Power density improvement of three phase flux reversal machine with distributed winding

Published in IET Electric Power Applications Received on 4th January 2009 Revised on 2nd April 2009 ISSN 1751-8660 Power density improvement of three phase flux reversal machine with distributed winding D.S. More B.G. Fernandes Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India E-mail: dsmore@ee.iitb.ac.in Abstract: The three phase flux reversal machine (FRM) is a doubly salient permanent magnet machine with concentrated windings. This study proposes the distributed winding for this machine. This winding (i.e. full pitch winding) offers high-power density and improves the efficiency. The permanent magnet (PM) flux linking the stator winding has effectively two or four pole flux pattern, which depends on the number of stator poles and independent of number of rotor poles. Finite element method (FEM) analysis is performed on the concentrated stator pole winding FRM (CSPFRM) and proposed full pitch winding FRM (FPFRM) to obtain induced EMF, flux linkages and inductance of the winding. The inductance of both machines is also obtained using winding function approach and compared with FEM results. The effect of armature reaction is compensated by capacitive series compensation to improve the voltage regulation. FEM analysis is also carried out on both the compensated generators to evaluate the power density. Speed of the flux pattern and that of the rotor is different in FRM. The ratio of these two speeds is termed as fictitious electrical gear. FRM and permanent magnet synchronous machine (PMSM) have sinusoidal terminal voltage and surface mounted PMs. The power density of both machines is compared using the concept of fictitious electrical gear. To verify the above analysis, a 6/14 pole FRM with distributed and concentrated winding is designed and fabricated. The experimental results are in close agreement with simulated results. 1 Introduction Flux reversal machine (FRM) has the advantages of both switched reluctance machine and permanent magnet (PM) machine. The basic principle of operation, design, analysis and construction of single-phase FRM is introduced in [1]. This paper provides the qualitative comparison between the FRM and the other types of brushless machines in the same class. Single-phase FRM has potential applications in automobile, aerospace and defence sectors. The basic threephase FRM [2] has an eight salient pole rotor and six salient pole stator with concentrated windings. PMs are fixed to the stator pole. The design of the machine is optimised to ensure (i) high-pm flux linkage in the winding and (ii) low cogging torque and PM weight. The cogging torque of the machine is reduced by providing the skew to the rotor poles. Fig. 1a shows this machine configuration. Three-phase FRM for high performance, low-speed servo drive application is presented in [3]. The design procedure of the machine and design details of 28 rotor poles and 12 stator poles with two PM pairs on each stator pole are provided. This machine is designed for 128 rpm at 60 Hz. Hightorque density with less than 3% torque pulsation using vector control is achieved. Rotor teeth of unequal width are designed to reduce the cogging torque [4]. The performance of FRM is improved by optimising the stator/ rotor geometrical design parameters. Attempts are also made to reduce the leakage flux by providing flux barrier on the rotor poles [5]. Power density comparison of doubly salient PM electrical machines is made in [6] and it is shown that FRM has higher power density in comparison with other machines of the same class. Roof top wind energy generation has acquired importance because of environmental concerns and the need for increased use of renewable energy. These turbines have a potential to provide electricity to domestic and commercial applications and the generators should be directly driven thus eliminating the gear box. Depending upon the output IET Electr. Power Appl., 2010, Vol. 4, Iss. 2, pp. 109 120 109 & The Institution of Engineering and Technology 2010

