1364 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 20, NO. 6, NOVEMBER 2005 Analysis of Multiphase Space Vector Pulse-Width Modulation Based on Multiple d–q Spaces Concept Hyung-Min Ryu, Member, IEEE, Jang-Hwan Kim, Student Member, IEEE, and Seung-Ki Sul, Fellow, IEEE Abstract—Multiphase motors are usually designed to have the concentrated winding and nonsinusoidal airgap flux density distribution in order to maximize the torque per ampere. This means that the phase voltage of a multiphase motor has the nonsinusoidal waveform. Accordingly, the conventional analysis on a multiphase space vector pulse-width modulation (SVPWM), which is confined to a sinusoidal phase voltage, should be extended to a nonsinusoidal phase voltage. In this paper, based on a multiple – spaces concept a novel analysis on a multiphase SVPWM to synthesize an arbitrary nonsinusoidal phase voltage is proposed. Throughout this paper, a five-phase inverter is used as a practical example. The basic concepts can be easily extended to an -phase inverter. multiphase SVPWM scheme should be extended to synthesize a nonsinusoidal phase voltage. In this paper, a new analysis on a multiphase SVPWM is presented. Using a multiple – spaces concept, we explain how to realize an arbitrary nonsinusoidal phase voltage in terms of the applying times of available switching vectors. Based on the proposed SVPWM scheme, the analysis on the maximum modulation index of a multiphase inverter is also presented. Throughout this paper, a five-phase inverter is used as a practical example. The basic concepts can be easily extended to an -phase inverter. Index Terms—Multiphase motors, multiphase space vector pulse-width modulation (SVPWM), multiple – spaces concept, nonsinusoidal voltage. II. MULTIPLE – SPACES CONCEPT I. INTRODUCTION K ELLY et al. [1] mentioned an -phase unified pulse-width modulation (PWM) scheme by the extension of a three-phase unified PWM. In an -phase system, the th harmonic voltage offset can be calculated by averaging between the maximum and minimum reference phase voltages. This voltage offset equally divides the applying times of two zero switching vectors within the sampling interval so that the dc-link voltage utilization can be increased compared to a sine-triangle PWM. In [1], Kelly et al. also verified that an -phase space vector PWM (SVPWM) scheme can be described in terms of the applying times of available switching vectors on the basis of the space vector concept. However, the paper only focuses on how to realize a sinusoidal phase voltage. As is widely known, most multiphase motors are designed to have the nonsinusoidal back-EMF voltage in order to increase the torque per ampere [2]–[8]. For example, multiphase permanent magnet motors are usually designed to have the nearly rectangular air-gap flux density distribution and the concentrated winding with a full-pitch in order to maximize the torque per ampere [2]. Toliyat et al. [3] verified that the injection of the third harmonic current in a five-phase induction motor with the concentrated winding enables the air-gap flux density to be nearly trapezoidal so that the torque per ampere can be increased by 10%. In both cases, the back-EMF voltages have the nonsinusoidal waveforms. Therefore, the conventional analysis on a Manuscript received August 25, 2004; revised February 25, 2005. Recommended by Associate Editor J. H. R. Enslin. H.-M. Ryu is with the Automation Laboratory, INTECH Factory Automation Company, Kyonggi-do 449-910, Korea (e-mail: hmryu@intech-fa.co.kr). J.-H. Kim and S.-K. Sul are with the School of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-742, Korea (e-mail: ghks95@eepel.snu.ac.kr; sulsk@plaza.snu.ac.kr). Digital Object Identifier 10.1109/TPEL.2005.857551 In a three-phase motor, all the (6 1)th harmonics of 3)th three-phase variables, with the exception of the (6 harmonics corresponding to zero-sequence components, can be equivalently represented by the following two-dimensional (2-D) complex space vector (hereafter referred to as space vector) on a stationary reference frame [9] (1) where 2 3 . From (1), the space vector on a synchronously rotating reference frame can be defined as follows: (2) where and denotes the fundamental frequency, or the angular velocity of reference field. Using the space vector of (2), the fundamental harmonic of three-phase variables can be regarded as a dc component. This idea is of key importance to the field-oriented vector control and synchronous frame current control of a three-phase motor. Likewise, this space vector concept can be extended to a five-phase motor. Since five-phase variables have four degrees of freedom with the exception of zero-sequence components, two space vectors should be used in order to define the equivalent transformation of five-phase variables. These two space vectors can be reasoned out because a five-phase motor has the following two kinds of phase displacements according to the harmonics of winding density distribution. 