Slot-less torus solid-rotor-ringed line-start axial-flux permanent-magnet motor.
Line-start permanent-magnet (LSPM) motors perform better than induction motors in applications that operate for a long period and require constant speeds (pumps, fans, compressors) . For starting and operation, they can be connected directly to the utility supply. Their main advantages over induction machines are their higher efficiency, higher power factor, higher torque density, better thermal behavior, and lower sensitivity to frequency variations . Their starting and synchronization remain important challenges needing improvement, each depending on the motor and the load parameters .
Permanent-magnet machines generally can be axial-flux or radial-flux . Advantages of axial-flux permanent-magnet (AFPM) motors over conventional radial-flux permanent-magnet (RFPM) motors include higher torque-to-weight ratio, good efficiency, adjustable air-gap, balanced rotor-stator attractive forces, and better heat-removal [5,6]. Their air-gaps are planar and easily adjustable . AFPM machine size and shape are a plus in limited-space applications, earning them their place in military and transport applications and motivating researchers to develop novel designs . The starting and synchronization of LSPM motors have been the subject of much research, besides the effects of selected parameters on their performance [9-21]. In all these works, however, the focus has been on new methods for line-start radial-flux permanent-magnet (LSRFPM) motors while practically there is no publication on line-start axial-flux permanent-magnet (LSAFPM) motors.
Use of solid rotors instead of laminated rotors has advantages. The weakness and insufficient rigidness of the usual laminated rotor render it unsuitable for use in high-speed machines. A solid-rotor motor electromagnetically is weaker than a laminated rotor, though it is superior physically [22, 23]. At higher speeds, the solid-rotor-core can be used with mechanical bearings, because it easily maintains its balance. Also, research has shown that use of solid rotor in a poly-phase induction machine increases torque per ampere at standstill . Moreover, high speeds and direct connection of load to the solid-rotor shaft do not affect the solid-rotor's ability to withstand the natural mechanical vibrations that threaten to damage the bearings.
AFPM machines can be single-sided or double-sided, with or without armature slots, with or without armature core, with internal or external permanent-magnet rotors, with surface-mounted or interior permanent-magnet, and single-stage or multi-stage machines . Topologies of double-sided AFPM machines can be one-stator-two-rotor (TORUS) or two-stator-one-rotor (AFIR) . This paper chooses the slot-less TORUS LSAFPM motor configuration for design, analysis, and prototyping. Leakage and mutual inductances in a slot-less motor are lower, and slot-due effects (flux ripple, cogging torque, and high-frequency rotor losses) are absent, making a low-noise machine . With slot-less, too, the end windings are shorter, so copper losses reduce. The slot-less configuration has lower cogging torque, which makes for better synchronization.
The design and modeling of a solid-rotor-ringed line-start axial-flux permanent-magnet motor that incorporates all the above mentioned benefits is hereby presented. It reports a successful design of an auto-start AFPM motor that at high speeds retains high efficiency and has sufficient starting ability when connected to the large loads. For auto-starting, a spaced and raised ring is added to the inner radii of the rotor disc that partially covers the inner yoke of the stator. The proposed design is novel, and the motor's structure unique. The interaction between the induced eddy-currents in the solid rotor rings and the rotating field in the air-gap between the rings and the inner end winding produces electromagnetic torque. In an AFPM motor, radial windings usually produce torque, whereas end windings usually are redundant, but in this work, inner end windings, too, produce torque.
The paper is organized as follows: Section 2 extracts the basic equations of the raised rotor-disc rings of the proposed axial-flux motor, Section 3 presents the structure, specifications and fabrication of the proposed LSAFPM motor, Section 4 includes the 3D finite element analysis of the proposed motor, Section 5 discusses the simulation and experimental results of the prototype motor, and Section 6 contains the conclusion.