Figure 1 Cross-section of CSPFRM a 6/8 pole b 6/14 pole power, the variation in the rated speed of the rooftop wind turbine is 140 500 rpm. The corresponding generator capacity is in the range of 5 1 kw. Existing machines such as permanent magnet synchronous machine (PMSM) and induction machine for above power and speed range require large number of poles and stator slots. This results in machine with large air gap diameter. However, FRM can have large number of poles with less number of stator slots. Thus, FRM is suited for direct driven rooftop wind power generation. In this paper, full pitch distributed winding arrangement to increase the power density and to improve the efficiency of FRM is proposed. The concept of fictitious electrical gear based on flux pattern speed and rotor speed is introduced. Power density comparison of FRM and PMSM is carried out based on this fictitious electrical gear. In addition, two-dimensional FEM analysis [7, 8] is carried out on concentrated stator pole winding FRM (CSPFRM) and FPFRM. To validate simulation and analysis results, a prototype of 6/14 pole, three phase FRM is designed and fabricated for rooftop wind power application. The design details of this machine are discussed in Section 2. Section 3 describes the concept of fictitious electrical gear and full pitch stator winding arrangement. In Section 4 the results of 2D FEM analysis of CSPFRM and FPFRM, and winding function approach to determine the self inductance are discussed. Section 5 describes the FEM analysis to predict the voltage regulation of CSPFRM and full pitch winding FRM (FPFRM), and a method used to improve the voltage regulation. Power density and efficiency comparisons of both machines are also made. In Section 6 the experimental results of prototype machine, and in Section 7 the power density comparison of CSPFRM and FPFRM with PMSM are discussed. 2 Three-phase 6/14 pole FRM The cross-sections of three-phase 6/14 pole CSPFRM and FPFRM are shown in Figs. 1b and 2a, respectively. The 6/ 14 pole FRM consists of 14 salient pole rotor and 6 salient pole stator. Two pairs of PM are fixed on each stator pole. The difference between the CSPFRM and FPFRM is in the winding arrangement. Instead of fabricating two Figure 2 Cross-section of FRM a 6/14 pole FPFRM b 6/14 pole prototype FRM 110 IET Electr. Power Appl., 2010, Vol. 4, Iss. 2, pp. 109 120 & The Institution of Engineering and Technology 2010

Table 1 Dimensions of 6/14 pole, 214 rpm FRM Sr. no. Description Symbol Value 1 airgap, mm g 0.5 2 magnet thickness, mm h pm 2.5 3 rotor pole span angle, 8 mech. b r 8.568 4 stator pole span angle, 8 mech. b s 488 5 stator pole span, mm t ps 67.43 6 rotor pole span, mm t pr 11.96 7 stator pole height, mm h ps 22.5 8 rotor pole height, mm h pr 30 9 outer diameter of rotor, mm D i 160 10 outer diameter of stator, mm D o 270 11 number of turns/phase for CSPFRM N ph 66 of the machine is given in Table 1. The cross-section of the prototype is shown in Fig. 2b. The stator and rotor of the fabricated machine are shown in Fig. 3. 3 Fictitious electrical gear and full pitch stator winding The flux plot of 6/14 pole FRM on no load with PM excitation is shown in Fig. 4a. FRM machine has 6 stator poles and 14 pole variable reluctance rotor. The normal component of flux density at the middle of stator pole along the periphery of the machine is shown in Fig. 4b. It can be observed that this flux density plot is similar to that of two pole machine. In other words the machine has two effective poles. The number of poles corresponding to the flux pattern for various FRM configurations is given in Table 2. The frequency and speed relationship for FRM [3] is given by 12 number of turns/phase for FPFRM N ph 66 f ¼ n n r Hz (1) 60 13 stack length of machine, mm l s 150 14 stator pole width, mm W sp 25 15 stator core