Space Vector PWM Techniques for Sinusoidal Output Voltage Generation with a Five-Phase Voltage Source Inverter

Transcription

1 Electric Power Components and Systems, 34: , 2006 Copyright Taylor & Francis, LLC ISSN: print/ online DOI: / Space Vector PWM Techniques for Sinusoidal Output Voltage Generation with a Five-Phase Voltage Source Inverter ATIF IQBAL EMIL LEVI Liverpool John Moores University School of Engineering Liverpool, United Kingdom Five-phase variable-speed drives currently are considered for numerous applications, including electric and hybrid-electric vehicles, traction, and ship propulsion. If the machine is designed with a concentrated stator winding, the third stator current harmonic injection can be used to enhance the torque production and the machine needs to be supplied with the fundamental and the third harmonic of the voltage. On the other hand, if the machine is with a sinusoidally distributed winding, the supply should consist of the fundamental harmonic only. Since five-phase drives are invariably supplied from five-phase voltage source inverters (VSIs), adequate methods for VSI pulse width modulation (PWM) are required. This article analyzes different space vector PWM (SVPWM) schemes for a five-phase VSI, which can be used for five-phase motor drives with sinusoidal distribution of windings. A detailed model of a five-phase VSI is presented first in terms of space vectors and the existing technique of utilizing only large space vectors is elaborated. It is shown that this SVPWM method leads to generation of high amounts of low-order output voltage harmonics. Next, a novel SVPWM method is introduced, which enables operation with pure sinusoidal output voltages up to a certain reference voltage value, which is smaller than the maximum achievable with the given DC link voltage. To enable full utilization of the DC bus voltage, two different SVPWM schemes are further developed that can be used to extend the operation so that full utilization of the DC bus is achieved. This unavoidably leads to the generation of some low-order harmonics. These harmonics are however of significantly lower values than when only large vectors are used. A detailed performance evaluation of the existing and newly developed schemes is performed, based on the low-order harmonic content in the output voltages. Simulation results are included throughout the article to illustrate and verify the theoretical considerations. Keywords output five-phase voltage source inverter, space vector modulation, sinusoidal Manuscript received in final form on 6 May The authors gratefully acknowledge support provided for the work on this project by the EPSRC, under the standard research grant number EP/C007395/1, and by Semikron-UK, MOOG- Italiana and Verteco-Finland. Address correspondence to Emil Levi, Liverpool John Moores University, School of Engineering, JP Building, Byrom Street, Liverpool L3 3AF, UK. e.levi@livjm.ac.uk 119

2 120 A. Iqbal and E. Levi 1. Introduction Variable speed electric drives predominately utilize three-phase machines. However, since the variable speed AC drives require a power electronic converter for their supply, the number of machine phases is essentially unlimited. This has led to an increase in the interest in multiphase AC drive applications, especially in conjunction with traction, EV/HEVs and electric ship propulsion, since multiphase machines offer some inherent advantages over their three-phase counterparts. Supply for a multiphase variable-speed drive is in the majority of cases provided by a voltage source inverter (VSI). A number of pulse width modulation (PWM) techniques are available to control a three-phase VSI. However, space vector PWM (SVPWM) has become the most popular because of the ease of digital implementation and better DC bus utilization when compared to the rampcomparison sinusoidal PWM method. SVPWM for three-phase VSI has been discussed extensively in the literature [1]. The same does not apply to multiphase VSIs, since there are only a very few application specific SVPWM techniques available. In principle, there is a lot of flexibility available in choosing the proper space vector combination for an effective control of multiphase VSIs because of a large number of space vectors. A specific problem encountered in multiphase drive systems is that generation of certain low-order voltage harmonics in the VSI output can lead to large stator current harmonics since these, in essence, are restricted only by stator leakage impedance [2]. For example, if a five-phase machine with sinusoidal winding distribution is supplied with voltages containing the third and seventh harmonic, stator current harmonics of the third and seventh order will flow freely through the machine and their amplitude will be restricted by stator leakage impedance only. It is, therefore, important that the multiphase VSI output is kept as close as possible to sinusoidal and the SVPWM scheme of [2], for a dual three-phase machine, was developed with exactly this reasoning in mind. On the other hand, five-phase machines can be designed with concentrated windings and, in such a case, it is desirable to utilize the third harmonic stator current injection to enhance the torque production [3]. Since now both the fundamental and the third stator current harmonic are controlled, it is necessary to have a suitable PWM technique that enables control of both the fundamental and the third harmonic of the stator supply voltage. A space vector PWM method, proposed in Ryu et al. [4], has been developed for this type of a five-phase machine. It is based on the vector space decomposition technique and the inverter output voltages contain the fundamental and the third harmonic, which are both of controllable magnitudes. If a five-phase machine is with a sinusoidally distributed winding, the output voltages should contain only the fundamental component and they need to be free of low-order harmonics. A five-phase VSI offers a total of 2 5 = 32 space vectors, of which 30 are active state vectors, forming three concentric decagons, and two are zero state vectors. The simplest method of realizing SVPWM is to utilize only 10 large length vectors, belonging to the largest decagon in the d-q plane, in order to implement the symmetrical SVPWM [5, 6]. Two active space vectors neighboring the reference space vector and two zero space vectors are utilized in one switching period to synthesize the input reference voltage. This method is a simple extension of space vector modulation of three-phase VSIs. While being the simplest possible, it leads to generation of low-order output voltage harmonics of significant values, as shown in this article. The reason is that only two (instead of four) active space vectors are used. As discussed in Kelly et al. [7] in conjunction with a SVPWM technique developed for a nine-phase inverter with sinusoidal output, the number of applied active vectors in SVPWM of a multiphase VSI