2. FUNDAMENTAL DESIGN OF RAISED ROTOR-DISC RING
Presented are the details of the raised rings design at the inner radii of the rotor disc for the line-star axial-flux induction motor. The design aims to attain an appropriate solid-rotor-ringed axial-flux motor construction enabling the auto-start capability of the motor. The dimensioning is from the electromagnetic point of view only and hence the mechanical constraints are not considered. It is to be mention that the design procedure and parameter selection based on sizing equation and finite element method (FEM) for AFPM motor is not the subject of this work as it was presented in .
[FIGURE 1 OMITTED]
Figure 1 is a two-view cross-section of the proposed motor. Figures 1(a) and 1(b) are respectively the radial and the axial views. The winding configuration presented in Figure 1(a) is used for the designed motor. The analytical design of the rotor ring considered only the radial component of the flux density because it is what influences the motor's starting capability the most. Tangential tension [[sigma].sub.tan] estimates the rotor-ring size for the desired torque production. Alternative to air-gap tangential tension, the magnitude of the internal power from the rotor volume of the induction machine defines the machine constant C :
C = [[pi].sup.2]/[square root of 2] [k.sub.w1] A[B.sub.gr_max] (1)
where A is the linear current density, [B.sub.gr_max] the radial air-gap maximum flux density, and [k.sub.w1] the winding factor of the fundamental harmonic. The machine constant is written as a function of tangential tension :
C = [square root of 2][[sigma].sub.tan][[pi].sup.2][k.sub.w1] (2)
From Equations (1) and (2), the tangential tension is:
[[sigma].sub.tan] = A[B.sub.gr_max]/2 (3)
The tangential tension in the air-gap can be calculated from:
[[sigma].sub.tan] = [tau]/[R.sub.ave][S.sub.r] (4)
where [R.sub.ave] is the mean value of the rotor ring's inner and outer radii, [S.sub.r] the rotor ring's surface facing the air-gap between the rotor ring and the stator surface, and [tau] the torque. The tangential force [F.sub.tan] is:
[F.sub.tan] = [[sigma].sub.tan] [S.sub.r] (5)
The radial air-gap length between the rotor ring and the stator yoke surface should be selected with great care. Increased air-gap length increases the stator's copper losses and decreases the rotor's eddy current losses. Loss is minimum between the rising copper losses and the rotor's diminishing harmonic eddy current. The highest possible air-gap flux density ensures the attainment of a high-performing maximum-torque solid-rotor-ringed LSAFPM motor.
[FIGURE 2 OMITTED]
For acceptable calculation results of the analytical design, the rotor ring and the stator were divided into a number of parallel active circular sub-volumes as shown in Figure 2. By this, the curvature of the machine is assumed to have no further effect on the performance of an individual sub-machine. Densities of the rotor and the stator fluxes were studied in each area from 1 to n. Upon selection of the rotor ring slot width [b.sub.r] and rotor ring slot pitch [T.sub.r](1 ... n), the motor's inner parts must not be allowed to saturate. The flux density of the rotor-ring teeth is calculated from :
[B.sub.dr(1 ... n)] = [B.sub.gr_max][[tau].sub.r(1 ... n)]/[[tau].sub.r(1 ... n)] - [b.sub.r] (6)
The allowable rotor ring tooth flux density [B.sub.dr] of radial-flux induction machines is between 1.5 T and 2.2 T  (this range is also used here). Assuming the total flux of the slot pitch to flow along the teeth, no flux in either the slots or their insulation, and non-saturated motor teeth, allows calculation of the rotor-ring teeth length as:
[w.sub.dr] = [[tau].sub.r(n)][B.sub.gr_max]/[B.sub.dr(n)] (7)
Thickness of the rotor ring is:
[w.sub.th-r] = [[phi].sub.rr]/2[B.sub.rr][k.sub.rFe][l.sub.e] (8)
where [B.sub.rr] is the peak flux density of the rotor ring, [k.sub.rFe] the space coefficient of the iron (which depends on the relative thickness of the electric sheet insulation to the press fit of the stack), [l.sub.e] the equivalent rotor ring core length, and [[phi].sub.rr] the peak value of the main radial flux penetrating the rotor ring and the stator winding. The rotor yoke flux density allowable for radial-flux induction motor is 0.4 T to 1.6 T  (this range is also used here). The mechanical strength of a solid rotor ring is reliable, unlike that of a laminated rotor whose minimum thickness may only be determined by mechanical constraints.
