d-q Equivalent Circuit epresentation of Three-Phase Flux eversal Machine with Full Pitch Winding D. S. More, Hari Kalluru and B. G. Fernandes Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai - 4 76, INDIA. Email: dsmore@ee.iitb.ac.in khari@ee.iitb.ac.in bgf@ee.iitb.ac.in Abstract In this paper a d-q equivalent circuit for flux reversal machine (FM) is proposed. In order to improve the power density, full pitch winding is proposed. FM with this winding winding (FPFM) is compared with conventional concentrated stator pole winding FM (CSPFM). The output power of FPFM is twice that of CSPFM for the same machine dimensions, electrical and magnetic loadings. The results obtained using proposed d-q circuits are compared with those obtained from FEM analysis. Steady state and dynamic performance of FPFM and CSPFM is evaluated with proposed d-q circuits. I. INTODUCTION Single phase flux reversal machine (FM) was first introduced in 1997 by. P. Deodhar and et al for automobile application to replace the standard claw pole alternator [1]. It has numerious advantages such as simple construction, low inertia, high power density and is suitable for high speed application due to stationary permanent magnets and stator winding. This single phase configuration is fully explored as a high speed automotive generator. Three phase FM was introduced by C. Wang and et al in 1999 [2]. The design of the machine was optimized to ensure (i) high PM flux linkage in the winding, (ii) low cogging torque and PM weight. The basic machine configuration is 8 salient pole rotor and 6 pole stator with concentrated windings. Permanent Magnets are fixed to stator pole. Fig. 1 shows this machine configuration. FM for low-speed servo drive application was introduced by Ion Boldea and et al in 22 [3]. This low speed machine has 28 rotor poles and 12 stator poles with two permanent magnet pairs on each stator pole. This machine is designed for 128 rpm at 6 Hz. Using vector control high torque density with less than 3% torque pulsation was achieved. In order to reduce the cogging torque, rotor teeth pairing method has been proposed [4]. Attempts were made to reduce the leakage flux by providing flux barrier on the rotor poles at its edges [5]. Power density comparison of doubly salient permanent magnet electrical machines has been made. It is concluded that FM has higher power density in comparison with other machines in the same class [6]. Full pitch winding flux reversal machine (FPFM) was proposed to improve the power density of the machine as compared to conventional concentrated stator pole winding flux reversal machine (CSPFM) [7]. Fig. 1. Cross-section of 6/8 pole concentrated stator pole winding FM. In this paper, concept of fictitious Electrical Gear is proposed based on the flux pattern of the machine. The d- q equivalent circuits for FM based on this gear is proposed. In order to validate d-q equivalent circuits, two dimensional FEM analysis [8] is carried out on CSPFM and FPFM. Optimized machine dimensions are obtained from C. Wang and et al [2]. The important dimensions of FM are given in Table I for ready reference. Section II describes the flux linking to the stator winding of the machine and there from the concept of full pitch stator winding arrangement is discussed. Section III proposes the fictitious Electrical Gear concept applicable to FM. Section I proposes the d-q equivalent circuit for FM based on this fictitious gear. Section describes the FEM simulation results to validate the d-q equivalent circuit. Section I compares the power density of FPFM with CSPFM and finally conclusions are drawn. II. FU PITCH STATO WINDING FO FM Geometry of 6/8 pole three-phase FM (as per Table 1) and the flux distribution in this machine at no load is shown in Fig. 2. FM machine has 6 stator poles and 8 pole variable 978-1-4244-1668-4/8/$25. 28 IEEE 128 Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOOGY BOMBAY. Downloaded on December 4, 28 at 23:3 from IEEE Xplore. estrictions apply.