3 Five-Phase Space Vector Techniques 121 should be equal to n 1, where n is the number of phases. This translates into the need to apply not only large but medium length space vectors as well in the case of a five-phase VSI. An attempt to realize sinusoidal output voltages by means of SVPWM of a fivephase VSI is described in de Silva et al. [8], this being a special case of the concept of Kelly et al. [7]. By combining the utilization of large and medium length neighboring space vectors in an appropriate manner, perfectly sinusoidal output voltages are created. However, this situation can only be maintained up to a certain value of the input reference, which is considerably smaller than the maximum reference achievable with the given DC link voltage. In this article, an alternative SVPWM scheme is at first proposed, which provides the same quality of performance as de Silva et al. [8], while calculating the active space vector application times in a different manner. This scheme has the same limit of the realizable reference voltage with zero values of low-order harmonics as the one in de Silva et al. [8]. In order to enable full utilization of the available DC bus voltage, the scheme needs to be further complemented with a different SVPWM method, which enables smooth transition from application of four active vectors to application of only two vectors (since, in order to achieve full utilization of the DC bus, one eventually has to revert to application of large vectors only). Two possible methods for achieving this goal are proposed. The performance of the developed novel SVPWM schemes is compared to the existing schemes, with the emphasis placed on the low-order harmonic content in the output voltages. A combined SVPWM scheme, which enables operation at first with pure sinusoidal and later with near-sinusoidal output voltages and utilizes the DC bus voltage to the limit is formulated in this way. It should be noted that all the SVPWM schemes developed here are based on the appropriate explicit analytical expressions derived to calculate the required application times for various space vectors. This is similar to the approach used in previous studies [2, 7 8]. An alternative approach, discussed in Delarue et al. [9] and aimed at fast practical implementation, is beyond the scope of this article. 2. Modelling of a Five-Phase VSI Power circuit topology of a five-phase voltage source inverter is shown in Figure 1. The inverter input DC voltage is regarded as constant. The load is taken as star-connected Figure 1. Five-phase voltage source inverter circuit.

4 122 A. Iqbal and E. Levi and the inverter output phase voltages are denoted in Figure 1 with lower case symbols (a, b, c, d, e), while the leg voltages have symbols in capital letters (A, B, C, D, E). The model of the five-phase VSI is developed in space vector form in what follows, assuming an ideal commutation and zero forward voltage drop. The relationship between the machine s phase-to-neutral voltages and inverter leg voltages is given with (the inverter leg voltages take the values of ±0.5V DC ) v a = (4/5)v A (1/5)(v B + v C + v D + v E ) v b = (4/5)v B (1/5)(v A + v C + v D + v E ) v c = (4/5)v C (1/5)(v A + v B + v D + v E ) (1) v d = (4/5)v D (1/5)(v A + v B + v C + v E ) v e = (4/5)v E (1/5)(v A + v B + v C + v D ) Since a five-phase VSI is under consideration, one deals here with a five-dimensional space. Hence two space vectors have to be defined, each of which will describe space vectors in one two-dimensional subspace (d-q and x-y). The third subspace is singledimensional (zero sequence) and it cannot be excited due to the assumed star connection of the system. Space vectors of phase voltages are defined in the stationary reference frame, using power variant transformation, as: v dq = v d + jv q = 2/5(v a + av b + a 2 v c + a 2 v d + a v e ) (2) v xy = v x + jv y = 2/5(v a + a 2 v b + a v c + av d + a 2 v e ) (3) where a = exp(j2π/5), a 2 = exp(j4π/5), a = exp( j2π/5), a 2 = exp( j4π/5), and stands for a complex conjugate. In general, an n-phase two-level VSI has a total of 2 n space vectors. Thus, in the case of a five-phase VSI there are 32 space vectors, two of which are zero vectors. The remaining 30 active vectors form three decagons in both d-q and x-y planes and are easily calculated using (2) (3) in conjunction with inverter output phase voltages of (1) for each possible inverter state. The phase voltage space vectors in the d-q plane are summarized in Table 1 for all 32 switching states, while corresponding space vectors in the x-y plane are given in Table 2. The five-phase VSI space vectors in the d-q plane and in the x-y plane are shown in Figure 2, where it can be seen that the outer decagon space vectors of the d-q plane map into the inner decagon of the x-y plane, the innermost decagon of d-q plane forms the outer decagon of the x-y plane, while the middle decagon space vectors map into the same region. Further, it is observed from the above mapping that the phase sequence a, b, c, d, e of the d-q plane corresponds to a, c, e, b, d sequence of the x-y plane, which are basically the unwanted low-order harmonic voltages (for example, the already mentioned third and seventh harmonics). In general, the harmonics of the order 10n ± 1(n = 0, 1, 2, 3,...) map into the d-q subspace, while the harmonics of the order 10n ± 3(n = 0, 1, 2, 3,...) map into the x-y subspace [4]. Since all the available non-zero space vectors map into both subspaces, the problem of sinusoidal output voltage generation reduces in essence to the problem of an appropriate generation of d-q voltage components (commensurate with the reference voltage space vector) that simultaneously ensure zero resulting space vector in the x-y subspace.