1)th harmonics of winding density distribu1) The (10 tion have the displacement of , , , , and phases counterclockwise. 3)th harmonics of winding density distribu2) The (10 tion has the displacement of , , , , and phases counterclockwise. 0885-8993/$20.00 © 2005 IEEE RYU et al.: ANALYSIS OF MULTIPHASE SPACE VECTOR PULSE-WIDTH MODULATION 1365 Fig. 2. Circuit diagram of a five-phase PWM inverter. Fig. 1. Two orthogonal 2-D spaces of a five-phase motor. (a) d –q space. (b) d –q space. Therefore, two space vectors, which can equivalently express , , , , and phase variables except zero-sequence components, can be defined as follows: Fig. 3. Two equivalent load configurations of a five-phase PWM inverter. (a) 1–4. (b) 2–3. respectively, can be defined as follows: (5) (6) (3) (4) where 2 5. The space vectors of (3) and (4) have the following properties. 1)th harmonics 1) Using the space vector of (3), the (10 of five-phase variables can be equivalently expressed as 1) in the ac components with the frequency of (10 stationary – space depicted in Fig. 1(a). 3)th harmonics 2) Using the space vector of (4), the (10 of five-phase variable can be equivalently expressed as 3) in the ac components with the frequency of (10 stationary – space depicted in Fig. 1(b). The number of subscripts used to differentiate two – spaces means the lowest order among the harmonics equivalently transformed to each – space. From (3) and (4), the space vectors in the – and – spaces synchronously rotating with the frequency of and 3 , Using the space vectors of (5) and (6), the fundamental and third harmonics of five-phase variables can be regarded as dc components. 1 -phase motor, it By generalizing this basic concept to a 2 can be easily seen that there exist -orthogonal – spaces, where 1 of phase all the harmonics with the order of less than 2 variables can be equivalently represented as dc components. III. FIVE-PHASE SPACE VECTOR PWM A. Definition of Switching Vectors In a five-phase PWM inverter, as shown in Fig. 2, two different equivalent load circuits can be generated depending on the switching pattern [1]. For example, when the switching pattern is (10000) the equivalent load circuit consists of one impedance in series with a group of four parallel impedances, as shown in Fig. 3(a). Likewise, the switching pattern (11000) generates the two to three equivalent load configuration, as shown in Fig. 3(b). In the switching pattern depicted by a 1366 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 20, NO. 6, NOVEMBER 2005 five-digit binary number, each digit means the switching func, , and in order, where “1” indicates the tion , , upper switch of the corresponding leg is on, while “0” indicates the lower switch is on. , , , , and By (3) and (4), five phase voltages , as decided by the switching patterns, can be expressed as the following switching vectors in – and – spaces: (7) (8) By utilizing the zero-sequence voltage, the switching vectors can be calculated directly from the switching functions (9) (10) denotes dc-link voltage. where Using (9) and (10), thirty nonzero switching vectors, with the exception of the two zero switching vectors (00000) and (11111), can be depicted on – and – planes, as shown in Fig. 4(a) and (b), respectively. B. Selection of Switching Vectors According to the equivalent load configuration and the magnitude, thirty nonzero switching vectors can be classified into the following three sets. — {2–3,1} switching vectors (11001), (11000), (11100), (01100), (01110), (00110), (00111), (00011), (10011), (10001). — {1–4} switching vectors (10000), (11101), (01000), (11110), (00100), (01111), (00010), (10111), (00001), (11011). — {2–3,2} switching vectors (01001), (11010), (10100), (01101), (01010), (10110), (00101), (01011), (10010), (10101). In the name of the switching set, e.g., {2–3,1}, the number following a comma is additionally used in order to differentiate the switching sets with the same equivalent load configuration. And the number “1” means that the magnitude of switching vectors in the corresponding switching set is the greatest among the switching vectors with the same equivalent load configuration on the – plane. From the average vector concept during one sampling period [10], a reference voltage vector on the – plane can be realized by adjusting the applying times of the nearest two {1–4} switching vectors and two {2–3,1} switching vectors. The other combinations of switching vectors increase the number of switching or decrease the maximum magnitude of the realizable voltage vector. It should be noted that only the switching vectors with the maximum magnitude in each equivalent load configuration are normally used. Fig. 4. Thirty nonzero switching vectors on (a) d –q plane and (b) d –q plane. C. Calculation of the Applying Times of Switching Vectors For example, a – axes reference voltage vector located in sector 1, as shown in Fig. 5, can be realized by solving the following complex equation: (11) (12) RYU et al.: ANALYSIS OF MULTIPHASE SPACE VECTOR PULSE-WIDTH MODULATION 1367 Fig. 5. Realization of a reference voltage vector located in sector 1 on the d –q plane. where the unknown , , , and denote the applying times of the switching vectors of (10000), (11000), (11001), and (11101), respectively. The subscript of indicates the number “1” in the corresponding switching pattern, or the applying means order of nonzero switching vectors in ON sequence. the sampling period, or half of the switching period. indicates the applying time of zero voltage due to (00000) and (11111). and represent the magnitudes of {1–4} and {2–3,1} switching vectors, respectively, which can be calculated using (9) as follows: Fig. 6. Realization of a reference voltage vector located on the d –q plane. From (11) and (15), the following matrix equation between the given – – – axes voltages and unknown , , , and can be obtained as follows: (17) (13) (18) (14) Four switching vectors on the – plane corresponding to four switching patterns, which are selected to realize a reference voltage vector located in sector 1 on the – plane, are depicted as shown in Fig. 6. Note that the magnitude of {2–3,1} switching vectors is the greatest on the – plane but the smallest on the – plane. In the same manner as that of (11), a – axes reference voltage vector can be realized by solving the following complex equation: (15) where on the lows: denotes the magnitude of {2–3,1} switching vectors – plane, which can be calculated using (10) as fol- (16) where the subscript of matrix means sector number. of (18) has a full rank, the applying times Since the matrix of nonzero switching vectors are uniquely decided by the following matrix equation: (19) The generalized matrix according to the sector number can be derived like (20), shown at the bottom of the next page. The resultant switching signals during one switching period can be drawn, as shown in Fig. 7. IV. MAXIMUM MODULATION INDEX In this section, the analysis on the maximum realizable output voltage of a five-phase inverter in the linear modulation region is addressed. The modulation index can be defined as the ratio of the amplitude of the fundamental phase voltage to the dc-link voltage like [1] (21) 1368 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 20, NO. 6, NOVEMBER 2005 Fig. 7. Switching signals when d –q axes reference voltage vector is located in sector 1. Fig. 8. Trajectory of the d –q axes voltage vector with the maximum modulation index when the phase voltage is sinusoidal. When the phase voltage is sinusoidal, or the magnitude of the – axes voltage vector is zero, the following relation between the applying times of nonzero switching vectors can be derived from (15) and (16): Fig. 9. Trajectories of the voltage vectors with the maximum modulation index when the magnitude ratio and phase between the d –q and d –q axes voltage vectors are 0.236 and [rad], respectively. (a) d –q plane. (b) d –q plane. can be drawn as the dotted circle in Fig. 8. Then, the maximum modulation index can be calculated as (22) In this case, the trajectory of the – axes voltage vector with the maximum magnitude in the linear modulation region (23) (20) RYU et al.: ANALYSIS OF MULTIPHASE SPACE VECTOR PULSE-WIDTH MODULATION 1369 Fig. 10. Waveforms of the phase voltage, offset voltage, and pole voltage generated from the d –q and d –q axes voltage vectors of Fig. 9. The Y -axis is normalized by dc-link voltage. From (23), it can be seen that the maximum modulation index of a five-phase inverter is reduced by 9% compared with a threephase inverter with the maximum modulation index of 0.5774. However, when the – axes voltage vector rotates along the dotted trajectory of Fig. 9(a) and atan atan rad (24) – axes voltage vector can rotate along the dotted the inscribed circle of the decagon, at the vertexes of which the {2–3,1} switching vectors are located, in Fig. 9(b). Then, the modulation index can be calculated as 10 and the magnitude of the culated as – 0.6155 (25) axes voltage vector can be cal0.1453 (26) In other words, if the – axes voltage vector has the magnitude of 23.6% compared to the – axes voltage vector and defined in (24) is rad , the maximum moduthe phase lation index of a five-phase inverter can be increased by 6.6% compared to that of a three-phase inverter. In this case, the phase voltage has the flat-top waveform, as shown in Fig. 10, and thus in the viewpoint of the triangular comparison PWM scheme using the offset voltage [1], the increase of the maximum modulation index can be easily understood. On the contrary, when the – axes voltage vector rotates is zero, the along the dotted trajectory of Fig. 11(a) and – axes voltage vector will rotate along the dotted inscribed circle of the decagon, at the vertexes of which {1–4} switching vectors are located, in Fig. 11(b). Then, the modulation index can be calculated as 0.3804 and the magnitude of the culated as – (27) axes voltage vector can be cal0.2351 (28) Fig. 11. Trajectories of the voltage vectors with the maximum modulation index when the magnitude ratio and phase between the d –q and d –q axes voltage vectors are 0.618 and 0 [rad], respectively. (a) d –q plane. (b) d –q plane. In this case, the phase voltage has the peak-top waveform as shown in Fig. 12 and thus in the viewpoint of the triangular comparison PWM scheme using the offset voltage the decrease of the maximum modulation index can be easily understood. From the previous results, it is clear that the maximum modulation index of a five-phase PWM inverter depends on the phase voltage waveform, which is decided by the magnitude ratio and between the – and – axes voltage vectors. phase This means that the power density of a five-phase inverter depends on the phase voltage waveform as well as power factor. For example, a five-phase surface-mounted permanent magnet synchronous motor (SMPMM) [2] typically has the maximum modulation index of 0.57 0.59 because the phase voltage has is close to rad . a flat-top waveform, and so the phase On the other hand, a five-phase synchronous reluctance motor 1370 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 20, NO. 6, NOVEMBER 2005 Fig. 12. Waveforms of the phase voltage, offset voltage, and pole voltage generated from the d –q and d –q axes voltage vectors of Fig. 11. Y -axis is normalized by dc-link voltage. TABLE I ELECTRICAL SPECIFICATIONS OF A FIVE-PHASE IPMSM (SRM) with a salient-pole rotor [8] has the maximum modulation index of less than 0.45 because the phase voltage has is close to zero. a peak-top waveform, and so the phase Therefore, in the viewpoint of the power density of motor drive system, a five-phase SRM with a salient-pole rotor has a disadvantage compared to a five-phase SMPMM. V. EXPERIMENTAL RESULT The experiment on synchronous frame current control [11], the output voltage of which is synthesized by using the proposed SVPWM, has been carried out. As a tested machine, a five-phase interior permanent magnet synchronous motor (IPMSM) is used and a three-phase spindle induction motor operating a speed control mode is used as a load machine. Table I shows the electrical specifications of a five-phase IPMSM. Two intelligent power modules (IPMs) are used as the switching devices of a five-phase PWM inverter and the proposed SVPWM algorithm is fully digitally implemented by the control board based on digital signal processor (DSP) TMS320VC33. A serial digital-to-analog converter with four channels is employed in order to observe the control variables stored in memory on an oscilloscope. The switching frequency of SVPWM is 5 kHz and then the current control bandwidth is set to 2000 rad/s. Fig. 13. Experimental result of the proposed SVPWM method when the current of a five-phase motor is regulated at the synchronously rotating d –q and d –q axes. (a) Waveforms of u-phase reference current, u-phase sampled current, and u-phase current error. (b) Trajectories of the sampled current vectors and reference voltage vectors on the stationary d –q and d –q axes. Fig. 13 shows the experimental result when the – – – axes currents on synchronously rotating reference frames are regulated to be 20 [A], 20 [A], 10 [A], and 10 [A]. The motor speed is regulated to be a constant 1000 [r/min] by the load machine. Fig. 13(a) shows the -phase reference current, -phase sampled current, and -phase current error from top to bottom. Fig. 13(b) shows the trajectories of the sampled current vectors and reference voltage vectors on the stationary – and – axes. This experimental result reveals that by employing the proposed SVPWM method the nonsinusoidal phase current including the third harmonic component can be regulated without steady-state error. VI. CONCLUSION