Space-Vector PWM Techniques for Dual Three-Phase AC Machine ...

1112

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 4, JULY/AUGUST 2006

Space-Vector PWM Techniques for Dual Three-Phase AC Machine: Analysis, Performance Evaluation, and DSP Implementation Djafar Hadiouche, Lotfi Baghli, and Abderrezak Rezzoug

Abstract—The major drawback of dual three-phase ac machines (6φM), when supplied by a voltage-source inverter (VSI), is the occurrence of extra harmonic currents. These extra currents circulate only in the stator windings and cause additional losses. One solution to reduce their amplitude is to act on the supply side using dedicated pulsewidth modulation (PWM) control strategies. The aim of this paper is to investigate new space-vector PWM (SVPWM) techniques suitable for 6φM and to perform a detailed analysis and a performance evaluation. The proposed performance criteria, specific to VSI-fed 6φM, lead to analytical formulas and graphics, which aid the design of high-performance PWM techniques and demonstrate the superiority of the proposed SVPWM techniques. Experimental results carried out on a 15-kW prototype machine and using a low-cost fixed-point TMS320F240 digital signal processor are given and discussed. Index Terms—Analysis, digital signal processor (DSP), dual three-phase ac machines, multiwinding multiconverter systems, performance criteria, space-vector pulsewidth modulation (SVPWM), voltage-source inverter (VSI).

I. I NTRODUCTION

S

INCE the late 1920s [1], dual three-phase ac machines, which will be referred to as six-phase machines (6φM) throughout this paper, have been used in many applications for their advantages in power segmentation, reliability, and minimized torque pulsations. Such segmented structures are very attractive for high power applications since they allow the use of lower rating power electronic devices able to function at a high switching frequency. Nevertheless, when 6φM are driven by a voltage-source inverter (VSI), large stator circulating harmonic currents occur [2], [3], which add extra losses and require larger semiconductor device ratings. These large harmonic currents

Paper IPCSD-06-036, presented at the 2003 Industry Applications Society Annual Meeting, Salt Lake City, UT, October 12–16 and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Industrial Power Converter Committee of the IEEE Industry Applications Society. Manuscript submitted for review October 15, 2003 and released for publication April 28, 2006. D. Hadiouche is with GE Fanuc Automation Solutions Europe, 6468 Echternach, Luxembourg (e-mail: [email protected]). L. Baghli and A. Rezzoug are with the Groupe de Recherche en Electrotechnique et Electronique de Nancy (G.R.E.E.N.), CNRS UMR 7037, Université Henri Poincaré, 54506 Vandoeuvre-lès-Nancy Cedex, France (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TIA.2006.877737

are caused by the small machine impedance associated with these harmonics [2], [4], [5], which is composed of the stator resistance and leakage inductance only. This problem has been known for quite a long time and has already been studied in detail using matrix theory [2] or symmetrical component theory [4]. Several solutions have been proposed to reduce the amplitude of these circulating harmonic currents, and they can be classified into three categories. The first category acts between the inverter and the machine by adding harmonic filters [12]. The second category acts directly on the machine side by modifying its structure and/or its stator winding [9]–[11]. The third category acts on the supply side with dedicated pulsewidth modulation (PWM) control strategies [6], [7], [11], [18], [19] or special inverter structures [8]. The aim of this paper is to investigate new space vector PWM (SVPWM) techniques suitable for 6φM and to perform a detailed analysis and a performance evaluation. The proposed performance criteria, specific to VSI-fed 6φM, lead to analytical formulas and graphics that aid the design of highperformance PWM techniques and demonstrate the superiority of the proposed SVPWM techniques. Implementation on a lowcost fixed-point TMS320F240 digital signal processor (DSP) is discussed, and experimental results carried out on a 15-kW prototype machine are given. II. M ACHINE M ODEL To derive the machine model of a six-phase induction machine (6φIM), the following assumptions have been made: space harmonics, magnetic saturation, and core losses are neglected. In the same way as for three-phase systems, the use of a (6 × 6) transformation matrix [2], [6] allows the stator variables to be expressed in an orthogonal base. The transformation used here was proposed in [7] and [10]. Using this transformation, the original six-dimensional (6-D) stator system can be decomposed into three two-dimensional (2-D) uncoupled subsystems. These are the usual (d−q) one, a zero-sequence (o1 −o2 ) one, and another nonelectromechanical energy conversion related one named (x−y). The transformation has the property to separate harmonics into different groups and to project them into each subsystem. The (x−y) components are responsible for the large circulating harmonic currents. So, the impedance related to these components should be as high as possible, and/or the applied voltages should contain minimum amplitude

0093-9994/$20.00 © 2006 IEEE

HADIOUCHE et al.: SPACE VECTOR PWM TECHNIQUES FOR DUAL THREE-PHASE AC MACHINE

1113

(x−y) components. The final model expressed in the stationary reference frame is given by [10], [11]       0 0 0 Rs vsd •  isd •  Rr M θ1 Lr θ1   ird   vrd   0   =  vsq 0 Rs 0  isq  0 • • vrq irq −M θ −L θ 0 R 1

r

1



r

 0 isd Ls M 0 0  d  ird   M Lr 0 + (1)    0 0 Ls M dt isq irq 0 0 M Lr          d isx Rs 0 isx Llsxy 0 vsx = + vsy isy 0 Rs 0 Llsxy dt is y 