5 Five-Phase Space Vector Techniques 123 Table 1 Phase voltage space vectors in the d-q plane Space vectors Value of the space vectors v 1 phase to v 10 phase 2/5V DC 2 cos(π/5) exp(jkπ/5) (large) k = 0, 1, 2,...,9 v 11 phase to v 20phase 2/5V DC exp(jkπ/5) (medium) k = 0, 1, 2,...,9 v 21 phase to v 30 phase 2/5V DC 2 cos(2π/5) exp(jkπ/5) (small) k = 0, 1, 2,...,9 v 31 phase to v 32 phase 0 Figure 2 can be used to explain why application of large vectors alone cannot yield a sinusoidal output voltage. When reference voltages are sinusoidal, the space vector reference contains only d-q components (x-y components are zero). However, if only large vectors are used to create the desired d-q voltages, x-y voltage components inevitably will be created as well (each switching combination that gives a large vector in the d-q plane gives simultaneously a small vector in the x-y plane; thus, low-order voltage harmonics, such as the third and the seventh, are generated as well). There are 10 distinct sectors spanning 36 each in the d-q plane (the same applies to x-y plane). The outer and innermost decagon space vectors in the d-q plane are the result of three switches being on from upper (lower) VSI half and two switches being off from upper (lower) VSI half or vice versa. Thus, the innermost space vectors of the d-q plane are redundant and, therefore, are omitted from further discussion. This is in full compliance with observations of Kelly et al. [7]. As is obvious from Figure 2, these vectors have the smallest d-q components while simultaneously having the largest x-y components. The middle region space vectors correspond to four switches being on from upper (lower) VSI half and one switch being off from upper (lower) VSI half Table 2 Phase voltage space vectors in the x-y plane Space vectors v 21 phase to v 30 phase v 11 phase to v 20 phase v 1 phase to v 10 phase Value of the space vectors 2/5V DC 2 cos(π/5) exp(jkπ/5) k = 0, 1, 2,...,9 2/5V DC exp(jkπ/5) k = 0, 1, 2,...,9 2/5V DC 2 cos(2π/5) exp(jkπ/5) k = 0, 1, 2,...,9 v 31 phase to v 32 phase 0

6 124 A. Iqbal and E. Levi (a) Figure 2. Five-phase VSI phase voltage space vectors in the (a) d-q plane and (b) x-y plane. or vice versa. In what follows, the vectors belonging to the middle region are termed medium vectors and the vectors of the outermost region of the d-q plane large vectors. An ideal space vector modulator for a five-phase inverter should satisfy a number of requirements. First of all, in order to keep the switching frequency constant, each switch can change state only twice in the switching period (once on to off and once off to on, or vice versa). This requirement is satisfied for all the schemes considered here. Secondly, the RMS value of the fundamental output phase voltage must equal the RMS of the reference d-q space vector. Thirdly, the scheme must provide full utilization of the available DC bus voltage. Finally, since the inverter is aimed at supplying the load with sinusoidal voltages, the low-order harmonic content needs to be minimized, which asks for minimization of the x-y components. It is shown in the next section that application of large vectors only satisfies the first three but not the last requirement. It is for this reason that novel SVPWM schemes are proposed in the subsequent sections. 3. SVPWM Using Two Neighboring Large Vectors Only This SVPWM scheme considers only the outermost decagon of space vectors in the d-q plane. The input reference voltage vector is synthesized from two active neighboring and zero space vectors. To calculate the time of application of different vectors, Figure 3,