.6 Normal component of flux density in air gap 4th degree.4 Flux density ( Wb/m 2 ).2.2.4 5 1 15 2 25 Distance along the air gap (mm) Fig. 2. Flux distribution in 6/8 pole FM at no load Fig. 4. Normal component of armature flux density along the air gap with magnets are de-energized reluctance rotor. The normal component of flux density at the middle of stator pole along the periphery of the machine is shown in Fig. 3. The observation of this normal component of flux density plot reveals that the machine has two pole flux pattern. Phase flux linkage in FM is sinusoidal in nature and hence the induced voltage [2]. Considering a linear load the phase current is also sinusoidal. Normal component of armature reaction along the air gap at one instant of time is shown in Fig. 4. This flux pattern also reveals that machine has effective two pole flux pattern. In other words machine has two effective poles. FM has 6 slots and two pole flux pattern, hence electrical angle per slot is 6. CSPFM stator winding has a coil span of 6. Fundamental pitch factor of the stator winding is.5. As electrical angle between the slots is 6, full pitch winding.8.6 Normal component of flux density 4th degree is possible. The arrangement of this full pitch winding is shown in Fig. 5 and fundamental pitch factor of stator winding is unity. Hence, voltage induced in FPFM is twice that of CSPFM for the same number of turns. TABE I DIMENSIONS OF FM Sr. No. Description Symbol alue 1 Air gap (mm) g 1 2 Magnet thickness (mm) h pm 3 3 otor pole span angle β r 16.2 4 pole span angle β s 42.6 5 pole span (mm) τ ps 27.8 6 otor pole span (mm) τ pr 1.3 7 pole height (mm) h ps 15 8 otor pole height (mm) h pr 18 9 Outer dia. of rotor (mm) D i 72 1 Outer dia. of stator (mm) D o 129 11 Number of turns /phase N ph 52 12 Stack length (mm) l sk 86.4 Flux density ( Wb/m 2 ).2.2.4.6.8 5 1 15 2 25 3 35 Distance along the periphery of the machine at the middle of the stator pole ( mm) Fig. 3. Normal component of flux density along the periphery of machine at middle of the stator pole Fig. 5. Full pitch winding arrangement in FPFM 129 Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOOGY BOMBAY. Downloaded on December 4, 28 at 23:3 from IEEE Xplore. estrictions apply.
III. FICTITIOUS EECTICA GEA The frequency and speed relationship for FM is given by [3] n = 6 f n r (1) where, n = rotor speed in rpm. n r = number of rotor teeth (poles). f = frequency in Hz. The three-phase 6/8 pole FM has two effective poles, and hence flux pattern speed for supply frequency f Hz is given by n f =6 f (2) where, n f = flux pattern speed in rpm. Equations (1) and (2) reveal that rotor speed and flux pattern speed is different. The shaft speed is n r times less than flux pattern speed. In conventional machines, flux pattern speed and rotor speed is same. Pictorial representation of 6/8 pole FM motor is shown in Fig. 6 while a pictorial representation of 2 pole PMSM is shown in Fig. 7. The difference in speed between rotor and flux pattern speed is represented by a fictitious step-down gear and is called Electrical Gear. Electrical gear ratio (K) is defined as ratio of flux pattern speed to the shaft speed. Supply Frequency = f Hz Equivalent 2 pole PM otor Flux pattern speed = 6 x f Shaft speed n rpm TABE II GEA ATIO FO AIOUS FM CONFIGUATIONS Sr. No. Machine No.of Gear No. of Flux type magnets ratio pattern poles 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 12/4 pole 6 2 4 The generalised equation for electrical gear ratio is given as K = n r (3) P eq /2 where, P eq = no. of flux pattern poles. Hence 6/8 pole FM can be analysed as 2 pole PMSM with a gear ratio of 8. Gear ratios for various FM configurations are given in Table II. It can be observed that no. of flux pattern poles in FM are 2 for 6 stator poles and 4 for 12 stator poles. I. d q EQUIAENT CICUITS FO FM Permanent magnet synchronous machine (PMSM) is analyzied with d-q equivalent circuits [9]. Transient and steady state behaviour of the PMSM is obtained with these equivalent circuits. Fig. 8 shows the d-q equivalent circuit for PMSM. Back EMF in FM is sinusoidal in nature. Self inductance and mutual inductance is almost constant with rotor position. It requires sinusoidal stator current to produce a constant torque. Hence d-q equivalent circuit can be used to analysis the steady state and transient behaviour of FM. d-q equivalent circuits of FM are derived from PMSM. The major difference between PMSM and FM is the relationship between speed and frequency. Three phase 6/8 pole FM can be considered as 2 pole PMSM with gear ratio (K) of 8 as shown in Fig. 6. Mathematical model of FM is similar to PMSM except Fictitious electrical n = (6 x f)/n r Gear i d e λ q ld Fig. 6. epresentation of 6/8 pole FM Supply Frequency = f Hz d md I f Flux pattern speed = 6 x f i q eλ d lq 2 pole PM otor Shaft speed n rpm n = 6 x f q mq Fig. 7. epresentation of 2 pole PMSM Fig. 8. d-q equivalent circuit of PMSM 121 Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOOGY BOMBAY. Downloaded on December 4, 28 at 23:3 from IEEE Xplore. estrictions apply.