The length of the coil [l.sub.m] depends on the number of pole-pairs p, the stator core length [l.sub.ea], the pole pitch [[tau].sub.p], the chord factor [CHI] (the ratio of the coil span W to the pole pitch [[tau].sub.p]), and the stator core thickness [l.sub.er]:
[l.sub.m] = p(2[l.sub.ea] + [chi][[tau].sub.p_in] + [chi][[tau].sub.p-out] + 2[l.sub.er]) (9)
[[tau].sub.p_in] and [[tau].sub.p_in] are the pole pitches at inner and outer radius respectively. In rotor ring design of the LSAFPM motor, the radial windings which produce torque in the synchronous state are considered as the end windings of the axial windings during the starting period. The currents flowing into the radial windings produce end-winding leakage flux. Due to the difficulty of analysing the geometry of the windings in axial-flux machines and because the leakage flux is influenced by all the phases of poly-phase machines, the inductance factors [[lambda].sub.E] and [[lambda].sub.W] are determined empirically. Richter provided calculated values of the inductance factors valid for induction machines . The end-winding leakage inductances [L.sub.WS] in the inner ([L.sub.WS_in]) and the outer ([L.sub.WS_out]) circumferences of the proposed axial-flux induction machine are:
[L.sub.WS_in] = 4m/Q q x [N.sup.2.sub.ph] x [pi] x f x [[mu].sub.0](2 x [l.sub.er] x [[lambda].sub.E] + [chi] x [[tau].sub.p_in] x [[lambda].sub.W]) (10)
[L.sub.WS_out] = 4m/Q q x [N.sup.2.sub.ph] x [pi] x f x [[mu].sub.0](2 x [l.sub.er] x [[lambda].sub.E] + [chi] x [[tau].sub.p_out] x [[lambda].sub.W]) (11)
where m is the number of phases, [N.sub.ph] is the number of winding turns per phase, f the frequency, Q the number of slots, q the number of slots per pole and per phase, [[tau].sub.p_in] the pole pitch at inner radius, [[tau].sub.p_out] the pole pitch at outer radius, and [[mu].sub.0] the permeability of free space.
3. MACHINE STRUCTURE
The designed motor is an inside-out double-rotor single-stator LSAFPM motor (see Figure 3 for an exploded view). Table 1 lists the motor's dimensions and specifications.
To build the stator, silicon-iron paper with a thickness of 0.1 mm was used for lamination. The axial flux motor's stator is either laminated spirally or axially. As shown in Figure 4(a), in this work spiral configuration was implemented. Also, a simple 3-phase winding with star connection was adopted as stator winding (Figure 4(b)). The details of winding configuration were presented in Figure 1(a). To fix the windings into place, prevent it from vibration during motor operation and to increase its insulation capability, a type of resin was applied, giving the windings characteristics such as stiffness in working temperature and good thermal conductivity for heat-release.
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
Figure 5 shows the motor's rotor and its permanent magnets geometry. Figure 5(a) is the single disc of the rotor which was constructed from mild-steel with its inner ring enabling starting capability of the motor. In each rotor disc, 4 axially-magnetized Nd-Fe-B permanent magnets shown in Figure 5(b) are mounted on the disc surface facing the stator. The permanent magnets used in the machine prototype have 1.3 T remanence and 900 kA/m demagnetizing field.
4. FINITE ELEMENT ANALYSIS
The governing equations for the FEA are :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
where [??], [??], [??], [sigma], and [mu] are respectively the magnetic flux density, the current density, the magnetic vector potential, the electrical conductivity, and the magnetic permeability.