7 Five-Phase Space Vector Techniques 125 (b) Figure 2. (Continued). depicting the position of different available space vectors and the reference vector in the first sector, is considered. The times of active space vector application are from Figure 3 t a = v s sin(kπ/5 α) v l sin(π/5) t b = v s t s sin(α (k 1)π/5) t s v l sin(π/5) (4a) (4b) t o = t s t a t b (5) Here k is the sector number (k = 1 to 10), and large vector length is v al = v bl = v l = 2 5 V DC2 cos(π/5) (6a) Corresponding medium vector length, which will be needed in subsequent sections, is v am = v bm = v m = 2 5 V DC (6b)

8 126 A. Iqbal and E. Levi Figure 3. Principle of time calculation for active space vector application in a five-phase VSI. Symbol v s denotes the reference space vector, while x is the modulus of a complex number x. Switching period is denoted with t s and indices a and b denote the neighboring space vectors to the right and to the left, respectively, of the reference space vector. Indices l and m stand for large and medium space vectors, respectively. The largest possible fundamental peak output voltage that can be achieved using this scheme corresponds to the radius of the largest circle that can be inscribed within the decagon. This maximum fundamental peak output voltage V max is V max = (2/5)2 cos(π/5) cos(π/10)v DC = V DC. The sequence of vectors applied in the first sector and corresponding switching pattern are shown in Figure 4, where states of five inverter legs take values of ±0.5 and the five traces illustrate, from top to bottom, legs A, B, C, D, and E, respectively. The simulation is performed for different peak values of the five-phase sinusoidal reference input, ranging from maximum possible magnitude down to 10% of this value in 10% steps. The DC link voltage is set to 1 p.u. and the switching frequency is 5 khz. An analogue first order filter is used to filter the output voltages and the resulting filtered output phase voltages and leg voltages are shown in Figure 5 for reference equal to the Figure 4. Switching pattern and space vector disposition for sector I when only large vectors are used.

9 Five-Phase Space Vector Techniques 127 Figure 5. Output of the five-phase VSI with SVPWM, which utilizes only large vectors (input reference set to maximum achievable, p.u. peak): (a) filtered phase voltages; (b) filtered leg voltages; (c) unfiltered phase a voltage and its spectrum.

10 128 A. Iqbal and E. Levi maximum achievable peak fundamental output voltage (i.e., p.u.) with 50 Hz frequency. The shapes of the phase and the leg voltages are preserved for changes in the input reference voltage, except that there is a corresponding reduction in their amplitude. The variation of different low-order harmonics with input reference voltage is shown later in this article. It is observed from Figure 5 that the output phase voltages contain a considerable amount of the third harmonic. In addition, there is a certain amount of the seventh harmonic component as well. These harmonic components are the consequence of x-y components of the space vectors and they will exist regardless of the reference voltage value. By using only outer decagon set of space vectors, the x-y components are generated leading to the existence of the low-order harmonics in the output voltages. 4. Combined Application of Medium and Large Space Vectors, Sinusoidal Output Voltages The application of only large space vectors does not produce satisfactory results in terms of the harmonic content of the output phase voltage, since only two active voltage vectors are always applied. As emphasized in Kelly et al. [7], one needs to apply four active vectors in the case of a five-phase VSI, rather than two. A different scheme, therefore, is developed in this section by combining the large and the medium space vectors. The aim is to obtain sinusoidal output phase voltages. However, purely sinusoidal output voltages can only be obtained up to a certain value of the reference input voltage, which is smaller than the maximum reference voltage achievable with the large vectors only. Hence the SVPWM scheme of this section provides sinusoidal output voltages and is operational for reference voltage values smaller than or equal to the 85.41% of the maximum obtainable fundamental with large vectors only. A different SVPWM scheme is required to provide a smooth transition from to 100% of the maximum achievable output voltage, at the expense of reappearance of low-order harmonics in the output voltage. This issue will be dealt with in the next section. The switching pattern and the sequence of the space vectors for all schemes that utilize both large and medium space vectors are shown in Figure 6. It can be observed from the switching pattern in Figure 6 that the switchings in all the phases are staggered, in contrast to Figure 4, valid for application of large Figure 6. Switching pattern for sector I with utilization of both large and medium neighboring vectors.

