Optimal Design of an Axial-Flux Permanent-Magnet Middle Motor Integrated in a Cycloidal Reducer for a Pedal Electric Cycle

This paper proposes an optimal design of a middle motor integrated into a mid-drive unit for pedal electric cycles. This middle motor is an axial-flux permanent magnet brushless DC motor with a stator plate. Facing this plate is a rotor plate that acts as an eccentric component of a cycloidal reducer next to the stator. Both the middle motor and cycloidal reducer are easily installed on the same axle of the bike crankset. In the preliminary design, driving requirements are used to make the middle motor specifications. The numbers of stator slots and magnet poles on the rotor were chosen to achieve the best winding factor, and the winding layout was determined accordingly. A one-dimensional magnetic circuit model of the middle motor was built, and its shape was optimized utilizing a multifunctional optimization system tool. Finally, the resulting design was verified and refined by a finite element analysis. A prototype of the middle motor was fabricated with a stator made of a soft magnetic composite and integrated with a cycloidal reducer. Experimental results show that the motor has a sinusoidal back electromotive force. Additionally, it is easily controlled by sinusoidal currents with acceptable torque and speed curves.


Introduction
Electric bicycles (e-bikes) have gained increasing attention as a means of personal transportation and part of an urban lifestyle [1].People choose to purchase and use e-bikes for a variety of reasons, such as affordability, light weight, ease of riding, power assistance, savings on commuting and parking costs, the requirement for a small parking space, and a more flexible trip compared to a standard bike [2,3].The pedal electric cycle (pedelec) is a type of e-bike in which a small electric motor assists its rider with electric power while he or she is pedaling [4].This electric power is particularly useful in many circumstances, such as hilly areas or in the presence of a strong headwind.
Most e-bike motors are mounted on the front or the rear wheel hub [4,5], or in the center of the crank [6].Compared to a traditional hub motor, a middle motor in a mid-drive unit has a higher performance and torque.This is because the middle motor drives the crank instead of the wheel itself, allowing it to take better advantage of the bike's existing gears.The separation of the mid-drive set and wheel results in easier maintenance and service compared to a bike with a hub motor.This is especially the case when changing a flat tire.The location of the mid-drive unit is close to the center of gravity and situated low to the ground, thereby improving the overall handling of the e-bike.
Turner [7] patented a pedelec mid-drive set that was directly connected to the sun gear of the planetary gear reducer to deliver power to the rear wheel.However, this design was bulky, and the bike frame was drastically compromised.In addition, Li and Tu [8] designed a mid-drive power unit.However, its output shaft was not coaxial with the crankshaft.Therefore, an additional longitudinal space was necessary to install a reduction gear for transmitting power to the crankshaft.Li et al. [9] proposed a mid-drive unit that was coaxially connected to the crankshaft by a harmonic reducer.The advantages of a harmonic drive are its compactness, light weight, high gear ratio, and absence of backlash.However, it suffers low endurance to impact forces, a low rigidness of the flex spline, and a limited lifespan.The radial-flux permanent-magnet (RFPM) motors were used for the three applications of middle motors described above.The cylindrical shape of the RFPM motor results in the construction of a long and bulky mid-drive unit.
The present paper proposes an optimal design of an axial-flux permanent magnet (AFPM) middle motor in a mid-drive unit on a pedelec.Specifically, the middle motor rotor was designed on the eccentric disc of a cycloidal reducer, facing the stator plate.The axial length is saved due to the integrated configuration of the motor and reducer.The study is presented as follows: (1) Section 2 introduces the configuration and specifications of the pedelec middle motor; (2) Section 3 elaborates the sensitivity analysis and optimal design of the AFPM middle motor with zero and one-dimensional (0-D/1-D) magnetic circuit models; (3) the result of the design is verified and refined by the 3-D finite element (FE) analysis presented in Section 4; (4) Section 5 provides the experimental outcome of the AFPM middle motor prototype performance; and (5) finally, conclusions are made in Section 6.