width, mm W sc 19.5 16 shaft diameter, mm D sh 52 17 rotor skew angle, 8 mech. 4 machines (one with concentrated stator pole winding and other with full pitch winding) one machine with both windings is fabricated. These windings have equal number of turns and cross-section. The 6/14 pole FRM is designed as per the procedure given in [3]. The design data where n is the speed (rpm) and n r is the number of rotor teeth or poles. The three phase 6/14 pole FRM has two effective poles, and hence, flux pattern speed (n f ) is given by from (1)and (2) n f ¼ 60 f (2) n f ¼ n n r (3) The speed of flux pattern in 6/14 pole FRM is n r times the shaft speed. In synchronous machine, speed of flux pattern Figure 3 Photograph of 6/14 pole FRM machine a Stator b Rotor IET Electr. Power Appl., 2010, Vol. 4, Iss. 2, pp. 109 120 111 & The Institution of Engineering and Technology 2010

Figure 4 6/14 pole FRM a Flux plot b Normal component of flux density at the middle of the stator pole along the periphery of the machine Table 2 Poles corresponding to the flux pattern and gear ratio for various FRM configurations Sr. no. Machine type No. of magnets Gear ratio No. of poles corresponding to flux pattern 1 6/8 pole 12 8 2 2 12/16 pole 24 8 4 3 6/14 pole 24 14 2 4 12/28 pole 48 14 4 5 12/40 pole 60 20 4 and that of rotor is the same. Pictorial representations of FRM generator and equivalent PM synchronous generator are shown in Fig. 5. This representation is for same speed and output frequency. Flux pattern in FRM rotating at n r times the rotor speed is represented by a fictitious step-up gear. This gear is called as electrical gear. The electrical gear ratio (K ) is defined as the ratio of flux pattern speed to rotor speed. The FRM with 6/14 pole configuration has a gear ratio of 14. The electrical gear ratios for other FRM configurations are given in Table 2. It may be noted that the synchronous machine equivalent to 6/14 pole FRM should have 28 poles for same output frequency. Three-phase 6/14 pole FRM has six stator slots and two pole flux pattern. Hence electrical angle per slot is 608. Conventional concentrated stator pole winding has a coil span of 608, short pitched by 1208. Therefore, fundamental pitch factor of the stator winding is 0.5. As electrical angle between the slots is 608, full pitch winding is possible. The arrangement of full pitch winding is shown in Fig. 2a, which results in unity pitch factor. Figure 5 Generator representation of 6/14 pole FRM and equivalent PMSM a FRM b PMSM 4 FEM analysis of CSPFRM and FPFRM at no load A prototype of CSPFRM and FPFRM having same number of turns and conductor cross-section is designed and fabricated. FEM analysis is carried out to determine the phase flux linkage variation of both windings with rotor position. This variation for unskewed rotor is shown 112 IET Electr. Power Appl., 2010, Vol. 4, Iss. 2, pp. 109 120 & The Institution of Engineering and Technology 2010

Figure 6 FEM simulated plots of prototype 6/14 pole FRM a Phase flux linkages of CSPFRM and FPFRM for unskewed rotor b Cogging torque of prototype machine c Self-inductance variation of CSPFRM and FPFRM with rotor position in Fig. 6a. From this figure it can be inferred that stator flux linkage in FPFRM is approximately twice that of conventional CSPFRM. Hence, the open circuit voltage of FPFRM is twice that of CSPFRM. The flux linkage variation with rotor position is sinusoidal. FRM is a doubly salient machine and hence cogging torque is high. This torque can be reduced by providing skew to the rotor. The rotor of the prototype machine is fabricated with a skew of 48. The cogging torque of the machine for unskewed and skewed rotor is shown in Fig. 6b. It is observed that peak to peak cogging torque without skewing is 19.24 Nm while this torque reduces to 4.3 Nm for 48 rotor skew. Stator phase inductance is calculated using Flux 2D package as per the procedure given in [2]. The stator phase inductance variation with