11 Five-Phase Space Vector Techniques 129 vectors only. All switches change state at different instants of time. The total number of switchings in each switching period is still the same, thus, preserving the requirement that each switch is once on and once off in a switching period. Space vector diagram of Figure 3 is considered again, where the reference is in the first sector. For any given reference value and position, projections along the directions of the neighboring active space vectors are the same, regardless of whether only large or large and medium vectors are used. To achieve sinusoidal output voltage, proportional (to the vector length) subdivision of time of application of large and medium space vectors is utilized. Consider the SVPWM aimed at reference values between 0 and 85.41% of the maximum achievable. The total times of application of active space vectors remain to be calculated using (4), where the large vector length is utilized. In order to subdivide these times into times of application of medium and large vectors, the principle of proportionality is adopted. Subdivision is performed according to the lengths of the medium and large vectors. This subdivision of the time in essence allocates 61.8% of the total time to the large space vectors and 38.2% to the medium space vectors, since the ratio of moduli of the medium to large space vectors in Table 1 is 61.8%. The times of application of medium and large space vectors are calculated as follows: t al = t a v l v l + v m t bl = t b v l v l + v m while the zero space vector application time is t am = t a v m v l + v m t bm = t b v m v l + v m (7a) (7b) t o = t s t al t am t bl t bm (8) To verify the volt-second principle, the following expression for the first sector is considered v s t se jα = t al v al +t am v am +t bl v bl e jπ/5 + t bm v bm e jπ/5 (9) After substitution of (7) and (4) into (9) the following relationship is obtained: v s t se jα = v l 2 + v m 2 v l ( v l + v m ) v s t se jα (10) This expression indicates that the output fundamental phase voltages from this type of space vector modulator are only 85.41% of the input reference voltage value (this is the value of the coefficient on the right hand side, obtained by replacing the lengths of medium and large vectors from (6) or Table 1). Thus, in order to obtain the output fundamental equal to the input reference, the reference value should be % larger than the required output. This means that, for any given reference equal to v s = 2V e jα (11) the equality of the reference RMS value V and the fundamental RMS in the output will be ensured if the reference given to the modulator is scaled with the factor 1/0.8541, that is, v s = (1/0.8541) 2V e jα (12)

12 130 A. Iqbal and E. Levi Since both medium and large vectors are used and medium vectors are always applied, the maximum output voltage that can be obtained with this SVPWM modulator is p.u. (peak), which is 85.41% of the maximum achievable with large vectors only ( p.u. peak). The simulation is performed to obtain the maximum achievable output value ( p.u. peak), with the scaled commanded input equal to p.u. (according to (12)) of 50 Hz frequency, thus ensuring the equality of the fundamental output magnitude and the reference magnitude. The other simulation conditions are identical to those in the previous section. The filtered phase voltages and leg voltages are shown in Figure 7 along with the harmonic spectrum. It can be seen from Figure 7 that the spectrum contains only fundamental (0.372 p.u. RMS or p.u. peak, 50 Hz) component and harmonics around multiples of the switching frequency. The low-order harmonics (such as the third and the seventh) are completely absent. The output is sinusoidal (except for the PWM ripple) because time subdivision according to (7) is such that it cancels all the undesirable x-y components (as can be seen from Figure 2). For lower values of the input reference than the one used in production of Figure 7 the output phase voltages preserve the shape with a corresponding reduction in the amplitude, meaning that the low-order harmonics are absent throughout the operating region of this SVPWM modulator (0 to p.u. peak, or 0 to 85.41% of the maximum achievable output with large vectors only). It should be noted that the SVPWM method of (8) is characterized with an identical behavior to the method devised here, with the difference being in the expressions used to calculate application times for medium and large space vectors. 5. Combined Application of Medium and Large Space Vectors, Near-Sinusoidal Output Voltages In order to extend the operating region of the PWM modulator from p.u. to p.u. (peak), it is necessary to devise an alternative SVPWM method. Both large and medium space vectors are used again and two possible SVPWM methods are considered. In the first method any reference above p.u. (peak) is considered as being the maximum achievable for the purposes of calculation of application times for medium and large vectors. The principle here is to drive the inverter along the maximum possible output for all input reference voltage magnitudes. This is achieved by forcing the zero space vector application time to remain zero for each input reference magnitude in the middle of all sectors. Further, to obtain the maximum possible output from the space vector modulator, ultimately one has to revert to the large space vectors only. This method is applicable when the input reference voltage lies between V DC v s V DC (i.e., in between the inscribed circles determined with the medium and the large vectors). However its application is envisaged as viable only when the previously described method cannot be used any more (i.e., for references 2V > p.u.). Let the fractions of application times of the medium and large space vectors are denoted as γ and δ, respectively. These factors vary with the magnitude of the input reference voltage in such a way that the inverter is always driven along the circle of the maximum available voltage. Thus, the times of application of the neighboring vectors in sector I are given as t a = t b = v s sin(π/5 α) [γ v m +δ v l ] sin(π/5) t s (13) v s sin(α) [γ v m +δ v l ] sin(π/5) t s (14)

13 Five-Phase Space Vector Techniques 131 Figure 7. Output of the inverter for the maximum achievable output fundamental voltage (85.41% of the one obtainable with large vectors only) with SVPWM using proportional subdivision of time for application of medium and large space vectors: (a) filtered phase voltages; (b) filtered leg voltages; (c) full PWM waveform and harmonic spectrum of the phase voltage.