Considering [??] as the magnetic field intensity, the above mentioned Equations (12)-(14) lead to the following equation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
The result is a formula whose vector fields are represented by first-order edge elements and scalar fields by second-order nodal unknowns. The circuit equation during motoring is given by:
u = [R.sub.ph]i + [L.sub.e] di/dt + e (16)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
where u is the applied source voltage, [R.sub.ph] the winding resistance per phase, Le the end winding inductance per phase, e the electromotive force (EMF) per phase, i the phase current, l the radial length of iron core, S the conductor area of each turn of phase winding, and [[Ohm].sub.e] the total cross-sectional area of conductors of each phase winding. The motion of the machine is given by:
[T.sub.em] - [T.sub.L] - [C.sub.v] [[omega].sub.r] = [j.sub.m] d[[omega].sub.r]/dt (18)
where, [j.sub.m] is the rotor momentum of inertia, [[omega].sub.r] the rotor speed, [T.sub.L] the load torque, [T.sub.em] the electromagnetic torque, and [C.sub.v] the damping coefficient.
Therefore, after discretization, the above mentioned three types of equations can be solved at each time step. Consequently both steady-state and transient performances of the proposed machine can be calculated. The proposed LSAFPM motor has a unique construction; its lack of symmetry makes 3D-FEA a design requisite. Note that 3D-FEM facilitates field analysis of electromagnetic problems with complex geometries. The model's complex magnetic circuit was analysed for an overall view of the saturation levels in various parts of the motor, and to extract the motor's characteristics. An advantage of 3D-FEA is that various components of flux density can be calculated highly accurately [32-40]. The design was simulated on commercial ANSYS 13.0 software . Corresponding materials and circuit currents were assigned to each model segment. Figure 6 shows the FEA structure of designed LSAFPM motor, which is a 3D auto-mesh: tetrahedral elements with 6 nodes fitting the circular shape of the layers starting from the inner to the outer diameter of the motor. Figure 6(a) includes the entire structure of the motor while Figure 6(b) only includes the rotors and the permanent magnets.
Evaluation of the magnetic flux density is important because the flux density saturation of the stator or the rotor affects the motor performance and reduces the efficiency. Figure 7 depicts the post-synchronization magnetic flux density distribution in various sectors of the designed LSAFPM motor. Figure 7(a) is the flux density plot in the stator showing higher flux density at the inner yoke of the stator due to condenser winding in those areas. Figure 7(b) is the flux density plot in the rotor disc and the inner rotor ring showing higher flux density around the permanent magnets and the lowest flux density on the rotor ring.
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
Figure 8 shows the magnetic flux density and back EMF of the motor. Figure 8(a) is the air-gap flux density distribution, in average radius. The maximum flux density is obviously 0.6 T, averaging 0.3 T. Due to the slot-less type AFPM motor, air-gap magnetic field distribution is smooth, and consequently the variation of back-EMF is also smooth and almost sinusoidal. Figure 8(b) shows the no-load back EMF at rated speed (1500 rpm), beside the FEA-calculated THD and back-EMF RMS. It should be noted that the optimization of the designed motor is not the subject of this paper.
5. RESULTS AND DISCUSSION
Figure 9 gives a schematic view of the d-q equivalent circuits of the proposed LSAFPM motor. The LSAFPM motor starts as an induction motor by the resultant of two torque components, i.e., cage torque and magnet opponent torque (braking torque). When the motor approaches synchronous speed, synchronization begins and its operation becomes synchronous. No eddy current except harmonic field current then flows into the solid rotor-ring.