    Rs 0 iso1 vso1 0  vso2  =  0 Rs 0   iso2  vro iro 0 0 Rr     Llso iso1 0 0 d + 0 Llso 0   iso2  dt iro 0 0 Llr 

T e = pM (isq ird − isd irq ) dθ1 = ΩM dt 1 dΩM = (T e − T c) dt J

Fig. 1. Six-phase VSI-fed 6φIM.

(2)

(3) (4) (5) (6)

√ where Ls = Llsdq + 3Lms , M = 3Msr / 2, and Lr = Llr + (3/2)Lmr . Llsdq is the stator leakage inductance of the (d−q) equivalent circuit, and Llsxy for the (x−y) equivalent circuit [10], [11]. III. SVPWM C ONTROL OF D UAL T HREE -P HASE AC M ACHINE The SVPWM techniques developed in this paper, which will be referred as 6φSVPWM, are based on the powerful vector space decomposition proposed in [6], where much of the theoretical foundation for this concept and the associated PWM control strategy were established. This paper extends the results given in [6] and addresses new discontinuous PWM techniques and a complete analytical study on performance evaluation. The drive system is a six-phase VSI-fed 6φIM, as shown in Fig. 1. A combinatorial analysis of the inverter switch state shows 64 switching modes. So, 64 different voltage vectors can be applied to the machine. By using the (6 × 6) transformation, we can decompose them into the (d−q), (x−y), and (o1 −o2 ) voltages. The (o1 −o2 ) ones are all equal to zero because the two three-phase windings are wye connected with isolated neutrals. So the aim of the PWM is to control four variables at the same time during each sampling period and generate maximum (d−q) voltages and minimum (x−y) voltages. The choice of particular switching modes allows satisfying these two conditions. By choosing the switching modes that permit to have the maximum amplitude (d−q) voltage vectors, we

Fig. 2. Inverter voltage vectors on (d−q) (x−y) planes corresponding to the chosen switching modes.

obtain the planes shown in Fig. 2, where each switching mode is represented by a decimal number corresponding to the binary number (Kc2 Kb2 Ka2 Kc1 Kb1 Ka1 ). This binary number gives the state of the upper switches. Since there are four variables to control, five voltage vectors V1 , V2 , V3 , V4 , and V5 need to be chosen with unique and positive solutions [6]. This can be done using   Vsd1 tV1  tV2   Vsq1     tV3  =  Vsx1    tV4 Vsy1 tV5 1 

Vsd2 Vsq2 Vsx2 Vsy2 1

Vsd3 Vsq3 Vsx3 Vsy3 1

Vsd4 Vsq4 Vsx4 Vsy4 1

 ∗   vsd T s Vsd5 −1 ∗ Vsq5   vsq T s   ∗   Vsx5  ·  vsx Ts  ∗   Vsy5 vsy T s 1 Ts (7)

where tVi is the dwell time of voltage vector Vi during each sampling period T s. IV. A NALYSIS OF PWM C ALCULATION The following analysis is made for a better understanding of 6φSVPWM and also for its optimal DSP implementation with reduced complexity and low execution time algorithm. The computational effort needed to solve (7) is known to be important [6], [18], [19]. Therefore, maximal offline calculation of (7) has been investigated. Taking into account the fact that the (x−y) reference voltage vector is chosen to be zero and that the

1114

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 4, JULY/AUGUST 2006

TABLE I COEFFICIENTS Aji IN MATRIX [V ]−1 OF (8)

TABLE III CALCULATION OF DWELL TIME tVi ACCORDING TO SECTOR NUMBER

TABLE II COEFFICIENTS Ki IN TABLE I

V. S WITCHING S EQUENCES

fifth vector is chosen from the four zero vectors, there are a total of 11 coefficients Aji in   tV1 A11 A  tV2   1  12    tV3  =  A13  det[V ]   tV4 A14 tVzero A15 

A21 A22 A23 A24 A25

× × × × ×

  ∗  vsd × 0 ∗ × 0   vsq     × 0  ·  0 Ts    × 0 0 × A55 1 (8)

which need to be calculated for each sector of the (d−q) plane. Detailed analysis (given in Appendix A) shows that these coefficients are made from the reduced set of five constants listed in Table II, arranged as shown in Table I. Evaluating (8) based on Tables I and II, it is found that only the following six different times need to be calculated for each sampling period:  √    −( 3 − 2) −1 T1 √ √  ( 3 − 1) −( 3 − 1)      T2  √    ∗ 1 ( 3 − 2)  Ts 1    vsd  T3  √   · ∗ . (9)  =  T4  VDC 2  1 −( 3 − 2)  vsq  √  √   T5  ( 3 − 1) ( 3 − 1)  √ T6 −( 3 − 2) 1 The dwell times of the four nonzero vectors can then be found be selecting from these six times according to Table III. This analysis leads to a significant reduction of the arithmetic operations needed to solve (7) and hence to program the necessary duty cycles for 6φSVPWM.

