A New Carrier-Based PWM Strategy with Minimum Output Current Ripple for Five-Phase Inverters D. Casadei, M. Mengoni, G. Serra, A. Tani, L. Zarri University of Bologna - Department of Electrical Engineering Viale Risorgimento, 2 40136, Bologna, Italy Tel.: +39 / 051 2093565 Fax: +39 / 051 2093568 E-Mail:
[email protected] URL: http://www.die.unibo.it
Keywords Multiphase drive, Voltage Source Inverters (VSI), Pulse Width Modulation (PWM).
Abstract In this paper a new PWM strategy which minimizes the output current ripple for five-phase inverters is analytically determined. The optimization procedure is applied in each switching period, therefore its validity is not restricted to sinusoidal operations. The effectiveness of the proposed PWM strategy is proved by numerical simulations and experimental tests.
Introduction Multi-phase motor drives are an emerging technology, which nowadays represents an attractive opportunity, particularly for high power applications. Compared to traditional three-phase drives, multi-phase motor drives have several advantages such as reduction of the amplitude and increase of the frequency of torque pulsations, and reduction of the stator current per phase. Furthermore, a greater number of phases implies a greater number of degrees of freedom, which can be successfully utilized to improve the fault-tolerant capability [1]-[2]. Several PWM strategies for multi-phase inverters, based on different approaches, have been developed in the last decade [3]-[12], but few rigorous attempts have been carried out in order to analyze the behavior of these strategies in terms of output current ripple. In [13], a new PWM technique with reduced output current ripple, for six-phase voltage source inverters, has been proposed. A rigorous analytical comparison among three specific modulation techniques for five-phase inverters has been presented in [14]. It has been demonstrated that the best modulation technique, in terms of output current ripple rms and in balanced sinusoidal operating conditions, is the sinusoidal PWM. In [15], the superiority of the sinusoidal PWM, once again in balanced sinusoidal operating conditions, has been proved for an odd number of phases up to nine. This relevant result has been obtained using the so called “polygon approach”. It should be noted that, in the case of high torque density [16] and multi-motor drives [17] applications the results of the ripple analysis presented in [14] and [15] are not applicable. In fact, in these specific drives, the inverter must work in non-sinusoidal operating conditions. In this paper the optimal value of the zero-sequence component of the modulating signals of a fivephase inverter, leading to the minimum rms value of the output current ripple in each switching period, has been analytically determined. As a consequence, a new carrier-based PWM technique for five-phase inverters, having the best performance in terms of output current ripple, is derived for any output voltage waveform. The result developed in this paper is based on the multiple space vector representation of five-phase
systems [18], [19], and confirms the one emphasized in [14] and [15] for the particular case of balanced sinusoidal output voltages. A quantitative comparison in terms of output current ripple among the proposed modulation strategy and the other well-known modulation techniques is presented. The results of some experimental tests, carried out by using a five-phase inverter prototype available in laboratory, are analyzed and confirm the effectiveness of the proposed optimal PWM strategy.
Multiple space vectors representation For a given set of 5 real variables x1 , ... , x k , ... , x5 a new set of complex variables y 0 , y1 , y 3 can be obtained by means of the following symmetrical linear transformations (Fortescue transformations):
y0 =
1 5 ∑ xk , 5 k =1
yρ =
2 5 ∑ xk α ρ(k −1) , 5 k =1
(1) (ρ = 1, 3),
(2)
where α = exp( j 2π / 5) . Relationships (1) and (2) lead to a real variable y 0 (zero-sequence component) and two complex variables y1 , y 3 (multiple space vectors). The inverse transformations are xk = y0 +
∑ yρ ⋅ α ρ(k −1) ,
(k = 1, 2, ..., 5),
(3)
ρ =1,3
where the symbol “ ⋅ ” represents the scalar product. According to (1)-(3), a general five-phase system can be represented by two space vectors and the zero-sequence component. The two space vectors are mutually independent and move arbitrarily in the corresponding α-β planes.
Generalized carrier-based PWM technique for five-phase inverters A schematic drawing of a five-phase voltage source inverter supplying a star connected balanced load is presented in Fig. 1. In a carrier-based PWM technique, the switching signals Sk (k = 1, 2, ..., 5) are obtained comparing a triangular carrier signal c, varying within the interval [0, 1], with five regular-sampled (i.e. assumed constant in each switching period Tsw) modulating signals mk, as shown in Figs. 2 and 3. The value of the modulating signals mk, in each switching period, can be obtained as [11] – [12]: mk = m0 +
1 Vc
∑ vρ,ref ⋅ α ρ(k −1) , (k = 1, 2, ..., 5).