i d i q d q K K e λ q eλ d ld md lq mq I f Phase flux linkages (Wb).5.4.3.2.1.1.2.3.4 FPFM CSPFM.5 5 1 15 2 25 3 35 4 45 otor position (mech. degrees) Fig. 9. d-q equivalent circuits of FM Fig. 1. Phase flux linkage of FPFM and CSPFM the gear ratio (K). This gear ratio (K) is considered in the modeling of FM. The following assumptions are made while deriving these equivalent circuits [9]. Saturation in the machine is neglected. The induced EMF is sinusoidal. Eddy currents and hysteresis losses are negligible. There are no field current dynamics. There is no cage on the rotor. With these assumptions, The d-q equations in synchronously rotating reference frame of FM are v d = i d pλ d Kω e λ q (4) v q = i q pλ q Kω e λ d (5) The electrical torque T e is given by λ q = q i q (6) λ d = d i d λ af (7) T e = 3 2 P eq 2 K(λ af i q ( d q )i d i q ) (8) where, = stator resistance (Ohm). i d,iq = d and q axes stator currents (A). d, q = d and q axes inductances (H). p = derivative operator. λ d,λ q = d and q axes flux linkages (Wb). λ af = mutual flux linkages due to PM (Wb). ω e = rotor speed (rad/sec). FM machine considered for simulation has 6/8 pole structure and has equal d and q axes inductance [2]. Therefore torque equation reduces to T e = 3 2 P eq 2 Kλ af i q (9) where, P eq = no. of flux pattern poles of the machine= 2 K = 8 FM is controlled with constant flux upto base speed by maintaining i d equals to zero. Under this condition and at steady state, (4) to (7) are reduce to v d = Kω e λ q (1) v q = i q Kω e λ d (11) λ q = q i q (12) λ d = λ af (13) A. Steady State Torque Calculation FM machine design data is obtained from [2] and is shown in Table I. Physical dimensions of the machine and number of turns/phase are kept same in FPFM and CSPFM, only the winding arrangement is changed. FEM analysis is carried out to determine the variation of phase flux linkage of both machines with rotor position. This variation for both the machines without skewed rotor is shown in Fig.1. Flux linkage variation is shown for one rotor pole pitch (i.e. 45 mech.). Figure clearly shows that FPFM stator winding flux linkage is approximately twice that to the CSPFM. alues of λ af obtained from Fig. 1 for CSPFM and FPFM. They are.21 Weber and.41 Weber respectively. The steady state torque of both machines for I ph = 15 A is obtained from (9). The calculated values of steady state torques of CSPFM and FPFM are 5.34 Nm and 1.43 Nm respectively.. FEM SIMUATION TO AIDATE THE d q EQUIAENT CICUITS FEM motor simulations of CSPFM and FPFM for constant torque operation are carried out. The linking between the FEM winding regions to coil components of the circuit for FM is shown in Fig.11. B1 to B6 are winding regions; where as b1 to b6 are corresponding coil components in the circuit. 1211 Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOOGY BOMBAY. Downloaded on December 4, 28 at 23:3 from IEEE Xplore. estrictions apply.
