Optimal Design and Simulation of an Axial Flux Machine for Unmanned Aerial Vehicles

Abstract

Currently, unmanned aerial vehicles (UAVs) are present in numerous sectors of the economy, which require them to carry out services with agility, safety and precision. This equipment can be designed in the multi-rotor concept, using several electric motors, normally with permanent magnets and radial flux. Another alternative would be to use axial flux electric motors, which have a potentially higher power density and help to reduce the weight of the UAV. In addition, the use of grain-oriented steel in the mo-tor stator contributed to reducing magnetic losses and increasing aircraft efficiency. Thus, this work proposes the development of a permanent magnet axial flux motor, using grain-oriented steel in a yokeless and segmented armature. The technical specifications for a UAV application in agriculture are presented, as well as the mathematical modeling required for optimized motor design. Finally, the developed project is validated through 3D finite element electromagnetic field simulations, demonstrating the feasibility of using rectangular teeth to simplify the stator assembly process.

Index Terms
Unmanned aerial vehicle; permanent magnet machine; yokeless and segmented armature; grain-oriented silicon steel.

I. INTRODUCTION

Unmanned aerial vehicles (UAVs), also known as drones, are aircraft that operate without the presence of a pilot on board, and are either autonomous or remotely piloted. UAVs have a wide range of applications in different sectors, such as environmental monitoring, surveying, infrastructure inspections, goods deliveries, agricultural spraying, aerial photography, military use, among others [¹]-[⁴]. Most commercial drones use an electric propulsion system to remain in the air and control movement through the use of batteries, electronic speed controllers (ESCs), electric motors, and propellers. This propulsion system must be optimized [⁵], finding a balance between physical dimensions, efficiency and reliability, allowing the UAV to transport the payload with greater flight autonomy.

The thrust required to maintain the UAV hovering is typically provided by a set of propellers, connected to 4, 6 or 8 electric motors [⁶]. These components represent a larger part of the propulsion system mass. Furthermore, electrical and mechanical losses of the motors cause a major impact on energy efficiency, affecting the weight and volume of the batteries. Therefore, electric motors play a key role in the optimized design of the UAV , with various configurations proposed [⁷].

For applications in multi-rotor UAVs, it is common to use radial flux permanent magnet synchronous machines (PMSM) [³], [⁸], with numerous companies selling electric motors that use this configuration. However, the presence of axial-flux permanent magnet (AFPM) synchronous machines becomes rele-vant, especially with respect to the electrification of transport systems [⁹]-[¹²]. According to [¹³], [¹⁴], AFPM machines with a high number of poles offer greater torque density than their radial counterparts, resulting in smaller volume and mass for the same power demand, making them suitable for drones.

In an effort to reduce mass, magnetic losses, and cogging torque, some research address the possibility of AFPM motors without ferromagnetic material in the stator (coreless) [¹⁵], [¹⁶]. An example is proposed in [¹⁷], in which a multiphase high power density drive is evaluated for UAV propulsion. An air-cored AFPM machine is used, with two rotors and a stator positioned between them. The use of an ironless stator enables the possibility of manufacturing motor coils on printed circuit boards (PCBs) [¹⁸]-[²⁰]. However, the absence of magnetic material in the stator exposes the copper conductors to fluctuations in the airgap’s magnetic flux density. This exposure can cause considerable losses due to eddy currents. Furthermore, machines with a wide magnetic airgap suffer from an uneven distribution of magnetic flux density and flux fringing. Consequently, parallel conductors experience varying voltages, resulting in circulating current losses [¹⁶], [²¹].

For AFPM machines with cored stator, materials such as silicon steel, soft magnetic composite (SMC) and amorphous magnetic metal (AMM) are used [²²]. SMC is made up of iron powder that is covered with insulation material, allowing the manufacture of complex and irregular three-dimensional parts through machining or pressing processes [²³]-[²⁵]. Due to the high resistivity of SMC compared to silicon steel, eddy current losses can be maintained at a low level, and SMC becomes a more efficient material for applications whose operating frequency is higher than 1 kHz. However, in terms of maximum permeability and coercivity, SMC shows poorer performance [²²], [²⁶], [²⁷], resulting in lower levels of magnetic saturation and higher losses than electrical steel in the low-frequency range.

