in identifying an appropriate range of variable values for the optimization study, they are frequently employed as preludes to optimization solutions.

Current optimization methods usually choose to optimize all design parameters simultaneously, and such optimization methods require a lot of computational time and are complex. Therefore, all design parameters are broken down into five phases and optimized individually based on the results of the sensitivity analysis to address the issue of the increase in calculation consumption brought on by the increase in design parameters. Figure 5 shows a suggested optimization framework for the investigated BLDC motor based on the multistep parametric optimization strategy. As shown, the optimization progress has been divided into 5 steps. Different objectives and constraints will be considered for each optimization level. The objectives are to minimize total loss (Pcore and Pcu), maximize torque (Tmech), rate power (Pmech), and increase efficiency, as well as minimize cogging torque (Tcog).

Equation (30) provides the generalized form of the multi-objective function utilized. In this case, y or f(x) are objective functions that reach their minimum or maximum value while preserving the ranges of the other parameters. x is a vector variable stating structural or sizing parameters in the iteration related to the machine size and restricted to take a value within a lower (x_L) and an upper (x_H) bound. The g_m(x) and h_k(x) are constraint functions, M is inequality, and K is equality constraint. The parameters are as follows:

✓

n_cs is the conductor per slot;

✓

L_stk is the stack length;

✓

h_m is magnet thickness;

✓

α_m is the pole pitch/magnet pitch ratio;

✓

b_s0 is slot opening.

$$\left.\begin{array}{l}\max \hspace{0.17em}or\hspace{0.17em}\min \hspace{0.17em}\left\{\begin{array}{l}{y}_j=\hspace{0.17em}f(\overrightarrow{x}),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,2,\dots ,J\\ \overrightarrow{x}=\left[x_1,x_2,\dots ,x_D\right],\overrightarrow{x}\in R^D\end{array}\right.\\ subject\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left\{\begin{array}{l}{x}_L\le x_i\le x_H,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,2,\dots ,N\\ g_m(x)\ge 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m=1,2,\dots ,M\\ h_k(x)=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k=1,2,\dots ,K\end{array}\right.\end{array}\right\}$$

(30)

For example,

$$\left\{\begin{array}{l}\mathrm{minimize}\to P_{core}=f(h_m,{\alpha }_m,L_{stk},n_{cs},b_{s0})\\ \mathrm{minimize}\to P_{cu}=f(h_m,{\alpha }_m,L_{stk},n_{cs},b_{s0})\\ \mathrm{maximize}\to P_{mech}=f(h_m,{\alpha }_m,L_{stk},n_{cs},b_{s0})\\ \mathrm{maximize}\to T_{mech}=f(h_m,{\alpha }_m,L_{stk},n_{cs},b_{s0})\\ \mathrm{maximize}\to \eta =f(h_m,{\alpha }_m,L_{stk},n_{cs},b_{s0})\\ \mathrm{minimize}\to T_{cog}=f(b_{s0})\\ \mathrm{related} \mathrm{to}\to \left\{\begin{array}{l}20mm\le L_{stk}\le 60mm,++1mm\\ 1mm\le h_m\le 5mm,++0,2mm\\ 0.5\le {\alpha }_m\le 1,++0.01\\ 6\le n_{cs}\le 18,++1\\ 2mm\le b_{s0}\le 4mm,\hspace{0.17em}++0.1mm\end{array}\right\}\end{array}\right\}$$

(31)

In addition, there are several constraints, such as

$$\mathrm{related} \mathrm{to}\to \left\{\begin{array}{l}{h}_1(x)=\hspace{0.17em}10\hspace{0.17em}kA/m\le A_m\le 25kA/m\\ h_2(x)=\hspace{0.17em}0.7T\le B_{mg}\le 8.5T\\ h_3(x)=\hspace{0.17em}{4 \mathrm{A}/\mathrm{mm}}^2\le j_{con}\le 6.5A/m{m}^2\\ h_4(x)=\hspace{0.17em}{k}_{fill}\le \%60\end{array}\right\}$$

(32)

Five goal optimization problems with multiple solutions are analyzed, as illustrated in Figure 6. Finding the best solution for all five goal functions is typically challenging because of the significance of five goal functions on motor performance. For the desired motor performance at constant power, a particular increase in the number of conductors and length increases the efficiency, while magnet thickness, pole arc to pole pitch ratio, and slot opening decrease the efficiency, as shown in Figure 6. The initial and optimum values of some dimensional parameters determined to meet the desired targets are in Table 1. The performance values of the motor are presented in

Table 3. Comparison data of the BLDC motor.