Since five vectors must be applied during the sampling period T s, there are many choices for the arrangement of these vectors. However, many of them do not minimize the (x−y) harmonic currents and instantaneous switching frequency. The method proposed in this paper is to choose switching sequences in such a way that on the (x−y) plane, two consecutive nonzero vectors are practically opposite in phase [14]. In this way, each change of applied vectors will lead to a succession of increase and decrease in (x−y) currents around zero. The remaining differences between possible switching sequences are the selection and placement of zero vectors during the sampling period. The switching sequences proposed in this paper lead to continuous and discontinuous modulation techniques, and consequently, to different harmonic distortion characteristics. A modulation technique is continuous when on/off switchings occur within every sampling period for all inverter legs and all sectors. A modulation technique is discontinuous when one (or more) inverter leg stops switching, i.e., the corresponding phase voltage is clamped to the positive or negative dc rail for at least one sector [21], [24]. The selection and placement of zero vectors are shown in Table IV. A. Continuous Modulation When the reference voltage vector v ∗sdq is in sector IV, for example, continuous modulation (C6φSVPWM) is obtained with the switching sequence |0−11−27−63−26−18−0|0−18−26−63−27−11−0|. A deeper look into Table IV shows that additional switchings occur in the first three-phase legs (Kc1 Kb1 Ka1 ) when transitioning from an even sector number to an odd sector number, and in the second three-phase legs (Kc2 Kb2 Ka2 ) when transitioning from an odd sector number to an even sector number. For example, from sector II to sector III, the zero vector changes from V63 to V56 , and hence (Kc1 Kb1 Ka1 ) switch from 1 to 0. A total of six such switchings (three pulses) occur over a

HADIOUCHE et al.: SPACE VECTOR PWM TECHNIQUES FOR DUAL THREE-PHASE AC MACHINE

1115

TABLE IV SELECTION AND PLACEMENT OF ZERO VECTORS ACCORDING TO SECTOR NUMBER

fundamental period T = 1/f of v ∗sdq . Therefore, to achieve an inverter average switching frequency over fundamental period T that is the same as the carrier frequency of sine-triangle PWM (STPWM), the sampling period of C6φSVPWM is given by T sC6φSVPWM = T sSTPWM /(1 − 3T sSTPWM /T ) ≈ T sSTPWM ,

T sSTPWM T

can be simply determined by counting all phase leg switchings within only one sampling period instead of counting them over a complete fundamental period. Fig. 3 shows that C6φSVPWM has a total of 12 switchings per sampling period, D6φSVPWM-A has eight, and D6φSVPWM-B has six. Therefore, the coefficients kf are given by kfA = 8/12 = 2/3

kfB = 6/12 = 1/2.

where T sSTPWM is the carrier period of STPWM. B. Discontinuous Modulations 1) Sequence A: When reference voltage vector v ∗sdq is in sector I, for example, discontinuous modulation (D6φSVPWM-A) is obtained with the sequence |0 − 11 − 27 − 26 − 18 − 0|0 − 18 − 26 − 27 − 11 − 0|.

To maintain the same average switching frequency as for STPWM, the sampling period of D6φSVPWM-A is given by T sD6φSVPWM-A = kfA T sSTPWM /(1 − 3T sSTPWM /T ) = kfA T sC6φSVPWM ≈ kfA T sSTPWM ,

T sSTPWM T.

Similarly, the sampling period of D6φSVPWM-B is given by 2) Sequences B1 and B2: D6φSVPWM-B1 is obtained with the sequence |0 − 11 − 27 − 26 − 18|18 − 26 − 27 − 11 − 0|. D6φSVPWM-B2 is obtained with the sequence

T sD6φSVPWM-B = kfB T sSTPWM /(1 − 3T sSTPWM /T ) = kfB T sC6φSVPWM ≈ kfB T sSTPWM ,

T sSTPWM T.

|11 − 27 − 26 − 18 − 7|7 − 18 − 26 − 27 − 11|. Since D6φSVPWM-B1 and B2 have the same harmonic characteristics, they will be referred to as D6φSVPWM-B. The switching sequences of C6φSVPWM, D6φSVPWM-A, and D6φSVPWM-B for sector IV are depicted in Fig. 3. 3) Sampling Period Reduction Coefficients kf : When comparing the performances of C6φSVPWM and D6φSVPWM, to account for the reduction in the total number of per fundamental cycle switchings of the D6φSVPWM techniques, a sampling period reduction coefficient kf is introduced as kf =

Total Nb of switchings D6φSVPWM . Total Nb of switchings C6φSVPWM

Since the number of switchings is symmetrically distributed among all sectors and inverter phase legs, the coefficient kf