14 132 A. Iqbal and E. Levi The constraints on the expressions for applications times (13) (14) are: if v s V DC then γ = 1, δ = 0 (15a) if v s =0.6155V DC then γ = 0, δ = 1 (15b) Using either (13) or (14) to find the values of γ and δ, with the constraints t a = t b = 0.5t s, t o = 0, one obtains: γ v m +δ v l =2 v s sin(π/10) sin(π/5) (16) Substitution of the values of lengths of the medium and large vectors from (6) into (16) yields (γ + δ2 cos(π/5)) 2 5 V DC = 2 v s sin(π/10) sin(π/5) (17) This condition ensures that the maximum available fundamental peak voltage of the inverter equals input reference voltage for every value of the input reference voltage. Factors γ and δ are subject to 0 γ 1, 0 δ 1, and γ + δ = 1. This gives the expressions for γ and δ as: γ = δ = 2 cos(π/5) 5 v s sin(π/10) V DC sin(π/5) 2 cos(π/5) 1 5 v s sin(π/10) V DC sin(π/5) 1 2 cos(π/5) 1 (18) Simulations are further done to determine the performance of the space vector modulator based on this principle. It is seen from (18) that, for the maximum achievable output voltage, the result is identical to the one with large space vectors only (shown in Figure 5), since only large vectors are eventually used. A sample of simulation results, therefore, is shown in Figure 8 for input reference voltage equal to 90% of the maximum achievable (i.e., of p.u.). It can be observed from the phase voltage spectrum that low-order harmonics (the third and the seventh) are now present, although they are both of rather small values. The variation of low-order harmonics with the input reference is shown in the next section, where it will be seen that the method produces phase voltages with a variable harmonic content. The second SVPWM scheme is again aimed at complementing the method of Section 4, that is, at voltage references between p.u. and p.u. (peak). However, it is operational for all reference values (in contrast to the first method of this section) and is based on a predefined trajectory of zero space vector application time. The zero space vector trajectory is modified in such a way that it yields a positive time of application for zero space vector for reference input vector varying from zero to the maximum achievable value. The time of application of zero space vector is guaranteed to be positive throughout the range of the input voltage reference if it is made positive in the middle of each sector.

15 Five-Phase Space Vector Techniques 133 Figure 8. Output of the five-phase VSI when both medium and large vectors are used and the input reference equals 90% of the maximum achievable: (a) filtered phase voltages; (b) filtered leg voltages; (c) unfiltered waveform and harmonic spectrum of a phase voltage.

16 134 A. Iqbal and E. Levi The time of application for the zero space vector in the first sector, at α = π/10, denoted with t oo, is shown in Figure 9 for three different situations, as function of the input reference voltage: when only large vectors are used (t ool ), when only medium vectors are used (t oom ), and when both large and medium vectors are used (t oo ). The last one is the modified trajectory that is further to be discussed. In Figure 9, V ml and V mm represent the maximum achievable output voltage with large space vectors and medium space vectors, respectively. The expressions for time of application of zero space vector, if only large vectors and if only medium vectors are used, are: t ool = t s t oom = t s v s v l cos(π/10) t s v s v m cos(π/10) t s (19) These equations represent straight lines, which vary linearly with the length of the input reference vector. The aim is now to apply both the large and medium vectors and, thus, the curve defining the time of application of zero space vector should lie between these two curves. The curve should pass through the two endpoints and should be a tangent to the lower curve for zero reference input voltage in order to apply predominantly medium vectors when the reference vector is small. The expression for the modified curve is assumed as t oo = a v s 2 + b v s +c (20) To determine the constants, the boundary conditions are specified as: at v s =0, t oo = t s ; at v s =V ml, t oo = 0; further, the slope of the curve at v s =0 should be equal to the t slope of line t oom. The slope at this point is found as slope = s v m cos(π/10). Thus, the constants are determined as: a = b t s V ml V 2 ml b = t s v m cos(π/10) = t s V mm (21) c = t s It is further necessary to determine the expression for time of application of zero space vector at an arbitrary angle α. In general, t o = t s t a t b, where for sector I t a = v s sin(π/5 α) t s, v sin(π/5) t b = v s sin(α) v sin(π/5) t s (22) Hence t o = t s v s (sin(π/5 α) + sin(α)) t s (23) v sin(π/5)

17 Five-Phase Space Vector Techniques 135 Figure 9. Variation of zero space vector application times at α = π/10 for different SVPWM schemes, as function of the reference input voltage (switching frequency = 5 khz). where v represents v l or v m depending on whether (22) (23) are used to determine the time of application of large or medium space vectors, respectively. Further, vs (t o ) α=π/10 = t oo = t s v cos(π/10) t s and, therefore, By substituting (24) into (22) (23) one finds: t s v s / v =(t s t oo ) cos(π/10) (24) t o = t s (t s t oo ) cos(π/10 α) (25) Equation (25) is used to calculate the time of application of zero space vector and t oo used here is the one of Eq. (20). The times of application of active space vectors need to be modified as well. A parabolic curve is again assumed for large active space vector application time. Figure 10 shows the variation of application times of zero and active state vectors at α = π/10. The expression for time of application of large active space vectors is found as ( v ) t ol = t s 2 s (26) V ml and the times of application of medium active space vectors are then t om = t s t oo t ol (27) Expressions for time of application of neighboring space vectors at an arbitrary angle α are further obtained as follows. In general, t al = v s sin(π/5 α) v l sin(π/5) t s and t bl = v s sin(α) v l sin(π/5) t s. Therefore, t al + t bl = v s t s(sin(π/5 α) + sin(α)) 2 v l cos(π/10) sin(π/10) (28)