(4)
ρ =1,3
In (4), v1,ref and v3,ref are the space vectors of the reference output voltages, Vc is the input voltage, whereas m0, which is a degree of freedom, represents the zero-sequence component of the modulating signals. By introducing the space vectors of the modulating signals m1 and m3 , (4) can be rewritten as mk = m0 +
∑ mρ ⋅ α ρ(k −1) , (k = 1, 2, ..., 5),
(5)
ρ =1,3
where m1 =
v1,ref Vc
, m3 =
v3,ref Vc
.
This PWM technique allows any voltage waveform to be easily synthesized.
(6)
Vc 1
0
Sk
5
k N
Fig. 1. Structure of a five-phase VSI. +1
mk m1 m5
0
Tsw
c
mk
+
−
1
Sk
c
t
Fig. 2. Carrier and modulating signals.
0
Fig. 3. Switching signals generation.
Analysis of the current ripple in five-phase inverters Let us consider a five-phase ac machine supplied by a five-phase PWM inverter, and analyze the behavior of the stator currents in a switching period. Let us denote with i1,ref and i3,ref the desired current vectors flowing through the load, and with v1,ref and v3,ref the corresponding reference voltage vectors. The actual values of the voltage vectors v1 and v3 differ from the reference ones because of the switching behavior of the inverter. It is then possible to define the voltage and the current ripple vectors as follows:
v1,rip = v1 − v1,ref ,
v3,rip = v3 − v3,ref ,
(7)
i1,rip = i1 − i1,ref ,
i3,rip = i3 − i3,ref ,
(8)
where i1 and i3 are the actual values of the stator current vectors. The mean values of v1,rip and v3,rip during a switching period Tsw are zero, owing to the principle of operation of PWM technique. Under the assumption of high frequency ripple components, the following relationships can be determined: di1,rip dt
=
v1,rip L1,hf
di3,rip
,
dt
=
v3,rip L3,hf
,
(9)
where L1,hf and L3,hf are the high frequency equivalent inductances of the load, respectively in the α1-β1 and α3-β3 planes, measured from the inverter output terminals. The parameter commonly used to represent the distortion of three-phase waveforms is the total RMS value of the ripple in a switching period. In order to make more straightforward the mathematical developments, the square of the total RMS value of the ripple, expressed in (10), is evaluated: 2 I rip ,RMS ,Tsw
5
=∑
k =1
ik2,rip ,RMS ,T sw
1 = Tsw
Tsw 5
ik ,rip dt , ∫∑ k =1 2
(10)
0
where ik,rip is the current ripple of the k-th phase. Taking into account the symmetry of the double-sided switching pattern, and introducing the current ripple vectors, (10) can be rewritten as 2 I rip ,RMS ,Tsw
5 = Tsw
Tsw 2
∫ ( i1,rip i1,rip + i3,rip i3,rip ) dt , 0
*
*
(11)
where the symbol “*” represents the complex conjugate. Furthermore, the quantities i1,rip and i3,rip can be calculated integrating (9), and taking (7) into account, as follows: iρ ,rip =
∫ (vρ − vρ ,ref )dτ , t
1 Lρ ,hf
(ρ = 1, 3).
(12)
0
Space vectors vρ ,ref (ρ = 1, 3) can be expressed as
(
)
(13)
)
(14)
2 vρ ,ref = Vc m1 + m2 α ρ + m3 α 2 ρ + m4 α 3 ρ + m5 α 4 ρ , 5 whereas vρ =
(
2 v + v p ,2 α ρ + v p ,3 α 2 ρ + v p ,4 α 3 ρ + v p ,5 α 4 ρ . 5 p ,1
The behavior of the k-th pole voltage vp,k can be calculated from the basic principle of carrier-based PWM, i.e. the comparison of carrier and modulating signals, leading to ⎧ ⎪⎪v p ,k = Vc ⎨ ⎪v = 0 ⎪⎩ p ,k
if
0≤τ≤
if
Tsw mk 2
Tsw mk 2
.
(15)
(k = 1, 2, 3, 4, 5).
(16)
m0 ,DMAX then m0 = m0 ,DMAX . ⎪ ⎪⎩Otherwise m0 = m0 ,OPT
(30)
2 Note that, these rules retain the minimum allowable value for the current ripple since I rip ,RMS ,Tsw is a parabolic function of m0.