VI. P ERFORMANCE E VALUATION A. Performance Criteria for VSI-Fed 6φIM 1) Maximum Modulation Index: The modulation index m is defined as the ratio of the fundamental component magnitude of the line to neutral inverter output voltage to the fundamental component magnitude of the six-step mode voltage [22], [24]. For three-phase PWM techniques, the maximum modulation index mMAX is one criterion used for performance evaluation. STPWM and 3φSVPWM start √ to saturate at mMAX1 = π/4 ≈ 0.785 and mMAX2 = π/(2 3) ≈ 0.907, respectively [22]. The maximum modulation index of 6φSVPWM can be obtained by solving tVzero = 0. In the range [−π/12; +π/12] of sector I, for example, tVzero is minimum for θ = 0; therefore, tVzero = 0 is solved for θ = 0. It is obtained that 6φSVPWM

1116

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 4, JULY/AUGUST 2006

Fig. 3. Switching sequences of continuous and discontinuous 6φSVPWM when the reference voltage vector vsdq ∗ is located in sector IV.

starts to saturate at mMAX2 . Note that if only two adjacent nonzero vectors and one zero vector are selected, mMAX reaches a higher value [3], but in this case, there is no control on the (x−y) plane, so considerable fifth and seventh harmonic currents appear [3], [6], [19]. 2) Local rms Harmonic Currents and Harmonic Fluxes: The total per-switching cycle (“local”) square rms harmonic current is defined as [20], [21], [23], [24] 2 Ihrms,T c (m, θ)

1 = Tc

(n+1)T c

i2sa1 +i2sb1 +i2sc1 +i2sa2 +i2sb2 +i2sc2 dt

(10)

nT c

where isa1 = isa1 − ¯isa1 , isb1 = isb1 − ¯isb1 , . . . , are the harmonic components of stator currents. T c, the switching period, is equal to T s for 6φSVPWM and to 2∗ T s for STPWM. It is demonstrated that after applying the (6 × 6) transformation to stator harmonic currents, (10) becomes 2 Ihrms,T c (m, θ) =

1 Tc

(n+1)T c

i2sd + i2sq + i2sx + i2sy dt. (11)

nT c

Using harmonic current space vectors, (11) becomes 2 Ihrms,T c (m, θ) =

1 Tc

(n+1)T c

isdq · i∗sdq + isxy · i∗sxy dt.

(12)

nT c

Since the (d−q) equivalent circuit of 6φIM is the same as the three-phase induction machine, the (d−q) harmonic current space vector trajectory during T s can be calculated with [23]–[25] 

1  isdq = 1 (13) Vsdqk − v ∗sdq dt = λsdq σLs σLs where σLs is the total referred-to-stator leakage inductance, and λsdq is the (d−q) harmonic flux vector introduced in [24]. From (2), the (x−y) harmonic current space vector trajectory during T s can be approximated with 1 1 λsxy (14) (Vsxyk ) dt = isxy = Llsxy Llsxy where λsxy is the (x−y) harmonic flux vector. Equations (13) and (14) are accurate approximations provided that the sampling frequency fs is much higher than the fundamental frequency f [23]–[25], and that the smallest stator

time constant (τxy = Llsxy /Rs in 6φIM) is higher than the sampling period [25]. Equations (12)–(14) result in 2 Ihrms,T c (m, θ) 2 (n+1)T  c

1 1 ∗ +k 2 ·λsxy ·λ∗ sdq · λ = λ sdq σxy sxy dt T c σLs nT c  2

2  λb 2 = λdqhrms,T c (m, θ)+kσxy ·λ2xyhrms,T c (m, θ)  σLs 2 λb λ2hrms,T c (m, θ) (15) = σLs √ 2 where kσxy = (σLs /Llsxy )2 , and λb = (2 3VDC /(πf s)). This result is important since it shows that the harmonic characteristics of a VSI-fed 6φM cannot be determined by the PWM technique alone without taking into account the machine. Whereas for the three-phase case the modulation itself is sufficient [22], [24], for 6φIM, the parameter kσxy is necessary to evaluate and compare the performances of PWM techniques. kσxy characterizes the importance of harmonic distortion due to (x−y) currents. Its value varies essentially with inductance Llsxy , which strongly depends on the coil pitch of stator windings [10]. For a full-pitch winding, Llsxy is large because the leakage coupling between the two stator three-phase windings is weak. Consequently, the value of kσxy is small. For contrary reasons, it is large for a 5/6-pitch winding. 3) Global rms Harmonic Currents and Harmonic Fluxes: Averaging (15) over a fundamental period [20], [21], [23], [24] results in the total per-fundamental cycle (“global”) square rms harmonic current  2 λb 1 2 λ2hrms,T c (m, θ)dθ Ihrms (m) = σLs 2π 2π  2

2  λb 2 = λdqhrms (m) + kσxy · λ2xyhrms (m) σLs  2 λb = λ2hrms (m). (16) σLs

B. Performance Comparison The per-sampling cycle and per-fundamental cycle square rms normalized harmonic fluxes, defined in (15) and (16), have been calculated for each PWM technique using the software Mathematica. This involves significant algebraic computations and yields the analytical formulas summarized in Appendix A. The harmonic fluxes are illustrated in Figs. 4–6. Fig. 6 reveals that, for a 6φIM with low value of kσxy (full-pitch

HADIOUCHE et al.: SPACE VECTOR PWM TECHNIQUES FOR DUAL THREE-PHASE AC MACHINE

1117

Fig. 5. (d−q) and (x−y) per-fundamental cycle normalized square rms harmonic fluxes as a function of m for kfA = 2/3 and kfB = 1/2 (same inverter average switching frequency).