Specifications of the Pedelec Middle Motor
The proposed pedelec middle motor was designed to have its rotor exist as an eccentric component of a one-stage cycloidal reducer.This reducer possesses a high reduction ratio of 35:1, as shown in Figure 1.The eccentric plate is comprised of an original portion of the cycloidal reducer driven by a traditional motor attached to the input shaft.The traditional motor typically consists of a radial flux permanent magnet motor with a cylindrical configuration.However, this may result in a bulky crankset with a large axial length.The proposed AFPM motor has a disc-type configuration.Its rotor is integrated on the eccentric disc of the cycloidal reducer facing the stator plate opposite to the air gap.This placement ensures that the axial length of the motor and reducer assembly is condensed.As shown in Figure 1c, the rotor of radius R o is centered at point E. Outside of the rotor, a bearing of radius R c is installed, with the center of the bearing C at a separate distance of e from the center E of the rotor.This eccentric bearing connects to a cycloidal wheel.On the radius R w of the cycloidal wheel centered at C, a group of pin holes with radius d h drive the output shaft through a group of pins with radius d p .When the rotor on the eccentrically placed disc rotates about its center E, the cycloidal disc is driven in the opposite direction and pushes against the ring gear pins or rollers at the cycloidal disc lobes.
The reduction ratio of the cycloidal reducer is determined by: where n p is the number of ring gear pins; and n L is the number of lobes on the cycloidal wheel.In this paper, n p is 36, n L is 35, and N r equals 35.Specifications of the pedelec middle motor are determined by torque and speed under the driving requirements, as well as the constraints of battery and drive, as described in Table 1.The air temperature of 24 ˝C is specified for a moderate driving environment, as well as for a nominal ambient temperature for battery range tests.The first step is to determine the pedelec traction power that is required to overcome the road load resistance.The rider's power P b is achieved by: where C m is the tire loss coefficient; v b is the bike speed; C w is the aerodynamic coefficient; A f is the frontal area of the pedelec and rider; ρ is the air density; and v w is the wind speed.The rolling friction plus the slope pulling force F s on an inclined plane is obtained by: where M is the mass of rider, m is the mass of bike, g is the gravitational constant, C r is the rolling friction coefficient, and θ is the slope angle.
To determine the maximum torque, we assigned a power-assist ratio (PAR) and transmission gear ratio.The PAR is defined as the ratio between the human pedaling power and the output power of the motor, and is set to one.The transmission gear ratio was determined by dividing the number of teeth on the front chain-wheel by the number of teeth on the rear sprocket.In this paper, the maximum transmission gear ratio is set to three.The highest motor speed required for the pedelec is approximately 3000 rpm when the bike reaches a top speed of 25 km/h and the transmission gear ratio is three.The maximum motor torque required for the pedelec is approximately 1.314 Nm at point A (Figure 2) for the bike to be travelling on a maximum slope of 25% when the transmission gear ratio is one.

Optimal Design of the AFPM Motor
Figure 3 shows the engineering drawing and design parameters of the AFPM motor.All of the design parameters are simplified and defined based on the cross-section of the average radius of the rotor (AA) and stator (BB).

Preliminary Design (0-D)
From a 0-D magnetic circuit model [10], the output torque of a PM motor per phase is expressed as: where N m is the number of poles; k w is the winding factor; and N spp is the number of slots per pole, per phase.The magnetomotive force (mmf) n s i is produced by a coil of n s turns per slot, each carrying a phase current of i.When the phase back electromotive force (back emf) is sinusoidal and the motor is driven by three-phase sinusoidal currents with a 120 ˝E difference, the total torque production becomes T total = (3/2)K p i.The air gap cross-sectional area A g between the rotor and the stators is: where R i and R o are the inner and outer radii, respectively.In addition, α m is the magnet with a fraction, defined as the ratio between the magnet width w m and the magnet pitch τ m .The flux density of air gap B g is presented as: where µ R is the recoil permeability of the magnet; g is the minimal air gap length; k c is Carter's coefficient; B r is the magnet remanence; and l m is the magnet length.The magnet leakage factor is determined by: where τ f = τ m ´wm .The flux concentration factor is defined as Ishak [11] found that if the number of poles N m and the number of slots N s satisfied N s = N m ˘2, the motor would provide a high torque with an extended winding factor and low torque ripple.According to the AFPM motor specification, the number of slots and poles was chosen to be 12/14 with a corresponding winding factor of 0.933.
The motor parameters obtained by the 0-D design equations were selected for the 1-D optimal design described below.To minimize the angular spread of the phase winding, Table 2 illustrates the double layer winding layout for all three phases.The phase offset is 120 ˝E, but each phase has three coils spread by different and unsymmetrical electric angles.This unsymmetrical winding structure will cause complex d-and q-axis flux linkages so that the corresponding d-and q-axis inductances vary according to the phase current and angle.