In the context of electrical steel, the majority of radial flux PMSM rely on laminated non-oriented (NO) silicon steel [²⁸]. However, grain-oriented (GO) steel, typically used in transformer and reactor cores, has been incorporated into motors [²⁹], [³⁰]. GO steel offers benefits over NO steel, such as reduced core loss and enhanced saturation magnetic flux density. Nonetheless, the varying permeability and core loss characteristics in different directions, pose challenges in designing AFPM machines with a yoke, making it difficult to fully harness the advantages of GO material [³¹].

An alternative to exploring the benefits of GO steel is to use it in a yokeless and segmented armature (YASA) AFPM machines [³²]. In this type of electric motor, the stator consists of several independent teeth, each with its own coil and without a common connection to a magnetic yoke. In the YASA concept, a significant reduction in magnetic losses is expected when GO steel is used instead of NO steel [²⁹], [³⁰] due to the unidirectional magnetic flux in most of the stator volume.

Considering the information presented, the scientific contribution of this work lies in the design of a high-efficiency axial flux machine for drone propulsion, an application area where radial flux machines are predominantly used. Additionally, the work proposes the use of grain-oriented silicon steel to minimize magnetic losses, along with an investigation into the use of a stator with rectangular teeth, aiming to simplify the machine’s manufacturing and assembly processes. Section II presents an overview of the application, defining the specifications and operating conditions of the machine. The mathematical modeling and the adopted optimization process are described in Section III. Section IV presents the results of electromagnetic simulation using the finite element method. Finally, the conclusion are outlined in Section V.

II. DESIGN OVERVIEW

A. Agriculture Application

With the advancement of precision agriculture, drones have become essential tools in the field. They offer numerous benefits, such as saving time, reducing costs and increasing efficiency in crop monitoring. Their applications include land mapping, pest detection, precise irrigation, soil analysis, spraying and agricultural planning [³³]. Among the various UAV concepts, multirotor drones are valued for their high stability and ease of operation, making them the most used in agricultural activities. They are particularly useful for tasks that require greater precision and maneuverability, are easier to operate and require less operational training time [³⁴].

B. Propulsion System Sizing

A drone application for spraying and dispersing chemical or biological products in plantations is considered. For these purposes, it is common to find commercial drones that operate with an approximate take-off load of 50 kgf [³⁵], [³⁶]. Furthermore, a hexacopter-type UAV is defined, using six propulsion units to provide a thrust of 8.3 kgf each. However, a thrust-to-load ratio of 1.6 is applied to guarantee the propulsion system’s capacity for agile vertical movements and to maintain the drone’s stability in the presence of wind. Therefore, each propulsion unit is dimensioned to provide a nominal and maximum thrust of 8.3 kgf (81.7 N) and 13.3 kgf (130.8 N) respectively.

Although the necessary thrust (T_p) can be provided by different propeller sizes, each diameter represents a disc loading (Z_p) and an estimate of the required mechanical power (P_p) as shown in (1). The larger the diameter D_p, the larger the propeller area (A_p), resulting in less P_p to keep the aircraft hovering in the air with ρ_air density [³⁷].

(1) $$Z_p=\frac{T_p}{A_p},A_p=\frac{\pi }{4}{D}_p^2,P_p\approx \sqrt{\frac{T_p^3}{2{\rho }_{\text{air}}{A}_p}}$$

According to the correlation diagram between disc loading and hover lift efficiency of the aircraft shown in [³⁸], UAVs are considered to have a disc loading of approximately 100 to 300 N/m², which corresponds to a range of values observed in helicopters. Therefore, a commercial 32" (81 cm) propeller is chosen [³⁹], resulting in a disc loading between 158 and 252 N/m², due to the nominal and maximum thrust values expected for the drone. Based on the available data for the propeller [⁴⁰], the graphs in Fig. 1 are generated, relating thrust and torque (τ_p) to speed (ω_p), as show in (2) and (3).