Parameter	Initial Design	Optimum Design
Rated power	2.5 kW	2.5 kW
Rated speed	1284 rpm	873 rpm
Rated torque	18.58 Nm	27.36 Nm
Efficiency	88%	91.74%
Armature current density	9.15 A/mm2	4.83 A/mm2
Stack length	35 mm	38 mm
Stator inner diameter	196.88 mm	193 mm
Magnet thickness	2 mm	3 mm
Stator tooth width	12.19 mm	9 mm
Rotor yoke height	6.64 mm	5.5 mm
Stator yoke height	4.66 mm	6.6 mm
Slot opening	2.5 mm	3 mm
Conductors per slot	11	13
Wire diameter	1.93 mm	2.58 mm
Slot fill factor	30.11%	44.9%

.

Although the proposed initial design ensures some criteria, such as good efficiency and output power, it fails to provide three other vital measures: armature current density, rated speed, and torque. The optimization acts to obtain the best results under the desired circumstances.

2.8. Two-DimensionalFem Analysis

More analytical sizing and optimization studies are needed to conclude the design. With electromagnetic analysis, the flux density and flux distributions in the whole model, as well as the operating point of the magnet according to the demagnetization characteristic should be investigated. The 2D motor model is time stepped as a function of the magnetic field solutions created by the windings fed with the external voltage source and the permanent magnets.

Figure 7 shows the B-H curve of the stator and rotor cores respectively. Figure 8 shows the flux distribution and effective flux line of the entire motor surface, the field strength in the magnets, current density, phase currents, and output torque for the 15.48-degree rotor position, respectively. The rotor yoke flux density is low at the center of the pole arc. In contrast, at the junction of the two poles, it is around 1.7 T. While the flux density of the stator teeth is about 1.4 T, with a maximum of about 2 T on the shoe tips, the average flux density on the entire stator surface is 0.72 T. As shown in Figure 7, and these flux values, the saturation does not occur in the stator steel. The magnetic field lines are closed circuits through expected paths, and the periodicity conditions are appropriate. As is apparent from Figure 8c, the magnet demagnetization does not occur at operated conditions with the ambient temperature. In actual working conditions, excessive load will increase the temperature. Irreversible demagnetization may occur in permanent magnets with the temperature. Therefore, temperature analysis should be performed [1,35].

The current density presented in Figure 8d for overall conductor regions is a maximum of 4.68 A/mm². It is an acceptable current density value for this scale machine. Figure 9a–e show the simulated phase currents, back EMF, currents FFT, cogging torque, and corresponding output torque. The rms value of the currents was obtained at 34 A. When the FFT of the current is examined, it is observed that the 5th and 7th harmonic values are high. This is an expected situation in BLDC motors. The average value of the output torque was obtained at 36 Nm, which provides the objective function for the optimization design. To improve the torque quality, the cogging torque value must be low. The least common multiple (LCM) and slot opening width affect the cogging torque value [36]. A very low gear torque can be obtained if the slot and pole numbers are selected, so the LCM between these numbers is enormous. The cogging torque for a design with 32 poles and 36 slots is represented in Figure 9d. It can be seen that, for this design, the peak-to-peak value of the cogging torque is 0.12 Nm.

2.9. Electromagnetic–Thermal Coupling

Heat generation in electrical machines is caused by power losses such as ohmic (including eddy loss) and iron (hysteresis loss, eddy loss, and excess loss) present in the machine. It can cause its components to reach temperatures that affect their performance and life. Balancing cooling system efficiency with energy losses is essential to optimize motor performance [37]. Thermal analysis of electrical machines is a critical study for determining and optimizing the cooling method of the machine before production. Thanks to numerical-based analysis, machine temperatures can be reached very quickly, and the analysis of work that cannot be performed on real machines is possible. The thermal model required for thermal analysis must include the three modes of heat transfer: conduction, convection, and radiation.

Heat transfer by conduction is expressed with Fourier’s law equation as [38]

$$q_{con}=-kA\nabla T$$

where q is the heat flux density (W/m²), k is thermal conductivity W/(m·°K) of the material, A is the surface area (m²), and ∇T is the temperature gradient (°K/m).

Convection is the heat transfer caused by fluid motion, as opposed to conduction. It can be accomplished by natural convection due to density differences in the fluid, or by forced convection by external devices such as fans. Convection heat transfer depends on the temperature difference between the surface and the fluid and is expressed by Newton’s law.

$$Q_{conv}=hA(T_s-T_{\infty })$$

(33)

where h is the convection heat transfer coefficient (W/m²K), T_s is the surface temperature, and T_∞ is the fluid temperature (°K or °C).

Heat transfer by electromagnetic waves is defined as a radiation phenomenon. The heat flux rate for the radiation quad is derived from the Stefan–Boltzmann law, which determines the emissivity of radiation emitted by a surface:

$$q_{rad}=\epsilon \sigma A(T_s^4-T_{sur}^4)$$

(34)

where ε is the surface emissivity, σ is the Stefan–Boltzmann constant (W/m²K⁴), and T_sur is the surrounding temperature.