In synchronous state, two torque components (reluctance torque and synchronous torque) mobilize the rotor.  describes the dynamic performance of line-start permanent-magnet motors in a stationary d-q reference frame:
[V.sub.sq] = [r.sub.s][i.sub.sq] + [[omega].sub.r] [[lambda].sub.sd] + d[[lambda].sub.sq]/dt (19)
[V.sub.sd] = [r.sub.s][i.sub.sd] - [[omega].sub.r] [[lambda].sub.sq] + d[[lambda].sub.sd]/dt (20)
[V'.sub.rq] = [r'.sub.rq][i'.sub.rq] + d[[lambda]'.sub.sd]/dt = 0 (21)
[V'.sub.rd] = [r'.sub.rd][i'.sub.rd] + d[[lambda]'.sub.rd]/dt = 0 (22)
where [V.sub.sq], [V.sub.sd], [V'.sub.rq], and [V'.sub.rd] are the stator voltages and the rotor voltages referred to stator, [[lambda].sub.sq], [[lambda].sub.sd], [[lambda]'.sub.rq], and [[lambda]'.sub.rd] the stator linkage fluxes and the rotor linkage fluxes, and [i.sub.sq], [i.sub.sd], [i'.sub.sq], and [i'.sub.sd] the stator currents and the rotor currents, [[omega].sub.r], [r.sub.s], [r'.sub.rd], and [r'.sub.rq] respectively the rotor speed, stator resistance, and rotor resistances (all relative to the stator). The stator linkage flux and the rotor linkage flux are:
[[lambda].sub.sq] = [L.sub.sq][i.sub.sq] + [L.sub.mq][i'.sub.rq] (23)
[[lambda].sub.sd] = [L.sub.sd][i.sub.sd] + [L.sub.md][i'.sub.rd] + [[lambda]'.sub.m] (24)
[[lambda]'.sub.rq] = [L'.sub.rq][i'.sub.rq] + [L.sub.mq][i.sub.sq] (25)
[[lambda]'.sub.rd] = [L'.sub.rd][i'.sub.rd] + [L.sub.md][i.sub.sd] + [[lambda]'.sub.m] (26)
where [L.sub.sq], [L.sub.sd], [L'.sub.rq], are the stator self-inductances and the rotor self-inductances, [L.sub.mq] and [L.sub.md] the mutual inductances, and [[lambda]'.sub.m] the permanent-magnet flux. The electromagnetic torque is then:
[T.sub.em] = 3p/2 ([L.sub.sd] - [L.sub.sq])[i.sub.sd] [i.sub.sq] + 3p/2 ([L.sub.md] [i'.sub.rd] [i.sub.sq] - [L.sub.mq] [i'.sub.rq] [i.sub.sd]) + 3p/2 [[lambda]'.sub.m] [i.sub.sq] (27)
with the first term being the reluctance torque, the second term the cage torque, and the third term the magnet's synchronous torque.
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
In order to test the performance of the prototype motor, an experimental setup was built. The hardware schematic for the experimental setup is shown in Figure 10. Since the speed in not very high, in-line torque transducer with suspended installation and single-element coupling to create shorter drive train were used. Hysteresis brake on the motor shaft provides the desired torque load. Back-EMF and speed/time response were the main performance parameters to obtain. It should be noted that during cruising-speed test, secondary measurement such as temperature rise in the motor's critical sections were also monitored.
To analyze the proposed model in transient state during auto-starting, the motor was run as an induction motor (i.e., without any permanent magnets) with the aid of FEA. The stator was excited via a three-phase voltage supply; the motor then began rotating as an induction motor and at sub-synchronous speeds. Field equations were coupled with circuit equations for conductors, because in transient simulations, supply voltages are applied and currents are unknown. Classical Newton-Raphson algorithm was used for the nonlinearities. A major difficulty for transient simulation is that induced currents on the rotor rings must be calculated in each step before calculating the torque.
Line-start synchronous motor is a low-cost solution that works reasonably well if the motor load and power supply voltage remains mostly constant. The prototyped motor was also tested in transient state during auto-starting without permanent magnet, just like an induction motor. The performance comparison of the experimental results versus the FEA simulations agrees well. Figure 11 gives the proposed motor induced-torque curves both experimentally and via simulation. It should be noted that the torque-speed characteristic of this type of motor does not follow the typical torque-speed characteristic of radial induction motors. It is clearly seen that the torque reduces smoothly from its maximum value at the speed of zero and reaches to zero at the synchronous speed.