Fig. 1
Thrust and torque characteristics provided by the 32x11" propeller [³⁹], [⁴⁰].

From (2) and (3), it is determined that the propeller requires 2560 rpm and 3.0 Nm for nominal thrust and 3200 rpm and 4.8 Nm for maximum thrust.

(2) $$T_p\left({\omega }_p\right)=1.4382\cdot {10}^{-5}{\omega }_p^2-8.0000\cdot {10}^{-3}{\omega }_p+7.9389[N]$$

(3) $${\tau }_p\left({\omega }_p\right)=5.8306\cdot {10}^{-7}{\omega }_p^2-5.3679\cdot {10}^{-4}{\omega }_p+0.5649[Nm]$$

C. Motor Specifications

Among the numerous structures and concepts of electric motors, this work considers the use of an AFPM machine to be part of the UAV propulsion system. Although single-sided construction is the simplest and most cost-effective solution, it results in unbalanced axial forces. This, associated with mass imbalance and geometric tolerances, creates challenges in maintaining mechanical stability [¹¹], [¹⁸]. To mitigate this problem, a structure with two rotors and a stator placed between them is adopted, as illustrated in Fig. 2.

Fig. 2
Representation of the AFPM machine with two rotors evaluated in the research.

When silicon steel is used in the AFPM motor stator, radial lamination is required to reduce eddy current losses, but this presents manufacturing challenges due to the stator´s asymmetrical radial cross-section. A common solution is to wind a silicon steel strip, pre-stamped with the stator’s teeth and yoke profile [²²], [³¹]. To simplify the manufacturing process, a YASA-type motor is developed, with each stator segment made using radial lamination. The main design specifications are presented in Table I.

TABLE I
KEY SPECIFICATIONS OF AFPM MOTOR

Parameter	Value
Number of segments (Q), poles pair (p) and phases (Nph)	12, 8, 3
Max. mechanical torque and speed (τm,mx, ωm,mx)	4.8 Nm, 3200 rpm
Nom. mechanical torque and speed (τm,nm, ωm,nm)	3.0 Nm, 2560 rpm
Magnets material and grade	NdFeB (N45UH)
Magnets rel. permeability (µm) and remanent induc. (Bm,r )	1.04, 1.25 T (80◦C)
Mechanical air gap (hgap) and tooth tip (ht) thickness	1 mm, 3mm
Clearance between coils (wc,cl) and magnets (wm,cl)	2 mm, 2 mm
Teeth material (GO silicon steel grade and thickness)	27QG095, 0.27 mm
Massive magnetic losses (kloss,F e)	25 W/kg (1.7 T, 400 Hz)
Copper resistivity at 80◦C (ρCu,80C )	2.276E-8 Ωm
winding (kw), stacking (kstk) and fill factor (kfill)	0.88, 0.97, 0.35
Copper (σCu), steel (σFe) and magnet (σm) densities	8900, 7650, 7600 kg/m3

To maximize the power density of the motor, it is advantageous to select the highest possible number of poles in order to reduce the yoke thickness. However, a high number of poles increases the electrical frequency for a given speed, leading to higher magnetic losses. Therefore, 8 pole pairs are chosen, which provides a maximum frequency of 400 Hz. This choice represents an acceptable compromise given the type of magnetic lamination used for the stator teeth. Furthermore, the combination with 12 segments has the advantage of simplifying stator construction, providing a concentrated winding configuration with multiple parallel paths. This allows adjust the number of turns in the segments with a smaller wire cross-section, reducing the circulating currents that would occur if strands were placed in parallel.

Although other configurations with a greater number of poles and segments are theoretically possible, in practice it is necessary to maintain minimum dimensions that allow the cutting of steel sheets and provide mechanical resistance. Considering that the motor to be designed has a relatively small power and outer diameter (less than 2 kW and 180 mm respectively), the number of poles and segments chosen allows the teeth and coils of the stator to have physical dimensions that are feasible to manufacture. Furthermore, the chosen combination of segments and poles also speeds up the electromagnetic field simulation process, due to the periodicity of the geometry every 90 degrees.