8 6 Phase current Phase voltage oltage () and current (A) 4 2 2 4 Fig. 11. Coupling between FE regions and electrical circuit of FPFM. 6 TABE III FU OAD TOQUE (NM) OF CSPFM AND FPFM Parameter CSPFM FPFM Calculated using d-q circuits 5.34 1.43 Obtained from FEM simulation 5.18 1.14 1, 2 and 3 are the end turn leakage inductance/phase. 1, 2 and 3 is stator winding resistance/phase. Motor is supplied from three phase sinusoidal current source I 1, I 2 and I 3. ector control is obtained with i d equal to zero and i q is maintained in phase with back EMF of the machine. Simulated steady state torque of both machines using Flux 2D software is shown in Fig.12. The average value of steady state torque of CSPFM and FPFM is 5.18 Nm and 1.14 Nm respectively. The average torque obtained using d-q equivalent circuits and that obtained from FEM simulation at full load is shown in Table III and it can be seen that there is a good agreement between these results. Steady state waveforms of voltage and rated current supplied to the CSPFM and FPFM at 1995 rpm obtained 8.5 1 1.5 2 2.5 3 3.5 4 Time (seconds) x 1 3 Fig. 13. Phase voltage and phase current supplied to CSPFM. TABE I PAAMETES OF FM Sr. No. Parameter CSPFM FPFM 1 K 8 8 2 ω e (rad/sec.) 28.9 28.9 3 (ohm).5.17 4 d = q (mh).94 3.67 5 λ af (Wb).21.41 6 i q (A) 21.21 21.21 using FEM simulation are shown in Fig. 13 and Fig. 14 respectively. Peak value of fundamental component of supply voltage obtained from FEM simulation study for CSPFM and FPFM is 48.47 and 165.88 respectively. Peak value of fundamental component of supply voltage is calculated from (1) to (13). The data required for these equations is given in Table I. The relationship between peak value of supply 14 12 FPFM CSPFM 2 15 Phase voltage Phase current Torque (Nm) 1 8 6 4 2 oltage () and Current (A) 1 5 5 1 15 2 4 6 8 1 12 14 otor angle (mech. degrees) 2.5 1 1.5 2 2.5 3 3.5 4 Time (seconds) x 1 3 Fig. 12. FEM simulation of full load torque of CSPFM and FPFM Fig. 14. Phase voltage and phase current supplied to FPFM. 1212 Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOOGY BOMBAY. Downloaded on December 4, 28 at 23:3 from IEEE Xplore. estrictions apply.
TABE PEAK AUE OF SUPPY OTAGE () OF CSPFM AND FPFM Parameter CSPFM FPFM Calculated using d-q circuits 49.17 148.24 Obtained from FEM simulation 48.47 165.88 6 5 FEM Simulation d q equivalent circuit voltage s, v d and v q is given by s = vd 2 v2 q (14) Calculated peak value of supply voltage and value obtained from FEM simulation is shown in Table and they are in good agreement. Terminal voltage ( ) 4 3 2 1 A. oltage egulation of CSPFM and FPFM from d q Equivalent Circuits FEM based simulation is carried on CSPFM and FPFM generator at 2 rpm to determine the voltage regulation. The proposed d-q equivalent cicuit is used to caculate the terminal voltage of the machine. The d-q equations for FM generator are given below. v dg = Kω e λ q i d pλ d i d (15) v qg = Kω e λ af i q pλ q i q Kω e d i d (16) λ q = q i q (17) λ d = λ af d i d (18) The data required to calculate the values of v d and v q are given in the Table I. Terminal voltage regulation obtained from FEM simulation and d-q equivalent circuit for CSPFM is shown in Fig. 15, while these plots for FPFM are shown in Fig. 16. 1 2 3 4 5 6 7 8 oad current ( A ) Fig. 16. oltage regulation of FPFM I. POWE DENSITY COMPEISION OF CSPFM AND FPFM Physical dimensions and number of turns/phase are same in both machines, only winding arrangement is changed. Both machines have same rated current. FEM analysis at rated current is performed on both machines to determine the rated torque of both machines. Output torque of CSPFM and FPFM is 5.18 Nm and 1.14 Nm respectively. The torque constant of FM is given by K t = 3 2 P eq 2 Kλ af (19) 28 26 d q equivalent circuit FEM simulation 12 CSPFM FPFM 24 1 Terminal voltage () 22 2 18 16 Torque (Nm) 8 6 4 14 12 2 1 2 4 6 8 1 12 oad current (A) 1 2 3 4 5 6 Speed (rpm) Fig. 15. oltage regulation of CSPFM Fig. 17. Steady state speed torque capability curve of CSPFM and FPFM. 1213 Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOOGY BOMBAY. Downloaded on December 4, 28 at 23:3 from IEEE Xplore. estrictions apply.