[FIGURE 11 OMITTED]
[FIGURE 12 OMITTED]
The starting and the synchronizing of LSAFPM motors are challenging feats, primarily due to the brake torque of the magnet poles. Figure 12 shows the permanent-magnet braking torque calculated via FEA and experimentally for the proposed LSAFPM motor. This torque is developed between starting and synchronous speed. It reaches the maximum value at low speed. It is then reduced and finally reaches to a constant value at synchronous speed. For the initial period, the permanent magnet brake toque is negative, which causes vibration and noise during starting period of the LSAFPM motor. This phenomenon of oscillatory torques during the run-up period needs serious attention from the designers and users of the LSAFPM motors. The magnet braking torque Tb is given by :
[T.sub.b] = 3[pr.sub.s](1 - s)[E.sup.2.sub.0]/2[[omega].sub.s] x [r.sup.2.sub.s] + [X.sup.2.sub.sq] [(1 - s).sup.2]/[r.sup.2.sub.s] + [X.sub.sd][X.sub.sq] [(1 - s).sup.2]) (28)
with [E.sub.0], [X.sub.sd], [X.sub.sq], [[omega].sub.s], and s respectively being the magnet back-EMF, the d-axis stator reactance, the g-axis stator reactance, the synchronous speed, and the slip. This torque resists the cage torque that may cause failures in starting or synchronizing.
It is well known that a key factor in the working of line-start permanent-magnet motors is the starting performance. Interaction between the eddy-currents in the solid-steel rotor ring and the rotating field in the air-gap between the ring and the end winding produces electromagnetic torque. Simulation based speed-time responses of the designed LSAFPM motor under various load inertia are shown in Figure 13. In line-start motors, the load inertia clearly matters to synchronization: the higher the inertia, the more difficult it is to synchronize. The motors were started at no-load condition. The rotor momentum inertia of [j.sub.m] = 0.2 Nm was used during the simulations for comparison with the experimental results. The motor easily synchronizes, especially considering the capabilities of the starting torque.
[FIGURE 13 OMITTED]
The synchronization time increases to 1.1 sec for the load inertia of 1 Nm from 0.9 sec for the no-load condition. A noteworthy result is the maximum load torque that the motor can handle, which is clearly limited to 1.5 Nm. At 1.55 Nm load torque, the motor failed to synchronize and started to oscillate between 1140 rpm and 1480 rpm. Figure 14 shows the corresponding electromagnetic torque and input current versus time curves, for the load torques mentioned. It is evident that electromagnetic torque oscillates during run-up time and finally settles down at the value of the load torque in synchronous speed. The maximum current value during initial period reaches 5 to 6 times of its maximum value during synchronization. Figures 14(g) and 14(h) show that for the TL = 1.55 Nm load torque, both electromagnetic torque and input current never stop oscillation.
The back-EMF of the fabricated motor in various speeds was measured in open-circuit test. Figure 15 shows the voltage and its harmonic spectrum obtained experimentally in open-circuit test at rated speed. The finite element analysis simulation and experimental results comparison are tabulated in Table 2. Experimental results almost agree with those of simulation.
Figure 16 shows the measured starting and synchronizing performance of the prototype machine together with the simulated results from FEA. It is clear that the prototype line-start AFPM motor started the 1.3 Nm load, accelerated to the synchronous speed and finally locked-in and synchronized to supply line frequency (1500 rpm). The oscillatory nature of the speed response is due to the interaction between the rotor magnets and the stator supply current. The figure also shows a validation of the accuracy of the simulation results for the prototype LSAFPM motor.