Sensitivity Analysis (1-D)
The performance of the AFPM motor is evaluated based on its output torque, torque ripple, and torque density.These parameters are also known as objective functions or performance indices that describe the mechanical and electrical dynamics in terms of motor geometries, magnetic materials, and driving conditions.The sensitivity analysis is required to determine the derivatives of the objective functions with respect to the parameters of interest.Lastly, a set of design variables was determined.
The current excited by the stator windings is combined with the magnetic flux produced by rotor magnets that generate mmfs.This results in a magnetic flux density distribution and magnetic energy that are created over the 3-dimensional (3-D) air gap region between the stator and rotor plates.The variation of magnetic energy with respect to the rotor shift produces a torque distribution.However, a one-dimensional (1-D) magnetic torque distribution at the average radius of R avg = (R i + R o )/2 can be used to approximate the 3-D torque distribution for the following sensitivity analysis.
Based on the assumptions of material linearity, as well as the linear function of flux density and field intensity (i.e., the linear B-H curve), the magnetic co-energy stored in the air gap is expressed as: where µ 0 is the permeability of the vacuum; s denotes the rotor shift; F(x, s) is the mmf distribution produced by the stator windings and rotor magnet; and x represents the peripheral coordinate along the circle of the average radius R avg .The effective air gap distribution g(x, s) is the sum of the effective air gap length on both the stator and rotor sides.
Energies 2015, 8, 14151-14167 The torque resulting from the variation of the co-energy with respect to the rotor shift is denoted by: The four motor variables consisting of: (1) the inner motor radius R i ; (2) the magnet thickness l m ; (3) the minimum air gap length g; and (4) the slot opening o were chosen for investigating sensitivity to the following performance indices: Torque : max.T avg (10) Torque ripple : min pT max ´Tmin q{T avg (11) Torque density : max T avg {W We can infer that the torque decreases as the inner motor radius, slot opening, and air gap length increase, as shown in Figure 4a,b.Additionally, the torque increases with a rise in magnet thickness.Figure 4c,d show that the inner radius and magnet thickness have little effect on torque ripple.However, the air gap length and slot opening are inversely proportional to the torque ripple.In Figure 4e, the inner motor radius has a minimal effect on torque density, and this density dramatically declines as the air gap length increases.Figure 4f shows that the magnet thickness and slot openings are proportional to the torque density.Therefore, the air gap length is inversely proportional to the torque, torque ripple, and torque density, and was not selected for the optimal design presented below.

Multi-Objective Optimal Design (1-D)
The compromise programming method [12] in the multi-functional optimization system tool (MOST) was applied to the search for the optimal values of the design variables for optimizing performance indices Equation (10) to Equation (12).The average torque T avg , maximum torque T max , and minimal torque T min were obtained from the 1-D magnetic circuit model.The weight of the motor W was determined from the design variables.Different weightings were assigned to the performance indices for which the relative importance was addressed for the optimization.Listed in Table 3 are the top three results based on motor performance.The torque in the case of the weighting ratio of 5:1:5 was determined to be the best of the three.The weighting ratio of 5:5:1 has the lowest torque ripple and the highest torque density of the three performance cases.Therefore, it was chosen for subsequent FE verification and refinement.

Estimation of Torque versus Speed (TN) Curve
The performance of PM motors has been improved by the wide use of the flux-weakening method to extend the motor speed range.This can be achieved by estimating the middle motor torque versus speed (TN) curve, based on the theory of maximum torque per ampere (MTPA) and the flux linkage method [13] by FE analysis.Two-axis variables, such as d-and q-axis voltages, currents, and inductances transformed from three-phase variables are used to describe the motor performance.Due to the coupling nature, the d-and q-axis flux linkages are functions of both the d-axis current i d and the q-axis current i q .The torque production is expressed as: where λ d and λ q are obtained from the FE analysis; I is the phase current; i d = ´Isinγ, i d = Isinγ; and γ is the lead angle of the phase current.The voltages applied in the d-axis V d , and q-axis V q are: where R ph is the phase resistance; λ d = L d i d + λ m , λ q = L q i q , L d and L q are the d-axis and q-axis inductances, respectively; and λ m is the magnet flux linkage.Figure 7a,b show that the d-and q-axis flux linkages vary with the phase currents and lead angles.When the leading angle is zero, the input current does not have a component in the d-axis, and λ d should equal λ m .The cross-coupling effect is not significant in the q-axis, and the q-axis flux linkage λ q is approximately linear with respect to the maximum phase current for various phase lead angles, as shown in Figure 7b.In addition, at the same phase current in which the lead angle of the phase current is reduced, the d-axis flux linkage λ d increases as i d is reduced in magnitude.In contrast, the q-axis flux linkage λ q increases as i q rises.Here, the average d-and q-axis inductances are determined at β = 0, which are L d = 0.276 mH and L q = 0.279 mH, respectively.
After the maximum torque was obtained by Equation ( 13), the corresponding optimal currents i do and i qo were used over the constant torque operating region from Equation (14) to Equation (16).The corner speed of the TN curve is obtained as:

Thermal Analysis
The steady-state temperature distribution was investigated for the mid-drive unit with a middle motor driving an 80-kg rider on a 25% slope at a constant speed of 10 km/h.The gear ratio between the front and rear sprockets was set to one.The corresponding crank speed was 80.22 rpm, and the motor speed was 2811 rpm.At the maximum torque production of 1.4 Nm by the phase current of 14 A at 36 V, the stator copper loss was 41.44 W and the iron loss was 7.62 W. Figure 9 shows the steady-state temperature distribution on the mid-drive unit after 4100 s.The average stator winding temperature was 64.5 ˝C (337.5 ˝K), the average magnet temperature was 45.9 ˝C (318.9 ˝K), and the average housing temperature was 48.3 ˝C (321.3 ˝K).Since the mid-drive unit is installed on the metal frame of the bike, more heat can be dissipated by the frame's conduction and convection.