Fig. 6. Per-fundamental cycle normalized square rms harmonic flux as a function of m for kfA = 2/3 and kfB = 1/2 (same inverter average switching frequency). kσxy = 2.5: machine with low leakage coupling between stator three-phase windings. kσxy = 10: machine with high leakage coupling.

with D6φSVPWM-A in the lower half of the modulation range, and the best performance in the upper half.

VII. S IMULATION R ESULTS Fig. 4. (d−q) and (x−y) per-sampling cycle normalized square rms harmonic fluxes as a function of θ and m for kfA = 1 and kfB = 1 (same sampling frequency).

winding), STPWM and D6φSVPWM have comparable performance. D6φSVPWM-B exhibit the best performance in the high modulation range. An optimal PWM can be obtained with a transition between STPWM, D6φSVPWM-A, and D6φSVPWM-B. For a 6φIM with high value of kσxy (5/6-pitch winding), D6φSVPWM-B exhibit comparable performance

Fig. 7 illustrates an example of the simulation results obtained with the parameters of a 1.4-MW 5/6-pitch winding 6φIM (kσxy = 11). For a modulation index m = 0.736, it can be observed that discontinuous modulations (D6φSVPWM-A and D6φSVPWM-B) give better results than continuous modulations (C6φSVPWM and STPWM) in terms of phase current harmonic minimization, as previously shown in Fig. 6. This is due to their ability to control and maintain minimum harmonic current magnitude on the (x−y) plane. D6φSVPWM-B has smaller (x−y) harmonic currents than D6φSVPWM-A

1118

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 4, JULY/AUGUST 2006

Fig. 7. Simulation results for m = 0.736 and f = 44 Hz (constant V /f control), f p = 3.3 kHz, kfA = 2/3, kfB = 1/2, and kσxy = 11 (5/6-pitch winding). From left to right: STPWM, C6φSVPWM, D6φSVPWM-A, D6φSVPWM-B1, and D6φSVPWM-B2. From top to bottom: (d−q) plane current trajectory, (x−y) plane current trajectory, phase current, and electromagnetic torque.

Fig. 8. DSP PWM outputs. From top to bottom: Kc2 , Kb2 , Ka2 , Kc1 , Kb1 , and Ka1 . From left to right: C6φSVPWM (f s = 5 kHz), D6φSVPWM-A (f s = 7.5 kHz), D6φSVPWM-B1 (f s = 10 kHz).

but higher torque ripple magnitude. These results suggest the following: since the PWM control capability must be shared between (d−q) and (x−y) planes, an improvement of the control on the (x−y) plane inevitably affects the control on the (d−q)

plane, and vice-versa. D6φSVPWM-A therefore constitutes a good compromise between harmonic current and torque ripple minimization. Thus, in the experimental results, we will mainly focus on D6φSVPWM-A and C6φSVPWM.

HADIOUCHE et al.: SPACE VECTOR PWM TECHNIQUES FOR DUAL THREE-PHASE AC MACHINE

1119

Fig. 9. Phase currents isa1 and isa2 for m = 0.6 and f = 30 Hz (V /f control). Left: C6φSVPWM (f s = 5 kHz). Right: D6φSVPWM-A (f s = 7.5 kHz). Full-pitch winding (kσxy = 1.58).

Fig. 10. Phase current and FFT for m = 0.6 and f = 30 Hz (V /f control). Full-pitch winding (kσxy = 1.58). Top: C6φSVPWM (f s = 5 kHz). Bottom: D6φSVPWM-A (f s = 7.5 kHz).

VIII. E XPERIMENTAL R ESULTS The experimental bench is composed of a VSI feeding a 15-kW 6φIM prototype and a fixed-point TMS320F240 DSP for the control part. The 6φIM prototype (see Appendix C) allows many different connections of the stator coils; therefore, experimental tests were carried out with full-pitch and 5/6-pitch stator winding configurations, which, respectively, give the minimum and the maximum values of kσxy for this machine. Usually, a transition (from low to high or high to low) occurs in one of the DSP PWM outputs when the value of its associated compare register (loaded with the duty-cycle value) matches the value of the up/down counting timer, whose period equals the sampling period [27], [28]. This leads to one transition during each sampling period. Since for the proposed 6φSVPWM, power electronic devices need to switch more than once each sampling period in order to realize the

switching sequences, the compare value needs to be changed inside the sampling period, just after the first compare match. This is possible with additional compare match tests without additional calculation. Another way to obtain the proposed switching sequences is to program six PWM outputs with only one transition during the sampling period and to use external logic devices (for example, XOR functions). PWM outputs for C6φSVPWM, D6φSVPWM-A, and D6φSVPWM-B1 have been successfully tested and are given in Fig. 8. The phase currents for C6φSVPWM and D6φSVPWM-A are shown in Figs. 9 –11. Fast Fourier transform (FFT) analysis (Figs. 10 and 11) clearly shows that the harmonics are reduced when switching from a continuous PWM (top figures) to a discontinuous one (bottom figures). It is also confirmed that the 5/6-pitch configuration increases the current harmonics with respect to