[FIGURE 14 OMITTED]
[FIGURE 15 OMITTED]
[FIGURE 16 OMITTED]
This paper reported the design process, finite element analysis, and prototyping of a solid-rotor-ringed LSAFPM motor as a novel structure for the AFPM motor. A unique rotor comprised of a spaced raised solid ring that enables smooth line-start of the proposed motor was presented and its relevant equations were extracted. The steady-state and transient performance of the proposed design motor was examined via 3D finite element analysis. The motor was fabricated and tested. The experimental results of dynamic starting and synchronization performance from prototype motor agreed with those of simulation results. The paper verifies successful design of a 1/3 hP 4-pole auto-start slot-less inside-out axial-flux permanent-magnet motor, its good synchronization at rated speed, and its sufficient starting torque when connected to a maximum load torque of 1.5 Nm. Among the advantages of the solid-rotor motor are its physical ability to withstand natural vibrations that make it suitable for high-speed applications and increased torque per ampere at standstill.
A: linear current density
[??]: magnetic flux density
[B.sub.dr]: rotor ring tooth flux density
[B.sub.ga]: axial air-gap flux density
[B.sub.gr_max]: radial air-gap maximum flux density
[b.sub.r]: rotor ring slot width
[B.sub.rr]: peak flux density of rotor ring
C: machine constant
[C.sub.v]: damping coefficient
[D.sub.so]: stator outer diameter
[D.sub.si]: stator inner diameter
[D.sub.ro]: rotor-ring outer diameter
[D.sub.ri]: rotor-ring inner diameter
[E.sub.0]: magnet back-EMF
e: electromotive force (EMF) per phase
[F.sub.tan]: tangential force
[g.sub.a]: axial air-gap length
[g.sub.r]: radial air-gap length
[??]: magnetic field intensity
[i.sub.sd]: d-axis component of stator current
[i.sub.sq]: g-axis component of stator current
[i'.sub.rd]: d-axis component of rotor current referred to stator
[i'.sub.rq]: g-axis component of rotor current referred to stator
[??]: current density
[j.sub.m]: rotor moment of inertia
[j.sub.L]: load inertia
i: phase current
[k.sub.W1]: winding factor of fundamental harmonic
[k.sub.rFe]: space factor of the iron
[L.sub.cr]: rotor-yoke thickness
[L.sub.cs]: stator-yoke thickness
[L.sub.e]: end winding inductance per phase
l: radial length of iron core
[l.sub.e]: equivalent rotor ring core length
[l.sub.ea]: equivalent stator core length
[l.sub.er]: equivalent stator core thickness
[l.sub.m]: length of one turn of the winding
[L.sub.sd]: d-axis component of stator self-inductance
[L.sub.sq]: q-axis component of stator self-inductance
[L'.sub.rd]: d-axis component of rotor self-inductance
[L'.sub.rq]: q-axis component of rotor self-inductance
[L.sub.md]: d-axis component of stator mutual-inductance
[L.sub.mq]: q-axis component of stator mutual-inductance
[L.sub.pm]: magnet's axial length
[L.sub.r]: rotor ring axial length
[L.sub.WS]: end winding leakage inductances
[L.sub.SW_in]: end winding leakage inductances at inner radius
[L.sub.SW_out]: end winding leakage inductances at outer radius
[l.sub.w]: winding thickness
m: number of phases
n: number of concentric circular divisions for rotor ring and stator
[N.sub.ph]: number of winding turns per phase
p: number of pole pairs
[P.sub.r]: rated power
Q: number of slots
g: number of slots per pole per phase
[R.sub.ave]: average inner and outer radii of rotor ring
[R.sub.ph]: winding resistance per phase
[r.sub.s]: stator resistance
[r'.sub.rd]: d-axis component of rotor resistances referred to stator
[r'.sub.rq]: g-axis component of rotor resistances referred to stator
S: conductor area of each turn of phase-winding