18 136 A. Iqbal and E. Levi Figure 10. Variation of active and zero space vector application times at α = π/10 as function of the reference input voltage (switching frequency = 5 khz). At α = π/10, t oa = t ob = from (28) one obtains, v s 2 v l cos(π/10) t s and thus t oa + t ob = t ol = t al + t bl = v s v l cos(π/10) t s. Hence sin(π/5 α) 2 sin(π/10) t ol + sin(α) 2 sin(π/10) t ol (29) The first term in (29) is the time of application of large active space vector a and the second term corresponds to the time of application of large active space vector b. Similarly, for medium space vectors, the expression is found as t am + t bm = sin(π/5 α) 2 sin(π/10) t om + sin(α) 2 sin(π/10) t om (30) The first term in (30) is the time of application of medium vector a and the second term corresponds to the time of application of medium space vector b. To confirm that the output phase voltage magnitude is equal to the reference input phase voltage, the volt-second principle is verified using (9). Simulations are performed for input reference phase voltage magnitude values from maxmium achievable down to 10% of this value and the resulting plots for 70% of the maximum achievable are shown in Figure 11. Figure 11 depicts the filtered ouput phase voltages and leg voltages in addition to the harmonic spectrum of phase voltages. It can be seen that the output phase voltage practically does not contain low-order harmonics for this particular reference value. The variation of low-order harmonics generated by the modulator will be shown in the next section. This method also produces variable low-order harmonics in phase voltages. 6. Performance Comparison of Space Vector Modulators A comparative performance evaluation of the different SVPWM schemes, detailed in the previous three sections, is presented here. For this purpose the quality of the inverter output voltage, as related to the harmonic content and the range of applicability,

19 Five-Phase Space Vector Techniques 137 Figure 11. Output of the SVPWM controlled VSI for input reference equal to 70% of the maximum achievable: (a) filtered phase voltages; (b) filtered leg voltages; (c) phase voltage instantaneous waveform and harmonic spectrum.

20 138 A. Iqbal and E. Levi is analyzed. The emphasis is placed on the lowest order x-y harmonics, the third and the seventh. The limits on the maximum achievable fundamental output voltage are the following. If only large vectors are applied, any fundamental peak voltage between zero and p.u. can be obtained. The same holds true for the second method of Section 5, where large and medium vectors are used and a predefined trajectory for the variation of the zero space vector application time is applied. When large and medium vectors are used and the application times are determined proportionally to the vector length ratio, the realizable output fundamental voltage is between zero and p.u. If both medium and large vectors are used but the time subdivision is done according to the principle of driving the inverter along the circle of the maximum output, the fundamental output voltage can take any values between and p.u. As noted in conjunction with appropriate results shown in Sections 3 5, the simulations and the harmonic analysis of the inverter output phase voltage were performed for each method for a series of input reference voltage values. The low-order harmonic content of the four SVPWM schemes was obtained in this way and it is illustrated in Figure 12, by means of the third and the seventh harmonic percentage content (relative to the fundamental). The results shown in Figure 12 encompass the complete region of the reference input voltage for which each individual SVPWM scheme is operational. As can be seen from Figure 12, utilization of large vectors only produces approximately Figure 12. Comparison of the third and the seventh harmonic content in the output voltages for the four SVPWM techniques considered in the paper: (A) large vectors only; (B) medium and large vectors, proportional subdivision of times; (C) SVPWM that drives the inverter along the circle of the maximum output; (D) SVPWM with predefined trajectory of the zero vector application time.