1120

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 4, JULY/AUGUST 2006

Fig. 11. Phase current and FFT for m = 0.6 and f = 30 Hz (V /f control). 5/6-pitch winding (kσxy = 11.06). Top: C6φSVPWM (f s = 5 kHz). Bottom: D6φSVPWM-A (f s = 7.5 kHz).

the full-pitch configuration, as shown in Fig. 6. This is due to a higher value of kσxy (11.06 compared with 1.58). Of course, in order to compare these PWMs, the same average switching frequency was maintained thanks to the sampling period reduction coefficient kf (see Section V). It is particularly noted that the earlier analysis presented in this paper achieved a significant reduction in computational effort, making possible a PWM sampling frequency of 10 kHz with the experimental controller that was used. IX. C ONCLUSION In this paper, new SVPWM techniques suitable for VSIfed 6φM have been investigated. The proposed 6φSVPWM is based on the vector space decomposition technique in which four variables, namely 1) vsd , 2) vsq , 3) vsx , and 4) vsy , are controlled simultaneously. A detailed analysis of this technique reveals that the computational effort can be significantly reduced. According to the position of zero vectors during the sampling period, continuous and discontinuous modulations have been proposed. In order to evaluate and compare the performance of 6φSVPWM techniques, new performance criteria, specific to 6φM, have also been proposed. These criteria lead to analytical formulas and graphics, which aid the design of high-performance PWM techniques. It has been demonstrated that for a VSI-fed 6φM, the performance of a PWM technique depends on the machine parameter kσxy . This parameter is defined as the ratio of the total (d−q) referred-to-stator leakage inductance to the (x−y) leakage inductance. It is also shown that a transition between STPWM and the proposed D6φSVPWM-A and D6φSVPWM-B leads to optimal modulation for a low value of kσxy . For a high

value of kσxy , the proposed D6φSVPWM-B exhibits the best performance. It is hoped that this paper will stimulate new ideas and works in the area of multiphase PWM control strategies, multiphase vector control ac machine drives, and other multimachine multiconverter systems. A PPENDIX A This Appendix presents the derivation of the coefficients listed in Table I using the target vector shown in sector I of Fig. 2 as an example. When the reference voltage vector v ∗sdq is located in this sector, (7) becomes   Vsd45 tV45  tV41   Vsq45     tV9  =  Vsx45    tV11 Vsy45 tVzero 1 

Vsd41 Vsq41 Vsx41 Vsy41 1

Vsd9 Vsq9 Vsx9 Vsy9 1

Vsd11 Vsq11 Vsx11 Vsy11 1

  ∗  0 −1 vsd ∗ 0   vsq     0   0  · T s.    0 0 1 1

The voltage values in this matrix are obtained by applying the (6 × 6) transformation taken from [6] to the four phase voltage vectors so that VDC = √ 2 3 √ √ √  √   ∗ 3+2 3+2 √3+1 0 −1 vsd √3 + 1 ∗  − √3−1 √ −1 √ 1 √3+1 0  vsq      × −√3+1 − 3+2 − 3+2 −√3+1 0  ·  0  ·T s.     3−1 −1 1 − 3+1 0 0 1 1 1 11 1

HADIOUCHE et al.: SPACE VECTOR PWM TECHNIQUES FOR DUAL THREE-PHASE AC MACHINE

1121

After matrix inversion, this expression becomes =

6 4 2VDC  −V 3

√ 3−2) 6 √ 3 VDC ( 3−1) 6 √ 3 VDC ( 3−1) 6 √ 3 −VDC ( 3−2) 6 2V 4 − 6DC DC (

    ×    

3 VDC 6 √ 3 −VDC ( 3−1) 6 √ 3 VDC ( 3−1) 6 3 VDC 6

−

0

×

×

×

×

×

×

×

×

×

×

0



  v∗  sd 0  ∗   vsq     0 Ts 0 ·    0  0  1

4 2VDC 6

which defines the values of the 11 coefficients Aji in (8) for sector I, as listed in Table I. A similar calculation process can be repeated offline for all 12 sectors to obtain all the coefficients listed in Tables I and II.

Fig. 12. Machine prototype used for the PWM tests.