III. MODELING AND OPTIMIZATION

A. Mathematical Modeling

Fig.3 (a) illustrates the proposed AFPM motor and Fig.3 (b) highlighting the main dimensional parameters of interest. Due to periodicity and symmetry relative to the xy plane, only a part of the motor is modeled as shown in Fig.3 (c), to reduce the simulation time.

Fig. 3
Dimensional parameters used in the AFPM motor design: (a) electric motor overview, (b) main dimensional parameters of interest, (c) motor modeled part to be simulated by FEM.

Based on the operating theory of permanent magnet machines outlined in [⁴¹], [⁴²], a set of equations is proposed in [⁴³] that relate dimensional and electromagnetic parameters. Equation (4) defines the active motor area (S_act), where electromagnetic interaction results in torque generation. Thus, the slot and coil width (w_s, w_c) are defined by (5), using the variable k_s, which represents the concentration of magnetic flux in the segment. Additionally, the element heights are calculated in (6), which incorporates the stator copper area (S_c), conductor fill factor (k_fill) and the magnetic flux concentration factor in the rotor yoke (k_y).

(4) $$r_{s,o}=r_{s,i}+l_s{r}_{s,m}=r_{s,i}+\frac{l_s}{2}{S}_{act}=2\pi \cdot r_{s,m}\cdot l_s$$

(5) $$w_s=\frac{S_{act}\cdot (1-1/k_s)}{Q\cdot l_s}{w}_c=\frac{w_s-w_{c,cl}}{2}$$

(6) $$h_s=h_c=\frac{S_c}{k_{fill}\cdot Q\cdot 2w_c}{h}_y=\frac{\pi \cdot r_{s,m}}{2\cdot p\cdot k_y}$$

Considering the geometric parameters of the motor and the magnet specifications from Table I, the peak value of the magnetic flux density in the air gap (B_g,pk) is calculated as shown in (7). Here, the magnet embrace ratio (γ_m) indicates the reduction of B_g due to the space between the magnets (w_m,cl). Consequently, the electromagnetic torque (τ_elm), dependent on the electric loading (A_c,rms), is established in (8), in which J_c corresponds to the conductors RMS current density.

(7) $$B_{g,pk}=B_{m,r}\cdot \frac{{\gamma }_m}{1+\left(\frac{{\mu }_m\cdot {\gamma }_m\cdot h_g}{h_m}\right)}{\gamma }_m=1-\frac{p\cdot w_{m,cl}}{\pi \cdot r_{s,m}}$$

(8) $${\tau }_{elm}=2\pi \cdot k_w\cdot r_{s,m}^2\cdot l_s\cdot \frac{B_{g,pk}}{\sqrt{2}}\cdot A_{c,rms}{A}_{c,rms}=\frac{J_c\cdot S_c}{2\pi \cdot r_{s,m}}$$

Assuming that each phase of the motor has only one turn with current I_1t,rms, the flux (ϕ_1t,rms) and the induced voltage per turn (E_1t,rms) are determined in (9), where f_e is the electrical frequency.

(9) $$I_{1t,rms}=\frac{S_c\cdot J_c}{2N_{ph}}{\varphi }_{1t,rms}=\frac{{\tau }_{elm}}{N_{ph}\cdot p\cdot I_{1t,rms}}{E}_{1t,rms}=2\pi f_e\cdot {\varphi }_{1t,rms}$$

Finally, the mass and losses of the motor active parts must be determined, as they will form the objective function of the optimization process. Equation (10) calculates the mass is obtained from the volume of the teeth (V ol_teeth), yokes (V ol_yokes), coils (V ol_coils) and magnets (V ol_m) multiplied by the respective densities of the materials in Table I. Losses are divided into copper (P_Cu) and iron (P_{F e}) components, as shown in (11), with the magnetic losses in the yokes neglected. Massive magnetic losses (k_{loss,F e}) for the GO steel is defined in Table I. The Appendix A demonstrates the calculation of P_Cu.