In this paper, a novel analysis on a multiphase SVPWM to realize a nonsinusoidal phase voltage has been proposed. The multiple – spaces concept, which can equivalently analyze a multiphase ac motor like a dc motor, is presented. Based on this concept, it is explained that an arbitrary reference voltage vector of a five-phase ac motor can be synthesized in terms of the applying times of available switching vectors in a five-phase PWM inverter. RYU et al.: ANALYSIS OF MULTIPHASE SPACE VECTOR PULSE-WIDTH MODULATION In addition, through the analysis on the maximum modulation index it is shown that unlike a three-phase inverter the maximum modulation index of a five-phase inverter is dependent on the phase voltage waveform according to the third harmonic component. This result means that the power density of a five-phase inverter depends on the phase voltage waveform as well as power factor according to the motor type. The proposed SVPWM method is proved through an experiment using a five-phase IPMSM. 1371 Hyung-Min Ryu (S’00–M’05) was born in Kwangju, Korea, in 1975. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from Seoul National University, Seoul, Korea, in 1997, 2000, and 2004, respectively. He is currently a Research Engineer with INTECH Factory Automation Company, Kyonggi-do, Korea. His current research interests are power electronic control of electric machines and power converter circuits. REFERENCES [1] J. W. Kelly, E. G. Strangas, and J. M. Miller, “Multiphase inverter analysis,” in Proc. IEEE Int. Electric Machines Drives Conf. (IEMDC), 2001, pp. 147–155. [2] L. Parsa and H. M. Toliyat, “Multiphase permanent magnet motor drives,” in Proc. Industrial Application Soc. Annu. Meeting, Oct. 2003, pp. 401–408. [3] H. Xu, H. A. Toliyat, and L. J. Petersen, “Five-phase induction motor drives with DSP-based control system,” IEEE Trans. Power Electron., vol. 17, no. 4, pp. 524–533, Jul. 2002. [4] Y. Kats, “Adjustable-speed drives with multiphase motors,” in Proc. IEEE Int. Electric Machines Drives Conf. (IEMDC), May 1997, pp. TC2/4.1–TC2/4.3. [5] H. A. Toliyat, T. A. Lipo, and J. C. White, “Analysis of a concentrated winding induction machine for adjustable speed drive applications part 1 (motor analysis),” IEEE Trans. Energy Conv., vol. 6, no. 4, pp. 679–683, Dec. 1991. [6] H. A. Toliyat, T. A. Lipo, and J. C. White, “Analysis of a concentrated winding induction machine for adjustable speed drive applications part 2 (motor design and performance),” IEEE Trans. Energy Conv., vol. 6, no. 4, pp. 684–692, Dec. 1991. [7] H. A. Toliyat, T. A. Lipo, and J. C. White, “Analysis of a concentrated winding induction machine for adjustable speed drive applications-experimental results,” IEEE Trans. Energy Conv., vol. 9, no. 4, pp. 695–700, Dec. 1994. [8] H. A. Toliyat, L. Y. Xue, and T. A. Lipo, “A five phase reluctance motor with high specific torque,” IEEE Trans. Ind. Appl., vol. 28, no. 3, pp. 659–667, May/Jun. 1992. [9] D. W. Novotny and T. A. Lipo, Vector Control and Dynamics of AC Drives. Oxford, UK: Oxford Univ. Press, 1996. [10] H. W. Van der Broeck and H. C. Skudelny, “Analysis and realization of a pulse width modulator based on voltage space vectors,” IEEE Trans. Ind. Appl., vol. 24, no. 1, pp. 142–150, Jan./Feb. 1988. [11] H.-M. Ryu, J.-W. Kim, and S.-K. Sul, “Synchronous frame current control of multiphase synchronous motor—part I. Modeling and current control based on multiple d–q spaces concept under balanced condition,” in Proc. Industrial Application Soc. Annu. Meeting, Oct. 2004, pp. 63–70. Jang-Hwan Kim (S’02) was born in Kwangju, Korea, in 1975. He received the B.S. and M.S. degrees in electrical engineering from Seoul National University, Seoul, Korea, in 1999 and 2001, respectively, where he is currently pursuing the Ph.D. degree. His current research interests are power electronic control of electric machines, power quality systems, and power converter circuits. Seung-Ki Sul (S’78–M’87–SM’98–F’00) was born in Korea in 1958. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from Seoul National University, Seoul, Korea, in 1980, 1983, and 1986, respectively. He was with the Department of Electrical and Computer Engineering, University of Wisconsin, Madison, as an Associate Researcher from 1986 to 1988. He was then with Gold-Star Industrial Systems Company as a Principal Research Engineer from 1988 to 1990. Since 1991, he has been a member of the faculty of the School of Electrical Engineering and Computer Science, Seoul National University, where he is currently a Professor. His current research interests are power electronic control of electric machines, electric vehicle drives, and power converter circuits.