The principle diagram of the proposed analysis method is presented in Figure 10. This method provides the opportunity to perform direct coupling electromagnetic and thermal analyses in an integrated manner, as shown in Figure 11. The 1D machine model is converted into a 3D model in this proposed model. Numerical analysis determines the core and winding losses that create thermal loads and are directly connected to thermal analysis [13,39]. The resulting thermal values are coupled as feedback to the electromagnetic component to change the permeability and conductivity of the material. The correct definition of material engineering data needs to be considered here.

3. Results and Discussion

As is known from analytical sizing, optimization, and numeric analysis studies, the BLDC motor that was designed and considered in this paper to evaluate performance is a 2.5 kW 32-pole in-wheel motor, which is used in small-size electric vehicles. The prototype motor and test bed are given in Figure 12. Experimental studies have been conducted to verify the motor performance obtained by design and simulation studies.

Figure 13 and Figure 14 present the comparative performance evaluation of the optimal motor with simulation results and experimental results. As a result of experimental studies, the targeted performance values were obtained when the values presented in

Table 4. Performance data at rated operating condition.

Parameter	Optimum Design Simulation	Experimental	Deviation Rate
V [V]	96	96	0
I [A]	28.458	30.05	0.05
T [Nm]	27.36	26.61	0.028
n [rpm]	873	897	0.026
Pin [W]	2725	2737.372	0.004
Pout [W]	2500	2500.91	0
Efficiency	91.74%	91.36%	0.004

were examined. The motor performance has been analyzed according to the rated operation. At a 96 V operating voltage, the motor yields a nominal 26.61 Nm torque, nominal 2.5 kW power, and nominal 91.37%, efficiency. The deviation rate is quite small, and this indicates that the experimental results agree with the simulation results. The differences that occur may be due to several reasons. These are as follows: simulation may not fully include driver losses in the model; in experimental work, switching losses of MOSFETs or IGBTs, voltage drops, conduction losses, and heating effects may reduce efficiency; friction in bearings in experimental tests may significantly reduce efficiency at low speeds; and torque, current, and voltage sensors have a certain margin of error in actual experiments.

In Figure 15, the thermal analysis results of the motor and the experimental results are given comparatively. The temperature values of motor parts are given in

Table 5. Temperature values of motor parts.

Object	Temperature
Rotor	22 °C
Stator yoke	66.015 °C
Stator teeth	67.366 °C
Stator winding	65.975 °C
Magnet	22 °C
Rotor (minimum)	22 °C
Stator teeth (maximum)	72.874 °C
All body (average)	53.026 °C

. In the experimental study, the temperature distribution and values of the motor were obtained by thermal imaging after the motor was operated at full load for 5 min. Considering both simulation and experimental results, the winding temperature was approximately 72 °C, and the average temperature was 50 °C. As expected, the core’s temperature is in direct contact with the windings, which is higher than the yoke’s. There is a difference of approximately 1 °C between the temperature value obtained after the experimental measurement and the steady-state simulation estimate. There are some limitations in directly comparing the results of the short-term steady-state simulations with the experimental study. These limitations include transient thermal effects and potential sources of inconsistency (e.g., convection effects, material thermal properties, and environmental factors). In addition, although the simulations were performed based on steady-state conditions, the iteration value was kept small due to the long calculation times in 2D and 3D integrated electromagnetic–thermal analyses, and the analysis was performed in a relatively short run time. This situation caused a difference between the results obtained before the model reached full thermal equilibrium and the experimental measurements.

4. Conclusions

This paper proposes an analytic design and optimization method for electric propulsion motors specifically designed for small EVs. Based on the proposed analytical sizing and optimization model, a 36-slot, 32-pole, and a 2.5 kW outrunner BLDC motor was designed and simulated where essential issues such as output power, shaft torque, cogging torque, speed range, efficiency, slot fill factor, current density, specific magnetic loading, and cost were considered. The sizing parameters of the produced motor were determined from the optimization results. The experimental results of the motor were derived from the optimum design results. The simulation and experimental results are presented comparatively. The deviation ratio is quite small, indicating that the experimental results are in good agreement with the simulation results. A FEM multiphysics simulation structure is used to couple 3D FEM electromagnetic and steady-state thermal analysis. Also, using 2D FEM and experimental results validates the analytical design and analysis. These results show that the motor is suitable for the natural cooling method. The proposed in-wheel motor performs satisfactorily with experimental studies, and the results obtained show that the most appropriate structure is the outrunner, which allows many poles to achieve high efficiency at low speeds.