Comparison of PWM strategies in terms of output current ripple A comparison among different modulation strategies, in terms of the square rms value of the current ′ 2 , RMS ,T ), is presented in this Section. ripple in p.u. in a fundamental period T ( I rip The modulation strategies are identified with the names Sinusoidal Pulse Width Modulation (SPWM), Discontinuous MIN PWM (DMIN-PWM), Discontinuous MAX PWM (DMAX-PWM), Space Vector PWM (SV-PWM), Optimal PWM (OPT-PWM), and are defined in Table I by their zerosequence modulating signals. The modulation strategy referred to as S-PWM is the traditional sinusoidal PWM, and its zerosequence modulating signal is always 1/2. The value of m0 of DMIN-PWM is selected so that the minimum modulating signal among m1, . . . , m5 is always zero, whereas the maximum modulating signal of DMAX-PMW is always one. As a consequence, when these strategies are used, in every switching period, there is an inverter branch that does not commutate. The strategy called SV-PWM is often referred to as “symmetric modulation” and m0 is selected in order to satisfy the constraint 1-max (m1, ..., m5) = min (m1, ..., m5). Finally, the OPT-PWM is the proposed minimum ripple strategy.
Table I – Definition of m0 for some modulation strategies. m0 ,S = 1 2
S-PWM
⎞ ⎛ m0, DMIN = − min ⎜ ∑ mρ ⋅ α ρ(k −1) ⎟ ⎟ k =1, 2,...,5⎜ ⎠ ⎝ ρ=1,3 ⎞ ⎛ m0, DMAX = 1 − max ⎜ ∑ mρ ⋅ α ρ(k −1) ⎟ ⎟ k =1, 2,...,5⎜ ⎠ ⎝ ρ=1,3 1 m0, SV = (m0, DMIN + m0, DMAX ) 2 m0 ,OPT = Eq. (27 )
DMIN-PWM
DMAX-PWM SV-PWM OPT-PWM
Once the specific modulation strategy is defined, the square rms value of the current ripple in a switching period Tsw, expressed in p.u., is obtained as follows:
′ 2 , RMS ,Tsw (m1 , m3 ) = I rip
2 2 I rip , RMS ,Tsw L1, hf 2 Vc2 Tsw
.
(31)
It is worth noting that this quantity is strongly dependent on the parameter K L = L1,hf L3,hf . The value of this parameter depends on the specific inverter load. In the cases of multi-motor drives and synchronous motor drives K L = 1 is a reasonable assumption, whereas in the case of induction motor drives a realistic hypothesis is K L = 2 [14], [15]. ′ 2 , RMS ,T can be determined only if a specific operating condition of the inverter (i.e., the The value of I rip time behavior of m1 and m3 ) in the fundamental period T is defined. For example, in this paper, the following wide class of operating conditions is considered: m1 = M 1 e j θ ,
(32)
m3 = M 3 e j K F θ e j ϕ3 ,
(33)
where M1 and M3 are the modulation indexes, θ = ω1 t varies within the interval [0, 2π], and ϕ3 is the initial argument of m3 . The parameter KF represents the ratio between the constant angular speed ω3 of m3 and ω1 of m1 . It should be noted that (32) and (33) include all the steady-state operating conditions concerning conventional and advanced (i.e., multi-motor and high torque density) five-phase ac drives. ′ 2 , RMS ,Tsw can be considered a function of the angle θ. Then, the Taking (32) and (33) into account, I rip square rms value of the current ripple in a fundamental period T can be calculated as follows: ′ 2 , RMS ,T I rip
1 = 2π
2π
′ 2 , RMS ,T (θ) dθ . ∫ I rip
(34)
sw
0
Owing to the large number of parameters defining the possible operating conditions, a comprehensive comparison among the modulation strategies would be practically unfeasible. Therefore, only some specific operating conditions, named Case 1, ..., Case 12, and summarized in Table II, are analyzed.