(10) $$M_{Fe}=(Vo{l}_{\text{teeth}}+Vo{l}_{\text{yokes}})\cdot {\sigma }_{Fe}{M}_{Cu}=Vo{l}_{\text{coils}}\cdot {\sigma }_{Cu}{M}_m=Vo{l}_m\cdot {\sigma }_m$$

(11) $$P_{Cu}=Vo{l}_{\text{coils}}\cdot J_c^2\cdot {\rho }_{Cu,80C}{P}_{Fe}=Vo{l}_{\text{teeth}}\cdot {\sigma }_{Fe}\cdot k_{loss,Fe}$$

B. Optimization Procedure

The optimization process has the task of determining a set of design parameters that minimize the objective function (F_obj). In (12), F_obj is proposed, which considers the total mass (M_total) and the efficiency of the machine under nominal operating conditions (η_nm). To calculate efficiency, only magnetic or conduction losses are considered, disregarding mechanical losses in bearings or those caused by friction between moving parts and air. It is important to note that η_nm was chosen because it corresponds to the power condition (P_m,nm) at which the motor will remain running most of the time (τ_m,nm, ω_m,nm), while the maximum power condition (τ_m,mx, ω_m,mx) is used to calculate the mass.

(12) $$F_{obj}=\frac{M_{total}}{{\eta }_{nm}}{M}_{total}=M_{Fe}+M_{Cu}+M_m{\eta }_{nm}=\frac{P_{m,nm}}{P_{m,nm}+P_{Cu,nm}+P_{Fe,nm}}$$

To implement the optimization procedure, the specifications in Table I, the equations presented in Section III-A and the variable and constraint parameters of Table II are programmed in Python.

TABLE II
MOTOR DESIGN VARIABLES, BOUNDARIES AND CONSTRAINTS RANGE.

Variable parameters	Min. value	Max. value
Conductors RMS current density (Jc)	1 A/mm2	15 A/mm2
Conductors copper total area (Sc)	100 mm2	10000 mm2
Magnet thickness (hm)	3 mm	4 mm
Inner radius of the segment active part (rs,i)	40 mm	60 mm
Length of the segment active part (ls)	10 mm	30 mm
Magnetic flux concentration factor in segments (ks)	1.5	3.0
Magnetic flux concentration factor in rotor yoke (ky )	1.1	10.0
Constraint parameters	Min. value	Max. value
Electromagnetic torque error (ϵτ,elm)	0.0 Nm	0.25 Nm
Magnetic induction in stator segments (Bs,pk)	1.0 T	1.7 T
Magnetic induction in rotor yoke (By,pk)	1.0 T	1.7 T
Form factor of segments inner and outer radius (kfs)	0.80	0.85
Efficiency of motor nominal operation (ηnm)	88%	95%

In addition, the Artelys Knitro [⁴⁴], [⁴⁵] solver engine is employed, which uses a non-linear algorithm (interior point/barrier methods) to find a minimum value for the objective function. Fig. 4 shows a flowchart of the actions performed in the optimization process.

Fig. 4
Flowchart of tasks performed in the optimization process. The solver loop searches for the optimized solution for a given random initial condition, generated by the multistart loop.

The optimization procedure begins by loading of the design specifications as defined in Table I. Initial values for the variables of interest are then randomly defined within the ranges of Table II. The initial conditions, combined with the equations in Section III-A initiate the solver loop, to minimize the objective function F_obj. At each iteration, constraint satisfaction is checked (13), considering the torque error (ϵ_τ,elm), the magnetic induction in the segments (B_s,pk) and in the rotor yoke (B_y,pk), the teeth inner and outer radius form factor (k_fs) and the efficiency in motor nominal operation (η_nm).