D6φSVPWM-B    √  1 −465 + 167 3 2 √ m3 m + 3 18 2π 2  √   √ 63 − 21 3 + 36π − 8 3π m4 + 8π 3  √   3 −507 + 293 √ λ2xyhrms (m) = kfB2 m3 . 2 18 2π λ2dqhrms (m)

A PPENDIX B The per-fundamental cycle square rms normalized harmonic fluxes have been calculated for each PWM technique using (15) and (16). This results in the following m-dependent analytical formulas: STPWM  √      1 1 (6+5 2) 2 2 m3 + m4 λdqhrms (m) = m + − √ 48 4π 2 18 3π 2  √  2 −6+5 √ λ2xyhrms (m) = m3 . 2 18 3π C6φSVPWM   √ √  −4 3+3π 4(−96+53 3) 2 √ m + m3 16π 18 2π 2  √ √  3 (−27+27 3−22π+8 3π + − m4 2 8π 3    √  √ 3 3π) 7−4 4(−201+116 √ m2 + m3 . λ2xyhrms (m) = 24 18 2π 2 

λ2dqhrms (m) =

D6φSVPWM-A λ2dqhrms (m)    √  1 −141 + 59 3 2 2 √ m3 = kfA m + 12 18 2π 2    √ √ (−27 + 21 3 − 24π + 8 3π) + − m4 8π 3 λ2xyhrms (m)  =

kfA2

√   −507 + 293 3 √ m3 . 18 2π 2

= kfB2

A PPENDIX C The machine on which the tests were run is a rewound eightpole 15-kW squirrel-cage induction motor (Fig. 12). It has the leads from each of its 48 stator coils brought out to a terminal board. The 96 terminals can, therefore, be connected in many different ways for different pole numbers, phase numbers, and coil pitches, which make possible the full-pitch and 5/6-pitch connections used for the PWM tests. R EFERENCES [1] P. L. Alger, E. H. Freiburghouse, and D. D. Chase, “Double windings for turbine alternators,” Trans. AIEE, vol. 49, pp. 226–244, Jan. 1930. [2] M. A. Abbas, R. Christen, and T. M. Jahns, “Six-phase voltage source inverter driven induction motor,” IEEE Trans. Ind. Appl., vol. IA-20, no. 5, pp. 1251–1259, Sep./Oct. 1984. [3] K. Gopakumar, V. T. Ranganathan, and S. R. Bhat, “Split phase induction motor operation from PWM voltage source inverter,” IEEE Trans. Ind. Appl., vol. 29, no. 5, pp. 927–932, Sep./Oct. 1993. [4] E. A. Klingshirn, “High phase order induction motors: Part I—Description and theoretical considerations,” IEEE Trans. Power App. Syst., vol. PAS102, no. 1, pp. 47–53, Jan. 1983. [5] ——, “High phase order induction motors: Part II—Experimental results,” IEEE Trans. Power App. Syst., vol. PAS-102, no. 1, pp. 54–59, Jan. 1983. [6] Y. Zhao and T. A. Lipo, “Space vector PWM control of dual three phase induction machine using vector space decomposition,” IEEE Trans. Ind. Appl., vol. 31, no. 5, pp. 1100–1109, Sep./Oct. 1995. [7] D. Hadiouche, H. Razik, and A. Rezzoug, “Study and simulation of space vector PWM control of double-star induction motors,” in Proc. IEEECIEP, Acapulco, Mexico, Oct. 15–19, 2000, pp. 42–47. [8] K. Oguchi, A. Kawaguchi, T. Kubota, and N. Hoshi, “A novel sixphase inverter system with 60-step output voltages for high-power motor drives,” IEEE Trans. Ind. Appl., vol. 35, no. 5, pp. 1141–1149, Sep./Oct. 1999. [9] L. Xu and L. Ye, “Analysis of a novel winding structure minimizing harmonic current and torque ripple for dual six-step converter-fed high

1122

[10]

[11]