rotor position for both the machines is shown in Fig. 6c. This variation for FPFRM is from 18.74 to 19.27 mh, whereas for CSPFRM it is from 6.94 to 7.22 mh. Mutual inductance of CSPFRM winding is negligibly small, whereas in the case of FPFRM it varies between 5.72 to 6.2 mh. 4.1 Inductance calculation of prototype FRM Synchronous inductance component and harmonic inductance component of the self-inductance of prototype CSPFRM and FPFRM is calculated using (14) and (16), respectively. Stator and rotor of FRM have slotted structure. The Carter coefficient [9] for slotted stator or rotor is given by K c ¼ t t gg 0 (4) 8 g ¼ 4 b 0 p 2g 0 arctan b sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 < 0 2g 0 ln 1 þ b 2 = 0 : 2g 0 (5) ; where b 0 is the slot opening and t t is the tooth pitch. In case of FRM tooth pitch is equal to pole pitch. The Carter coefficient for stator (K c1 ) and rotor (K c2 ) is calculated using (4) and (5). The total Carter coefficient for FRM (K c ) is given by [10] t t K c ¼ K c1 K c2 (6) IET Electr. Power Appl., 2010, Vol. 4, Iss. 2, pp. 109 120 113 & The Institution of Engineering and Technology 2010

The values required to calculate the Carter coefficient are given in Table 3. The slot leakage inductance of FRM is determined using the procedure given in [11] and its value is small as slot width is higher than the slot depth. The total self-inductance is the sum of inductance calculated from winding function and slot leakage, as the end turn leakage inductance is neglected. The calculated values of total self-inductance and that obtained from FEM analysis are given in Table 4. 4.2 PM flux linkages of prototype FRM The expression for PM flux linkages of FRM is derived in the appendix. The maximum PM flux linkages for CSPFRM and FPFRM are given by (20) and (21), respectively. The machine data required for the calculation of PM flux linkages and the values of PM flux linkages for prototype machine obtained with analytical and FEM method are Table 3 Data to determine Carter coefficient Sr. Parameter Symbol Value no. 1 tooth pitch for stator, mm t ts 84.3 2 slot opening of stator, mm b 0s 16.86 3 tooth pitch for rotor, mm t tr 36.12 4 slot opening of rotor, mm b 0r 24.16 5 relative permeability of m r 1.05 the magnet 6 Carter coefficient for stator K c1 1.122 7 Carter coefficient for rotor K c2 1.726 Table 4 Self- and mutual-inductance of 6/14 pole CSPFRM and FPFRM Sr. Parameter CSPFRM FPFRM no. 1 Self-inductance with 6.94 7.22 18.74 19.27 FEM analysis, mh 2 Inductance from 6.2 18.6 winding function, mh 3 Slot leakage 0.66 0.66 inductance, mh 4 Total self-inductance (analytical value), mh 6.86 19.26 5 Mutual inductance with FEM analysis, mh 6 Mutual inductance from winding function, mh 0.13 0.32 5.72 6.2 6.2 Table 5 Data for PM flux linkage calculation Sr. Parameter Symbol Value no. 1 PM flux fringing K fringe 0.44 2 remanent flux density, Wb/m 2 B r 1.2 3 PM width, mm t pm 17 4 pole pairs per stator pole n pp 2 5 maximum PM flux linkages 0.1894 of CSPFRM (FEM), Wb 6 maximum PM flux linkages 0.171 of CSPFRM (analytical), Wb 7 maximum PM flux linkages 0.362 of FPFRM (FEM), Wb 8 maximum PM flux linkages of FPFRM (analytical), Wb 0.3421 given in Table 5. The value of K fringe is obtained from FEM simulation. It is observed that analytically computed values of PM flux linkages are in close agreement with those obtained from FEM computations. 