(13) $${ϵ}_{\tau ,elm}={\tau }_{elm}-{\tau }_{m,mx}{B}_{s,pk}=B_{g,pk}\cdot k_s{B}_{y,pk}=B_{g,pk}\cdot k_y{k}_{fs}=\frac{r_{s,i}}{r_{s,o}}$$

Once the solution is determined, defined as the local minimum value of the function F _obj, the process restarts with a new initial condition. When the maximum number of iterations (j_mx) is reached, and solutions that meet the design constraints were found, the stored results are used to generate the graph in Fig. 5 and summarizes the efficiency and mass of different solutions.

Fig. 5
Summary of the results obtained in the optimization process, showing the relationship between mass and efficiency of the solutions found.

An increase in efficiency is seen to require greater mass, primarily due to the need for more copper to reduce conduction losses, which become more significant at higher efficiencies. This also increases the need for more space (and mass) of steel to install the conductive material. Among the various solutions obtained, one is selected as the best compromise between mass and efficiency. The corresponding data are presented in Table III.

TABLE III
Value of design variables obtained in the optimization process.

Parameter	Value
Conductors current density (Jc) and copper total area (Sc)	10.2 A/mm2, 2010 mm2
Inner radius (rs,i) and length of the active part (ls)	44 mm, 11 mm
Magnet thickness (hm)	4 mm
Mag. flux conc. factor in segments (ks) and rotor yoke (ky )	2.2, 2.2
Mag. induc. in stator segments (Bs,pk) and rotor yoke (By,pk)	1.7 T, 1.7 T
Form factor of segments inner and outer radius (kfs)	0.80
Efficiency of motor nominal operation (ηnm)	91%
Mass of active parts (Mtotal)	1045 g

IV. RESULTS AND DISCUSSION

The values presented in Table III are used for the geometric representation of the AFPM motor, based on the equations in Section III-A. Subsequently, the 3D finite element simulation is performed using Altair Simlab 2025 software. To reduce computational effort, only 1/8 of the motor geometry is simulated, with the software responsible for automatically applying the symmetry and periodicity coefficients necessary to obtain the results for the complete machine. The motor phases are connected in a star (wye) configuration, with four coils in series per phase. Each coil is considered to have 30 turns, providing a motor phase voltage (V_ph,rms) at nominal power lower than 20 V. This value is feasible to be obtained with a voltage inverter powered by 60 V lithium batteries.

A. Teeth Geometry

Typically, studies exploring the YASA concept for AFPM motors adopt trapezoidal-shaped teeth, as shown in the Fig. 6 (a). Which can be manufactured without significant difficulty when SMC material is used [²²]-[²⁵]. However, when laminated steel is used for stator cores, the manufacturing process becomes more complex as producing the trapezoidal shape requires cutting and stacking sheets of varying widths [²⁹], [³⁰].

Fig. 6
Teeth geometry: (a) trapezoidal-shaped cross section, (b) rectangular-shaped cross section.

To overcome this difficulty, a minimal form factor of 0.80 between the internal and external radii (k_fs) is considered in the optimization process, allowing the tooth geometry to be approximated as a rectangular shape, as shown in the Fig. 6 (b). The width is then defined by r_s,i and maintained along l_s, resulting in an equal radial cross section of the entire tooth.

Teeth with a trapezoidal cross section provide more efficient use of the active area, resulting in enhanced magnetic coupling between the rotor and stator. Therefore, this geometry is used in the calculation and optimization of motor parameters, as it represents the ideal configuration. However, due to manufacturing constraints, the motor was simulated (and will be later constructed) using teeth with a rectangular cross section.

B. Grain Oriented Steel Parameters

In the simulations, the GO steel grade considered in the rotor yoke and teeth is 27QG095 [⁴⁶]. This material is modeled by a nonlinear B-H curve (14), where µ₀ is the vacuum permeability, µ_r is the initial relative permeability of the material, J_s is the saturation magnetization and a is the adjustment coefficient of the B(H) curve knee. The coefficients used in the model are in (15), obtained from data available in [⁴⁶]. Fig. 7 illustrates the BH model and the experimental data of material 27QG095.

Fig. 7
B-H curve of material 27QG095, considering the data available in [⁴⁶] and the model (14).