[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 4, JULY/AUGUST 2006

power AC machines,” IEEE Trans. Ind. Appl., vol. 31, no. 1, pp. 84–90, Jan./Feb. 1995. D. Hadiouche, H. Razik, and A. Rezzoug, “On the modeling and design of dual-stator windings to minimize circulating harmonic currents for VSI fed AC machines,” IEEE Trans. Ind. Appl., vol. 40, no. 2, pp. 506–515, Mar./Apr. 2004. D. Hadiouche, “Contribution to the study of dual stator induction machines: Modelling, supplying and structure,” (in French), Ph.D. dissertation, GREEN, Faculty Sci. Tech., Univ. Henri Poincaré-Nancy I, Vandoeuvre-lès-Nancy, France, Dec. 2001. E. A. Klingshirn, “Harmonic filters for six-phase and other multiphase motors on voltage source,” IEEE Trans. Ind. Appl., vol. IA-21, no. 4, pp. 588–594, May/Jun. 1985. T. A. Lipo, “A d-q Model for Six Phase Induction Machines,” in Proc. Int. Conf. Elec. Mach., Athens, Greece, 1980, pp. 860–867. D. Hadiouche, H. Razik, and A. Rezzoug, “Modelling of a double-star induction motor for space vector PWM control,” in Proc. ICEM, Espoo, Finland, Aug. 28–30, 2000, vol. 1, pp. 392–396. G. Gierse and W. Schuermann, “Microprocessor control for two magnetically coupled three-phase PWM inverters,” IEEE Trans. Power Electron., vol. PE-1, no. 3, pp. 141–147, Jul. 1986. J. W. Kelly, A. G. Strangas, and J. M. Miller, “Multi-phase inverter analysis,” in Proc. IEMDC, Cambridge, MA, Jun. 17–20, 2001, CD-ROM. Z. Ye, D. Boroyevich, and F. C. Lee, “Modeling and control of zerosequence current in parallel multi-phase converters,” in Proc. IEEE PESC, Galway, Ireland, Jun. 18–23, 2000, CD-ROM. A. R. Bakhshai, G. Joos, and H. Jin, “Space vector PWM control of a split-phase induction machine using the vector classification technique,” in Proc. IEEE APEC, vol. 2, pp. 801–808. R. Bojoi, A. Tenconi, F. Profumo, G. Griva, and D. Martinello, “Complete analysis and comparative study of digital modulation techniques for dual three-phase AC motor drives,” in Proc. IEEE PESC, 2002, CD-ROM. V. Blasko, “Analysis of a hybrid PWM based on modified space-vector and triangle-comparison methods,” IEEE Trans. Ind. Appl., vol. 33, no. 3, pp. 756–764, May/Jun. 1997. J. W. Kolar, H. Ertl, and F. C. Zach, “Influence of the modulation method on the conduction and switching losses of a PWM converter system,” IEEE Trans. Ind. Appl., vol. 27, no. 6, pp. 1063–1075, Nov./Dec. 1991. J. Holtz, “Pulsewidth modulation for electronic power conversion,” Proc. IEEE, vol. 82, no. 8, pp. 1194–1214, Aug. 1994. J. Holtz and B. Beyer, “Optimal pulsewidth modulation for AC servos and low-cost industrial drives,” IEEE Trans. Ind. Appl., vol. 30, no. 4, pp. 1039–1047, Jul./Aug. 1994. A. M. Hava, J. R. Kerkman, and T. A. Lipo, “Simple analytical and graphical methods for carrier-based PWM-VSI drives,” IEEE Trans. Power Electron., vol. 14, no. 1, pp. 49–61, Jan. 1999. S. Fukuda and K. Suzuki, “Harmonic evaluation of two-levels carrierbased PWM methods,” in Proc. EPE, Trondheim, Norway, 1997, vol. 2, pp. 2.331–2.336. D. Hadiouche, H. Razik, and A. Rezzoug, “Design of novel winding configurations for VSI fed dual-stator induction machines,” in Proc. ELECTRIMACS, Montreal, QC, Canada, Aug. 18–21, 2002, CD-ROM. “TMS320F/C240DSP controllers reference guide, peripheral and specific devices,” Texas Instruments, Dallas, TX, Literature Number SPRU161C, 1999. “Implementation of a speed field oriented control of three phase AC induction motor using TMS320F240,” Texas Instruments, Dallas, TX, Literature Number BPRA076, Mar. 1998.

Djafar Hadiouche received the Ph.D. degree in electrical engineering from the University Henri Poincaré, Nancy, France, in 2001. Until 2002, he was an Assistant Lecturer at the University Henri Poincaré and conducted research in the laboratory “G.R.E.E.N.,” where his main research interests concerned multiphase ac machines, their modeling, identification, PWM techniques, and vector control. Since 2003, he has been a Motion Specialist Engineer with GE Fanuc Automation Solutions Europe, Echternach, Luxembourg. His main activities include servo sizing, tools and motion programs development, electronic cam profiling, and motion technical training. Dr. Hadiouche received the Best Prize Paper Award from the Electric Machine, Committee at the 2001 IEEE Industry Application Society Annual Meeting.

Lotfi Baghli received the Electrical Engineer degree (with honors) from the Ecole Nationale Polytechnique of Algiers, Algiers, Algeria, in 1989, and the D.E.A. and Ph.D. degrees in electrical engineering from the Université Henri Poincaré, Nancy, France, in 1995 and 1999, respectively. He is currently a Lecturer with IUFM de Lorraine, Maxeville, France, and a Member of Groupe de Recherche en Electrotechnique et Electronique de Nancy, Université Henri Poincaré. His research interests include digital control using digital signal processing, particle swarm optimization, and genetic algorithms applied to the control and identification of electrical machines.

Abderrezak Rezzoug received the Electrical Engineer degree from ENSEM INPL, Nancy, France, in 1972, and the Dr.Ing. diploma and the Ph.D. degree from INPL, in 1979 and 1987, respectively. He is currently a Professor of Electrical Engineering at the Université Henri Poincaré, Nancy. As a member of the Groupe de Recherche en Electrotechnique et Electronique de Nancy, his main areas of research concern electrical machines, their identification, diagnostics and control, and superconducting applications.