5 FEM analysis of CSPFRM and FPFRM at full load FEM simulation is carried out at different load conditions to predict the voltage regulation of machines at 214 rpm. Terminal voltage variation with load current is shown in Fig. 7a. It can be seen that voltage regulation of FPFRM is poor as compared to CSPFRM because of the high value of self-inductance. Regulation of FRM is improved by series capacitive compensation [12]. The compensated machine operates at leading power factor. This results in magnetising armature reaction which improves the regulation. The value of capacitance is given by [13]. C ¼ 1 v 2 L s F (7) where v is the frequency of induced EMF (rad/s) and L s is the synchronous inductance of the machine (Henry). Synchronous inductance (L s ) for each winding is obtained from uncompensated regulation characteristics of respective winding is shown in Fig. 7a. The value of capacitance for CSPFRM and FPFRM are 1200 and 370 mf, respectively, when these simulations are driven at 214 rpm. FEM-based simulation study of series compensated CSPFRM and FPFRM is carried out at different loads to determine the regulation of the machine. Terminal voltage variation of 114 IET Electr. Power Appl., 2010, Vol. 4, Iss. 2, pp. 109 120 & The Institution of Engineering and Technology 2010

Figure 7 FEM simulated performance curves of prototype 6/14 pole FRM a Uncompensated terminal voltage characteristics b Compensated terminal voltage characteristics c Variation of efficiency with load compensated CSPFRM and FPFRM is shown in Fig. 7b.From these characteristics, it can be inferred that compensated FPFRM has approximately 150% output power as compared to compensated CSPFRM for same electrical and magnetic loading. The simulated efficiencies of both machines are shown in Fig. 7c. It can be observed that FPFRM has higher efficiency as compared to CSPFRM. Iron, friction and windage losses are approximately same in both machines. Although FPFRM has higher winding resistance than CSPFRM, in FPFRM the current is less for the same output power. Therefore copper loss in FPFRM is less as compared to CSPFRM. The overall efficiency of FPFRM is higher than that of CSPFRM. 6 Experimental results The three-phase 6/14 pole FRM is designed and fabricated to validate the simulation results. The machine generates 50 Hz supply at 214 rpm. Both windings are designed for same number of turns and cross-section. The prototype FRM is driven by DC motor through a reduction gear box. The experimental setup is shown in Fig. 8a. The open circuit voltage waveforms of both windings are shown in Figs. 8b and c, respectively. The variation of terminal voltage of uncompensated machine is shown in Fig. 8d. A close look at these characteristics show that no load terminal voltage of FPFRM is twice that of CSPFRM. But the voltage regulation is poor. Regulation of the machine is improved by series capacitive compensation. The value capacitors used for CSPFRM and FPFRM are 1200 mf, 50 V and 370 mf, 180 V, respectively. Experimental results are obtained by loading the machine with resistive load. The experimental results on compensated generator are shown in Fig. 8e. It can be observed that FPFRM has 1.8 times higher voltage than CSPFRM at a load current of 7 A. 7 Comparison of FRM and PMSM In case of FRM, the speed of rotation of flux pattern and that of the rotor is different. Hence, a term electrical gear ratio IET Electr. Power Appl., 2010, Vol. 4, Iss. 2, pp. 109 120 115 & The Institution of Engineering and Technology 2010

Figure 8 Experimental results a Experimental setup b No load terminal voltage of CSPFRM (10 V/div, 5 ms/div) c No load terminal voltage of FPFRM (20 V/div, 5 ms/div) d Uncompensated regulation characteristics e Compensated regulation characteristics (K ) is defined as the ratio of flux pattern speed to rotor speed. The FRM with 6/14 pole configuration has a gear ratio of 14. In case of PMSM, the speed of flux pattern is same as rotor speed and hence, the gear ratio is unity. The power density comparison between PMSM and FRM is possible with electrical gear ratio. This comparison for these machines is based on the following assumptions: 1. Stack length and air gap diameter is same for both machines. 2. Turns per phase are equal. 3. Speed and frequency are same for both machines. 4. Air gap flux density distribution is sinusoidal. 116 IET Electr. Power Appl., 2010, Vol. 4, Iss. 2, pp. 109 120 & The Institution of Engineering and Technology 2010