Furthermore, magnetic losses are calculated using the Bertotti model, which requires data obtained from steel samples previously tested at the frequency of interest. Due to the inaccessibility of information on losses in 27QG095 steel for frequencies above 60 Hz, data from a similar grain-oriented steel (20QG085) for a frequency of 400 Hz are used, according to the graph available at [³¹].

(14) $$B(H)={\mu }_{o}H+J_s\frac{H_a+1-\sqrt{{(H_a+1)}^2-4H_a(1-a)}}{2(1-a)}{H}_a={\mu }_{0}H\frac{{\mu }_r-1}{J_s}$$

(15) $${\mu }_r=171283J_s=1971.1\mathrm{mT}a=1.74145$$

C. Results

The AFPM motor is simulated with the current sinusoidal waveform imposed in phase with the back electromotive force (BEMF). Simulations are performed at 1^◦ intervals of rotor angular displacement up to 45^◦, covering a complete period of electrical quantities. Fig. 8 shows the simulation results.

Fig. 8
Magnetic induction in the active parts: (a) nominal operating condition (3.0 Nm, 2560 rpm), (b) maximum operating condition (4.8 Nm, 3200 rpm). Waveforms of voltage (V_ph), current (I_ph) and electromagnetic torque (τ_elm): (c) nominal operating condition (3.0 Nm, 2560 rpm), (d) maximum operating condition (4.8 Nm, 3200 rpm).

Fig. 8 (a) and Fig. 8 (b) show the magnetic induction in the active parts of the machine, under the nominal and maximum power conditions respectively. Waveforms of voltage (V_ph), current (I_ph) and electromagnetic torque (τ_elm) are shown in Fig. 8 (c) for the nominal operating condition and in Fig. 8 (d) for the maximum operating condition. While some regions exhibit magnetic induction above the expected level (between the magnets or in the corners of the teeth), most of the motor volume operates below the specified magnetic induction level of 1.7 T.

The voltage waveform in Fig. 8 (c) and (d) corresponds to the phase-to-neutral voltage. By harmonic analysis, there are the third, fifth and seventh harmonics in this signal. Considering that the terminal voltage depends on the motor inductance (L_s) and assuming that the supply current is imposed and free of harmonics, any harmonics observed in the voltage must result from variations in the L_s. Motor inductance is a parameter that depends on the number of turns and the magnetic circuit’s reluctance. Since the number of turns is fixed, any angular variation in inductance is due to changes in the magnetic reluctance. Consequently, voltage harmonics originate from variations in the magnetic reluctance between the stator and the rotor. These variations occur because of the presence of air gaps or non-magnetic materials between the stator teeth and between the rotor magnets. As a result, the magnetic flux distribution across the air gap becomes non-sinusoidal, introducing harmonic components in the induced electromotive force (emf), and, therefore, in the terminal voltage.

The torque waveforms show a ripple of approximately 1.0 Nm, with an average value that meets the specifications. The number of periods per turn of the cogging torque is equal to the least common multiple (LCM) between the number of slots and the number of rotor poles. In this case, LCM(12,16) results in 48 periods per turn or 6 pulses per 45 degrees, as seen in Fig. 8 (c) and (d).

The losses and efficiency for different operating conditions are presented in Fig. 9. The relevance of conduction losses in relation to magnetic losses is evident, with efficiency decreasing as power and current increase. The nominal power efficiency (92.6%) exceeds the 91% originally predicted in the optimization process, indicating the potential for further mass reduction of the active parts through improved design methodology.

Fig. 9
Copper losses, iron losses and efficiency under different operating conditions.

Table IV presents the simulation results, where the current amplitude varies depending on the desired torque. The designed machine meets the specifications in Table I, providing the required torque values.

TABLE IV
Simulation results using the finite element method.