Torque constant mainly depends upon λ af. The values of λ af for both machines are shown in Table I. λ af for FPFM is twice as that of CSPFM and hence torque production capability of FPFM is twice as that of CSPFM in constant torque zone. Steady state speed-torque capability curve at same rated current for both machines is deduced and is shown in Fig. 17. Speed higher than base speed is obtained with flux weakening. In the flux weakening region supply voltage and input current are maintained at rated values. i d is increased with speed to reduce the flux in the machine. As i d increases i q has to decrease resulting in reduction in torque capability of the machine. Speed range in constant power region is higher for CSPFM as compared to FPFM. II. CONCUSION Full pitch winding concept for FM is introduced which increases the output power of FM approximately twice that of FM with concentrated stator pole winding. Concept of fictitious electrical gear is introduced. The d-q equivalent circuits for FM are proposed and same are validated with steady state FEM analysis. EFEENCES [1]. P. Deodhar, Savante Anderson, Ion Boldea and T. J. E. Miller, The flux reversal machine : A new doubly salient permanent magnet machine, IEEE Trans. Industry Applications., vol.33, No. 4, pp. 925-934, July/August 1997. [2] C. Wang, S. A. Nasar, I. Boldea Three phase flux reversal machine (FM), IEE Trans. Electrical power application., vol. 146, No. 2, pp.139-146, March 1999. [3] Ion Boldea, Jichum Zhang, S. A. Nasar, Theoretical characterization of flux reversal machine in low speed servo drives-the pole PM configuration. IEEE Trans. Industry Applications., vol. 38, No. 6, pp. 1549-1557, November/December 22. [4] Tae Heoung Kim, Sung Hong Won, Ki Bong and Ju ee, eduction in cogging torque in flux reversal machine by rotor teeth pairing IEEE Trans. on Magnetics. vol. 41, No. 1, pp 3964-3966, october 25. [5] Tae Heoung Kim and Ju ee A study of the design for the flux reversal machine, IEEE Trans. on Magnetics., vol. 4, No. 4, pp. 253-255, July 24 [6] Jianzhong zhang, Ming Cheng, Wei Hua and Xiaoyong Zhu, New approach to power equation for comparison of doubly salient electrical machines, in in Proc.IEEE Industry Applications Annu. meeting., pp. 1178-1185, 26. [7] D. S. More and B. G. Fernandes. Novel three phase flux reversal machine with full pitch winding, Proc. of International conference on power electronics (ICPE 27 ) Daegu, South Korea. pp. 17-112, 27. [8] CEDAT, France Flux 2-D FEM Software,. [9] Pragasan Pillay and. Krishnan Modeling of permanent magnet motor drives. IEEE Trans. on Industrial Electronics.,vol. 35, No. 4, pp. 537-542, November 1998. [1] C. Wang, I. Boldea, S. A. Nasar Characterization of three-phase flux reversal machine as an automotive generator., IEEE Trans. on Energy Conversion., vol. 16, No. 1, pp. 74-8, March, 21. [11] Miller T. J. E. Brushless Permanent Magnet and eluctance Motor Drives. Clarendon Press. Oxford-1989. [12] Gieras J. F. and M. Wing Permanent Magnet Motor Technology. Design and Applications. Marcel Dekker Inc. 22. 1214 Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOOGY BOMBAY. Downloaded on December 4, 28 at 23:3 from IEEE Xplore. estrictions apply.