5. Similar voltage regulation characteristics. 6. Concentrated full pitch winding is assumed for PMSM (K w ¼ 1). The phase flux linkages in CSPFRM and FPFRM for prototype machine with skewed rotor are shown in Fig. 9a. The maximum values of phase flux linkage for CSPFRM and FPFRM are 0.126 and 0.25 Wb, respectively. The relationship between maximum flux linkage c 1 (max) and maximum flux density Bm 1 ( max ) is given by [14]. c 1 (max) ¼ Bm 1(max) l s t 2 N ph K w p where l s is the stack length of the machine (m), t is the pole pitch distance (m), N ph is the number of turns per phase and K w is the winding factor. 7.1 Power density of CSPFRM and PMSM CSPFRM has two effective poles (flux pattern of two poles), whereas equivalent PMSM has 28 poles. Hence, pole pitch (8) for equivalent PMSM and CSPFRM for an air gap diameter of 160 mm is 0.01795 and 0.251 m, respectively. Winding factor K w for CSPFRM and equivalent PMSM is 0.5 and unity, respectively. The value of Bm 1 (max) is determined using (8) and this value in CSPFRM and PMSM is 0.158 and 1.113 Wb/m 2, respectively. Flux density distribution is assumed to be sinusoidal as flux linkage is sinusoidal. Flux density distribution pattern of FRM rotates at n n r rpm, whereas that of PMSM, it rotates at n rpm. Both flux patterns generate same output voltage and frequency. Hence, PMSM should have a air gap flux density of 1.113 Wb/m 2 for the same output voltage as CSPFRM. The air gap flux density distribution for CSPFRM and equivalent PMSM is shown in Fig. 9b. The maximum possible air gap flux density in PMSM is 0.9 Wb/m 2 [15]. Hence, CSPFRM has 1.236 times higher power density than PMSM in low speed, low power range. 7.2 Power density of FPFRM and PMSM FPFRM is compared with PMSM on similar lines as CSPFRM. Winding factor K w for both machines is unity. Figure 9 Power density comparison based on flux density and flux linkages a Flux linkages of CSPFRM and FPFRM with skewed rotor b Air gap flux distribution in CSPFRM and equivalent PMSM c Air gap flux distribution in FPFRM and equivalent PMSM IET Electr. Power Appl., 2010, Vol. 4, Iss. 2, pp. 109 120 117 & The Institution of Engineering and Technology 2010

From (8), the value of Bm 1 (max) in FPFRM is 0.158 Wb/ m 2, whereas that in PMSM is 2.212 Wb/m 2. Hence, FPFRM has 2.45 times higher power density than PMSM. The air gap flux density distributions of FPFRM and equivalent PMSM are shown in Fig. 9c. 8 Discussion The no load induced voltage of FPFRM is twice that of CSPFRM for same number of turns. If voltage regulation of both machines is assumed to be same, then the power density improvement is double. However, the terminal voltage characteristics of compensated machines shown in Figs. 7b and 8e, show that both machines have different regulation, which decreases the power density of FPFRM. For the prototype machine with series capacitive compensation, the increase in power density is more than 150%. Power density improvement depends upon the method of compensation used. Power density of FRM is compared with PMSM by assuming that the regulation of both machines is approximately same. However, FRM has higher regulation than PMSM. This results in FRM having slightly less power density than the theoretical value obtained in Section 7. CSPFRM, FPFRM and PMSM are designed for same outer dimensions, magnet volume and speed. The comparison of these machines is given in Table 6. It can be seen that active weight/kva is less for FPFRM as compared to CSPFRM and PMSM. However, this machine requires higher compensating kvar. 9 FRM configurations and full pitch winding FRM machine configurations for low speed are presented in [3]. The configurations are 1. Low speed FRM with pole PMs on stator. 