[S.sub.r]: rotor ring's surface facing the air-gap between rotor ring and stator surface
[T.sub.b]: magnet breaking torque
[T.sub.em]: electromagnetic torque
[T.sub.L]: load torque
[T.sub.r]: rated torque
u: applied source voltage
[V.sub.sd]: d-axis component of stator voltage
[V.sub.sq]: q-axis component of stator voltage
[V'.sub.rd]: d-axis component of rotor voltage referred to stator
[V'.sub.rq]: q-axis component of rotor voltage referred to stator
[V.sub.r]: rated voltage
W: coil span
[w.sub.dr]: length of rotor-ring teeth
[w.sub.th_r]: thickness of the rotor ring
[X.sub.sd]: d-axis stator reactance
[X.sub.sq]: g-axis stator reactance
[[gamma].sub.p]: pole pitch
[lambda]: ratio of inner diameter to outer diameter of stator
[[lambda].sub.sd]: d-axis component of stator flux linkage
[[lambda].sub.sq]: q-axis component of stator flux linkage
[[lambda]'.sub.rd]: d-axis component of rotor flux linkage
[[lambda]'.sub.rq]: q-axis component of rotor flux linkage permanent magnet flux
[[lambda].sub.E]: empirical inductance factor
[[lambda].sub.W]: empirical inductance factor
[mu]: magnetic permeability
[[mu].sub.0]: permeability of free space
[sigma]: electrical conductivity
[[sigma].sub.tan]: tangential tension
[[tau].sub.p]: pole pitch
[[tau].sub.p_in]: pole pitch at inner radius
[[tau].sub.p_out]: pole pitch at outer radius
[[tau].sub.r(1 ... n)]: rotor ring slot pitch
[??]: magnetic vector potential
[[phi].sub.rr]: peak value for the main radial flux penetrating the rotor ring and the stator winding
[chi]: chord factor
[[Ohm].sub.e]: total cross-sectional area of conductors per phase
[[omega].sub.r]: rotor speed
[[omega].sub.s]: synchronous speed
The authors acknowledge the University of Malaya's provision of the High Impact Research Grant No. D000022-16001 funding the Hybrid Solar Energy Research Suitable for Rural Electrification.
Received 3 July 2012, Accepted 29 August 2012, Scheduled 16 September 2012
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A. Mahmoudi (1), *, S. Kahourzade (1), N. A. Rahim (1), W. P. Hew (1), N. F. Ershad (2)
(1) UM Power Energy Dedicated Advanced Centre (UMPEDAC), Wisma R&D UM, University of Malaya, Kuala Lumpur, Malaysia
(2) Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran
* Corresponding author: Amin Mahmoudi (email@example.com).
Table 1. The motor's dimensions and specifications. rated voltage (line-line RMS) [V.sub.r] 18 V rated power [P.sub.r] 250 W rated torque [[tau].sub.r] 15 Nm number of poles P 4 number of phases m 3 drive frequency f 50 Hz stator outer diameter [D.sub.so] 21 mm stator inner diameter [D.sub.si] 120 mm ratio of inner diameter [lambda] 0.57 to outer diameter of stator rotor ring outer diameter [D.sub.ro] 112 mm rotor ring inner diameter [D.sub.ri] 102 mm rotor ring axial length [L.sub.r] 10 mm magnet's axial length [L.sub.pm] 4 mm pole pitch [[gamma].sub.p] 100 [degrees] stator-yoke thickness + [L.sub.cs] + 36 + 4 mm winding thickness [l.sub.w] rotor-yoke thickness [L.sub.cr] 6 mm number of winding turns per phase [N.sub.ph] 4 x 90 axial air-gap flux density [B.sub.ga] 0.3 T axial air-gap length [g.sub.a] 2 mm radial air-gap length [g.sub.r] 2 mm Table 2. Back-EMF comparison between experimental and FEA results. Back-EMF [V.sub.max] [V.sub.RMS] THD % 1500 rpm experimental 153.5 108.5 5.7 FEA 153 108 5.7 750 rpm experimental 73.2 51.7 5.9 FEA 52.5 51.1 5.8
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|Author:||Mahmoudi, A.; Kahourzade, S.; Rahim, N.A.; Hew, W.P.; Ershad, N.F.|
|Publication:||Progress In Electromagnetics Research|
|Date:||Sep 1, 2012|
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