ωm	Iph,rms	Vph,rms	τelm	Pm	Sm	PF	PCu	PFe	η
[rpm]	[A]	[V]	[Nm]	[W]	[VA]	[1]	[W]	[W]	[%]
1000	3.8	6.9	0.71	74.3	78.7	0.945	2.8	2.4	93.5
1500	6.0	10.3	1.12	175.9	185.4	0.949	7.0	4.2	94.0
2000	10.5	13.9	1.95	408.3	437.9	0.933	21.3	6.2	93.7
2560	17.5	18.8	3.18	852.6	987.0	0.864	59.1	8.6	92.6
3000	25.0	24.5	4.40	1382.5	1837.5	0.752	120.6	10.7	91.3
3200	28.5	27.6	5.00	1675.5	2359.8	0.710	156.8	13.7	90.8
3400	33.0	31.9	5.70	2029.8	3158.1	0.643	210.0	17.0	89.9
3600	37.5	36.4	6.40	2412.8	4095.0	0.589	271.3	20.5	89.2

V. CONCLUSIONS

Through the optimization process and the benefits of GO steel with respect to the level of magnetic saturation and losses, the simulations demonstrated the potential to develop AFPM motors for drones, with high energy efficiency and torque density. Despite the challenges of working with steel sheets for the construction of the axial flux machine stator, a segmented armature with teeth of equal radial cross-section is proposed. This approach enables the tooth and its respective tip to be manufactured together through the electroerosion cutting of stacked steel sheets, simplifying the manufacturing process. A practical implementation of the motor is currently in progress to experimentally validate the simulated values and to assess whether the existing torque ripple affects the aircraft´s performance.

ACKNOWLEDGMENTS

This research was funded by the National Council for Scientific and Technological Development (CNPq) grant number 441907/2023-7 and by Fundação de Amparo à Pesquisa e Inovação do Estado de Santa Catarina (FAPESC) grant number 2023TR000925.

A. - Conduction losses calculation

Assuming a symmetrical and balanced electric motor, conduction losses are calculated according to (16), considering the number of phases (N_ph), conduction resistance (R_ph) and RMS current of each phase (I_ph,rms). If the coils of each phase are connected in series, I_ph,rms is rewritten as (17), where S_coil is the cross section area of the coil.

(16) $$P_{Cu}=N_{ph}\cdot R_{ph}\cdot I_{ph,rms}^2$$

(17) $$I_{ph,rms}=J_c\cdot S_{coil}$$

In turn, the phase resistance is defined by (18), where l_coil represents the average length of a turn and Q is the total number of segments in the machine.

(18) $$R_{ph}=\frac{Q}{N_{ph}}\cdot R_{\text{coil}}=\frac{Q}{N_{ph}}\cdot \frac{{\rho }_{Cu}\cdot l_{\text{coil}}}{S_{\text{coil}}}$$

Substituting (17) and (18) into (16), it is determined (19).

(19) $$P_{Cu}=N_{ph}\cdot \frac{Q}{N_{ph}}\cdot \frac{{\rho }_{Cu}\cdot l_{\text{coil}}}{S_{\text{coil}}}\cdot .J_c^2\cdot S_{\text{coil}}^2=Q\cdot S_{\text{coil}}\cdot l_{\text{coil}}\cdot J_c^2\cdot {\rho }_{Cu}$$

Since the total copper volume of the coils (V ol_coils) is defined by (20), the equation (19) is rewritten as (21), validating the calculation method for P_Cu presented in (11). Although a series connection of coils was considered, the same result is valid for the connection of the coils in parallel.

(20) $${\mathrm{Vol}}_{\text{coils}}=Q\cdot S_{\text{coil}}\cdot l_{\text{coil}}$$

(21) $$P_{Cu}={\mathrm{Vol}}_{\text{coils}}\cdot J_c^2\cdot {\rho }_{Cu}$$

DATA AVAILABILITY

Research data is available in the body of the document.

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• Editor:
  João C. W. A. Costa

• Associate Editor:
  Ursula do C. Resende

Publication Dates

• Publication in this collection
  17 Apr 2026
• Date of issue
  2025

History

• Received
  29 June 2025
• Reviewed
  07 July 2025
• Accepted
  23 Sept 2025