2. Low speed FRM with inset-pms on stator. Table 6 Comparison of CSPFRM, FPFRM and PMSM Machine type CSPFRM FPFRM PMSM KVA 1.92 2.88 1.4 copper weight, kg 4.9 11.6 4.4 core weight, kg 37.51 37.51 41 PM weight, kg 1.1 1.1 1.1 compensating capacitor, kvar 3.183 10.32 0.91 Figure 10 Three phase 12/16 pole FRM with full pitch winding These configurations have 12 stator poles (slots) and flux pattern of four poles. Full pitch winding arrangement is possible for both configurations. Full pitch winding arrangement for low speed FRM machine with 16 rotor poles and 12 stator poles is shown in Fig. 10. It may be observed that the end winding (overhang) length decreases with increase in stator poles. This decreases the end turn leakage inductance and copper loss. Hence, full pitch winding concept can be adapted for all three phase FRM configurations. 10 Conclusion This paper presents full pitch winding for three phase FRM to increase the power density and to improve the efficiency. The prototype 6/14 pole FRM for rooftop wind power generation is designed and fabricated. For the designed prototype machine, the output power of compensated FPFRM is approximately 1.5 times the conventional compensated CSPFRM. The value of self-inductance obtained from winding function approach closely matches with that of FEM-based simulation study. The power density comparison of CSPFRM and FPFRM with PMSM is made based on fictitious electrical gear. The gear ratios for various FRM configurations are presented. Power density of compensated CSPFRM is 1.236 times higher than PMSM, whereas the power density of compensated FPFRM is 2.45 times higher than PMSM. Hence, FPFRM topology is suitable for low speed low power applications. 11 References [1] DEODHAR R.P., ANDERSON S., BOLDEA I., ET AL.: The flux reversal machine: A new doubly salient permanent magnet machine, IEEE Trans. Ind. Appl., 1997, 33, (4), pp. 925 934 [2] WANG C., NASAR S.A., BOLDEA I.: Three phase flux reversal machine (FRM), IEE Trans. Electr. Power Appl., 1999, 146, (2), pp. 139 146 118 IET Electr. Power Appl., 2010, Vol. 4, Iss. 2, pp. 109 120 & The Institution of Engineering and Technology 2010

[3] BOLDEA I., ZHANG J., NASAR S.A.: Theoretical characterization of flux reversal machine in low speed servo drives the pole PM configuration, IEEE Trans. Ind. Appl., 2002, 38, (6), pp. 1549 1557 [4] KIM T.H., WON S.H., BONG K., ET AL.: Reduction in cogging torque in flux reversal machine by rotor teeth pairing, IEEE Trans. Magn., 2005, 41, (10), pp. 3964 3966 [5] KIM T.H., LEE J.: A study of the design for the flux reversal machine, IEEE Trans. Magn., 2004, 40, (4), pp. 2053 2055 [6] ZHANG J., CHENG M., HUA W., ET AL.: New approach to power equation for comparison of doubly salient electrical machines. Proc. IEEE Ind. Appl. Annu. meeting, 2006, pp. 1178 1185 [7] Flux 2-D FEM Software, CEDRAT, France [8] MORE D.S., FERNANDES B.G.: Novel three phase flux reversal machine with full pitch winding. Proc. Int. Conf. on Power Electronics (ICPE 2007), pp. 1007 1012 [9] GIERAS J.F., WANG R., KAMPER M.J.: Axial flux permanent magnet machines (Kluwer Academic Publishers, Dordrecht/Boston/London, 2004) [10] SHARIFIAN M.B.B., SHAARBAFI K., FAIZ J., ET AL.: Slot fringing effect on the magnetic characteristics of the electrical machine. IEEE Int. Conf. on Electronic Circuits and Systems (ICECS 2003), (2), pp. 1007 1012 [11] HANSELMAN D.C.: Brushless permanent-magnet motor design (McGraw-Hill Inc., 1994) [12] NAOE N.: Voltage compensation of permanent magnet generator with capacitors. Proc. IEEE Electrical Machines and Drives, 1997, pp. wb2.14.1 wb2.14.3 [13] WANG C., BOLDEA I., NASAR S.A.: Characterization of threephase flux reversal machine as an automotive generator, IEEE Trans. Energy Conver., 2001, 16, (1), pp. 74 80 [14] GIERAS J.F., WING M.: Permanent magnet motor technology: design and applications (Marcel Dekker Inc., 2002) [15] MILLER T.J.E.: Brushless permanent magnet and reluctance motor drives (Clarendon Press, Oxford, 1989) [16] EL-REFAIE A.M., ZHU Z.Q., JAHNS T.M., ET AL.: Winding inductances of fractional slot surface mounted permanent magnet brushless machines. Proc. IEEE Industry Applications Annu. Meeting, 2008, pp. 1 8 [17] EL-REFAIE A.M., JAHN T.M.: Optimalfluxweakeningin