Prototype Manufacturing
We fabricated a prototype of the AFPM motor.The stator, consisting of 12 fan-shaped teeth, was made of a soft magnetic composite (Somaloy 700 3P), as shown in Figure 10a.This was achieved under a stamping pressure of 800 MPa for the desired density of 7.5 g/cm 3 .A wire with a diameter of 0.8 mm was chosen for the windings concentrated on each tooth.Each tooth had a coil of 21 turns of conductor (0.9-mm diameter), and four coils were connected in a series and grouped into one phase on the stator plate.Figure 10b shows that the rotor, embedded with 14 magnets, is attached to the cycloidal disc by a ball bearing.The prototype motor was tested on a platform that was equipped with a motor drive, a dynamometer, an oscilloscope, a power meter, and a power supply, as shown in Figure 11a.The back emf was measured and compared with that from the FE analysis.Figure 11b shows that the peak of the back emf from the experiment is 3.22 V at 500 rpm, which is close to 3.2 V from the FE analysis.The fundamental harmonic of the back emf from the experiment is 79%, which is lower than the value of 94% obtained by the FE analysis shown in Figure 11c.This difference may be caused by manufacturing defects, meaning that the prototype represents higher amplitudes.Although they Energies 2015, 8, 14151-14167 are minor, they are on the high-order harmonics above the 5th order of those obtained by the FE analysis.However, the AFPM prototype can still be driven efficiently by sinusoidal currents for its major fundamental component.

Motor Performance
In the performance test, the TNE map of the prototype motor was obtained, as shown in Figure 12.Compared with the TNE map acquired by the FE analysis in Figure 8a, the maximum torque of the prototype motor is about 1 Nm, which is lower than 1.4 Nm obtained from the FE analysis.Moreover, this value was derived at a maximum current of 14 A. The corresponding torque constant was 0.07 Nm/A, which is 70% of the FE result.Accordingly, a maximum efficiency of 66% occurs near Energies 2015, 8, 14151-14167 optimal design with a 1-D magnetic circuit model; and (3) the 3-D FE analysis.The results of the FE simulation indicate that the driving efficiency of the pedelec propelled by the proposed middle motor with an ECE-40 driving cycle at the speed of 25 km/h is estimated to be as high as 71.3%.A prototype motor was fabricated, and the torque, speed, and efficiency performance were tested.Due to various manufacturing and sensory defects, the prototype motor represents a lower torque than that obtained by the FE analysis.However, it can still provide a maximum torque of 1 Nm and is expected to deliver 36 Nm at the output shaft of the cycloidal reducer.This would enable the motor to drive a pedelec with an 80-kg rider on a 15% slope.

Figure 1 .
Figure 1.(a) Configuration of the middle motor in the mid-drive unit; (b) its schematic assembly with a cycloidal reducer; (c) the cycloidal and eccentric discs; and (d) the assembly of the mid-drive unit, crank, and pedals.

Figure 2 .
Figure 2. The torque required at various bike speeds and road slopes.

Figure 4 .
Figure 4.The of torque versus: (a) the inner radius and air gap; and (b) magnet thickness and the slot opening.The sensitivity of torque ripple versus: (c) the inner radius and air gap; and (d) magnet thickness and the slot opening.The sensitivity of torque density versus: (e) the inner radius and air gap; and (f) magnet thickness and the slot opening.

Figure 6 .
Figure 6.Axial (a) over 360 degrees; and (b) its average versus the rotor speed.

Figure 7 .
Figure 7. (a) q-axis flux linkages; (b) d-axis flux linkages; (c) TN curves with and without the flux-weakening control; and (d) the corresponding current lead angle against the speed with a flux-weakening control.

Figure 8 .
Figure 8.(a) Driving points on the efficiency map estimated by the FE analysis; (b) The efficiency history along the ECE-40 driving cycle.

Figure 11 .
Figure 11.(a) Experimental setup; (b) comparison of the back emf (at 500 rpm) and (c) its harmonics from the FE analysis and experiment.

Table 1 .
Driving requirements and constraints.

Table 2 .
Winding layout and coil angles of the AFPM motor.

Table 3 .
Optimized motor parameters and performance.T avg : T avg {pT max ´Tmin q : T avg {W