21 Five-Phase Space Vector Techniques % of the third harmonic and around 5% of the seventh harmonic for all input reference values. On the other hand, SVPWM with proportional subdivision of application times provides zero values for low-order harmonics across the entire operating region, up to the p.u. (peak) output fundamental voltage. The SVPWM scheme which drives the inverter at all times along the circle of the maximum output voltage gives in the first part of the operating region (from p.u. onward) very high low-order harmonic content (higher than with large vectors only). However, as the reference voltage increases loworder harmonics decrease, reaching a minimum value very close to the operating limit of the proportional PWM, where all the values are considerably smaller than with large vectors only. The second scheme of Section 5, with predefined trajectory of the zero space vector application time, generates large low-order harmonics for small reference voltage values and harmonic content is worse than when only large vectors are applied. However, as the reference voltage increases low-order harmonics reduce and once when the reference voltage exceeds approximately 50% of the maximum achievable, low-order harmonics become smaller than when large vectors only are applied. Both SVPWM methods of Section 5 are designed primarily to enable smooth transition from the output voltage limit of the proportional SVPWM up to the maximum p.u. output voltage. Results shown in Figure 12 indicate that the characteristics of both methods in this region are much better than with utilization of large vectors only. This is further confirmed in Figure 13, where only the operating region from p.u. up to p.u. (i.e., 85.41% to 100% of the maximum achievable output fundamental) is shown. The proposed two SVPWMs for this operating region are characterized with significantly lower values of the low-order harmonics. In the limit, when the reference becomes equal to p.u., both schemes revert to application of large vectors only and hence the low-order harmonic values become the same. Comparison of the low-order harmonic content of the two methods of Section 5 in the region of the reference voltage values of interest (above p.u.), given in Figure 13, indicates that the scheme that drives the inverter along the circle of the maximum output voltage gives lower values for the low-order harmonics (especially for the third harmonic). It is concluded that this scheme is more favorable than the method based on the predefined trajectory of the zero space vector application time. Figure 13. Comparison of the third and the seventh harmonic content in the output voltage for reference voltage between and 100% ( and p.u. peak): (A) large vectors only; (C) SVPWM that drives the inverter along the circle of the maximum output; (D) SVPWM with predefined trajectory of the zero vector application time.

22 140 A. Iqbal and E. Levi 7. Conclusion This article discusses SVPWM techniques for five-phase VSIs. It is shown that the standard method, based on application of large vectors only, produces output phase voltages with a substantial low-order harmonic content. Three SVPWM techniques, therefore, are devised with the idea of achieving an output voltage waveform as close as possible to the sinusoidal, while simultaneously keeping the switching frequency constant and enabling full utilization of the available DC bus voltage. All rely on application of both medium and large vectors. The first of the three proposed SVPWM methods achieves sinusoidal output voltages but is operational only up to the 85.41% of the maximum available fundamental output voltage. The characteristics of this SVPWM method are the same as for the method of de Silva et al. [8] but the vector application times are calculated in a different manner. The second and the third SVPWM technique enable smooth transition from to 100% of the maximum available output voltage, while keeping the low-order harmonics at lower values when compared to the utilization of large vectors only. The method of driving the inverter along the circle of the maximum output voltage is more favorable than the method with predefined trajectory of the zero space vector application time, since the low-order harmonic content is smaller. By combining the two proposed SVPWM schemes it becomes possible to achieve sinusoidal or at least near-sinusoidal output voltages across the entire inverter operating range, up to the maximum achievable output voltage fundamental limited by the DC bus voltage. References 1. G. D. Holmes and T. A. Lipo, Pulse Width Modulation for Power Converters Principles and Practice, IEEE Press Series on Power Engineering, Piscataway, NJ: John Wiley and Sons, Y. Zhao and T. A. Lipo, Space vector PWM control of dual three-phase induction machine using vector space decomposition, IEEE Trans. on Industry Applications, vol. 31, no. 5, pp , H. Xu, H. A. Toliyat, and L. J. Peterson, Five-phase induction motor drives with DSP-based control system, IEEE Trans. on Power Electronics, vol. 17, no. 4, pp , H. M. Ryu, J. H. Kim, and S. K. Sul, Analysis of multi-phase space vector pulse width modulation based on multiple d-q spaces concept, Proc. 4th Int. Power Electronics and Motion Control Conf. IPEMC, Xian, China, CD-ROM Paper No. 2183, R. Shi and H. A. Toliyat, Vector control of five-phase synchronous reluctance motor with space vector pulse width modulation (SVPWM) for minimum switching losses, Proc. IEEE Applied Power Elec. Conf. APEC, Dallas, Texas, pp , H. A. Toliyat, R. Shi, and H. Xu, DSP-based vector control of five-phase synchronous reluctance motor, IEEE Industry Applications Society Annual Meeting IAS, Rome, Italy, CD-ROM Paper 40-05, J. W. Kelly, E. G. Strangas, and J. M. Miller, Multi-phase space vector pulse width modulation, IEEE Trans. on Energy Conversion, vol. 18, no. 2, pp , P. S. N. desilva, J. E. Fletcher, and B. W. Williams, Development of space vector modulation strategies for five-phase voltage source inverters, Proc. IEE Power Electronics, Machines and Drives Conf. PEMD, Edinburgh, UK, pp , P. Delarue, A. Bouscayrol, and E. S , Generic control method of multileg voltage-sourceconverters for fast practical implementation, IEEE Trans. on Power Electronics, vol. 18, no. 2, pp , 2003.