surface PM machines using fractional slot concentrated winding, IEEE Trans. Ind. Appl., 2005, 41, (3), pp. 790 800 12 Appendix 12.1 Self- and mutual-inductances from winding function The self-inductance of the winding consists of (i) synchronous inductance component, (ii) harmonic leakage component, (iii) slot leakage component and (iv) end turn leakage component. Winding function can be used to calculate the sum of synchronous inductance and harmonic leakage components of the phase winding [16, 17]. Turns function and winding function for CSPFRM are shown in Fig. 11a. The sum of synchronous inductance component and harmonic leakage component of the self-inductance of CSPFRM is given by L aa ¼ m ð 2p 0rl s g 00 N 2 a (w)dw (9) 0 Figure 11 a CSPFRM b FPFRM Winding function IET Electr. Power Appl., 2010, Vol. 4, Iss. 2, pp. 109 120 119 & The Institution of Engineering and Technology 2010

where r is the air gap radius (m), l s is the stack length of the machine (m), N a (w) is the winding function, N is the number of turns per phase and g 00 is the effective air gap length in the metre is given by g 00 ¼ K c g 0 (10) g 0 ¼ g þ h PM m r (11) where K c is the Carter coefficient due to stator and rotor slotting, g is the air gap in the machine, h pm is the magnet thickness and m r is the relative permeability of Nd Fe B magnet. L aa ¼ 2m 0 rl ð p s g 00 Na 2 (w)dw (12) 0 substituting the value of winding function L aa ¼ 2m ð p=3 0rl s N 2 g 00 dw (13) 0 4 Hence, the sum of the synchronous inductance and harmonic leakage component of self-inductance of CSPFRM is given by L aa ¼ m 0rl s N 2 p g 00 2 3 (14) Turns function and winding function for FPFRM are shown in Fig. 11b. Substituting the value of winding function in (9) L aa ¼ 2m 0rl s g 00 ð p 0 N 2 4 dw (15) L aa ¼ m 0rl s g 00 N 2 2 p (16) Comparing (14) and (16), and neglecting the slot leakage inductance, the ratio of self-inductance of FPFRM and CSPFRM is 3 for same dimensions and number of turns. An equation for mutual inductance between the phases of FPFRM can also be obtained from winding function. It is given by M ab ¼ m 0rl s N 2 p g 00 2 3 12.2 PM flux linkages (17) The flux density variation in the middle of the stator poles along the periphery of the machine at different rotor positions is shown in Fig. 12. The reference rotor position is shown in Fig. 4a, where the flux density in two stator poles is almost zero as the magnets under these two stator poles are shorted by the rotor poles. The flux density in the other four stator poles is equal and flux in these stator poles is because of PM of Figure 12 Flux density variation in stator poles at different rotor positions a 08 (electrical) b 308 (electrical) c 608 (electrical) corresponding stator poles alone (as shown in Fig. 12a). The flux density in the stator pole at this instant is termed as B s. The flux in the stator poles at 308 rotor position (electrical) is shown in Fig. 12b. At this instant the flux density in two statorpolesismaximum(b s (max)), whereas the other poles have half the maximum flux density. Assuming the stator and rotor core has infinite permeability, the flux density in the stator pole at reference rotor position is given by B r h pm B s ¼ K fringe h pm þ g n pp t pm W sp (18) where K fringe is ratio of flux linking the stator winding to air gap flux, B r is remanent flux density of PM (Wb/m 2 ), t pm is magnet width (m), n pp is number of PM pole pairs per stator pole, and W sp is stator pole width (m). The maximum flux density (B s (max)) in the stator pole is given by B s (max) ¼ B s sin(p=3) (19) The maximum PM flux linkage in the stator winding of CSPFRM is given by w pm (max) ¼ B s (max)l s W sp N ph (20) The maximum flux linkage of FPFRM is twice that of CSPFRM; therefore, maximum flux linkage in FPFRM is given by w pm (max) ¼ 2B s (max)l s W sp N ph (21) 120 IET Electr. Power Appl., 2010, Vol. 4, Iss. 2, pp. 109 120 & The Institution of Engineering and Technology 2010