Table II – List of the analyzed operating conditions. Cases
M1
M3
KL
KF
ϕ3
Case 1
0 → M1,max
0
1
-
-
Case 2
0 → M1,max
0
2
-
-
Case 3
0
0 → M3,max
1
-
-
Case 4
0
0 → M3,max
2
-
-
Case 5
0 → M1,max
0.2
1
1
0
Case 6
0 → M1,max
0.2
2
1
0
Case 7
0 → M1,max
0.2
1
3
0
Case 8
0 → M1,max
0.2
2
3
0
Case 9
0 → M1,max
0.2
1
1
π/5
Case 10
0 → M1,max
0.2
2
1
π/5
Case 11
0 → M1,max
0.2
1
3
π/5
Case 12
0 → M1,max
0.2
2
3
π/5
Note that, M1,max and M3,max are the maximum allowable values of M1 and M3, respectively, according to the voltage limit. These values depend on the operating conditions. The obtained results are illustrated in Figs. 4-7, where the square rms values of the current ripple in p.u. in a fundamental period T of the modulation strategies are shown as function of one of the two modulation indexes M1 and M3. The dashed vertical line emphasizes the maximum value of the variable modulation index for S-PWM. Some general remarks can be pointed out. As expected, the proposed OPT-PWM achieves the minimum current ripple in all the operating conditions and for all the values of the modulation indexes. DMIN-PWM and DMAX-PWM show the same behavior and are the worst modulation strategies. Obviously, this result was expected because the comparison assumes the same switching period. On the contrary, SV-PWM is a nearly-optimal modulation strategy in all the operating conditions. Its behavior degrades slightly for high values of the modulation indexes. Finally, S-PWM has optimal performance in the Cases 1-4 ( M 1 ≠ 0 & M 3 = 0 , or, M 1 = 0 & M 3 ≠ 0 ). In the other Cases, this characteristic is no longer valid, and the performance degrades as the modulation indexes increase. It is easily recognizable from the analysis of Figs. 4-7 that the voltage limit of S-PWM is more stringent than the voltage limits of the other modulation strategies.
0.005
0.015 SV
rip,RMS,T
DMAX - DMIN
0.002
2
0.003 S - OPT
I'
I'
2
rip,RMS,T
0.004
0.018
0.001
DMAX - DMIN
0.012
SV
0.009 0.006
S - OPT
0.003
0.000
0.000 0.0
0.1
0.2
M1
0.3
0.4
0.5
0.0
(a) Case 1. M3 = 0, KL = 1.
0.1
0.2
M1
0.3
0.4
0.5
(b) Case 2. M3 = 0, KL = 2.
Fig. 4. Square RMS value (p.u.) of the current ripple in T as function of M1. 0.005
0.015 DMAX - DMIN
SV
I' 2rip,RMS,T
I'
2
rip,RMS,T
0.004
0.018
0.003 0.002
S - OPT
0.001
DMAX - DMIN
0.012
SV
0.009 0.006 0.003
0.000
S - OPT
0.000 0.0
0.1
0.2
M3
0.3
0.4
0.5
0.0
(a) Case 3. M1 = 0, KL = 1.
0.1
0.2
M3
0.3
0.4
0.5
(b) Case 4. M1 = 0, KL = 2.
Fig. 5. Square RMS value (p.u.) of the current ripple in T as function of M3. 0.005
0.018
DMAX - DMIN rip,RMS,T
0.002
2
S OPT
I'
2 rip,RMS,T
I'
SV
0.003
0.001 0.1
0.2
M1
0.3
0.4
0.009
S
0.006 OPT
0.0
0.5
(a) Case 5. M3 = 0.2, KL = 1, KF = 1 0.005
DMAX - DMIN
0.018
SV
S
0.002
0.2
M1
0.3
0.4
0.5
DMAX - DMIN
SV
0.015
I' 2rip,RMS,T
0.003
0.1
(b) Case 6. M3 = 0.2, KL = 2, KF = 1
0.004 rip,RMS,T
0.012
0.000 0.0
2
SV
0.003
0.000
I'
DMAX - DMIN
0.015
0.004
OPT
0.001
0.012 0.009
S
0.006 0.003
OPT
0.000
0.000 0.0
0.1
0.2
M1
0.3
0.4
(c) Case 7. M3 = 0.2, KL = 1, KF = 3
0.5
0.0
0.1
0.2
M1
0.3
0.4
0.5
(d) Case 8. M3 = 0.2, KL = 2, KF = 3
Fig. 6. Square RMS value (p.u.) of the current ripple in T as function of M1 (ϕ3 = 0).
Experimental results The output current ripple of S-PWM and OPT-PWM strategies has been investigated by means of experimental tests. The experimental setup consists of a custom-designed five-phase voltage source inverter feeding a star-connected five-phase balanced R-L load (R=18.3 Ω, L=10 mH). With this type of load KL=1. The IGBTs are rated 30 A and 600 V. The dc bus voltage is about 100 V, obtained with a three-phase diode rectifier and filtered by a capacitance of 3300 µF.
0.005
0.018
DMAX - DMIN rip,RMS,T
0.003
2
S
0.002
OPT
0.001 0.1
0.2
M1
0.3
0.4
0.009
S
0.006 OPT
0.0
0.5
(a) Case 9. M3 = 0.2, KL = 1, KF = 1 0.005
DMAX - DMIN
0.018
SV
S
0.002
0.2
M1
0.3
0.4
0.5
DMAX - DMIN
SV
0.015
I' 2rip,RMS,T
0.003
0.1
(b) Case 10. M3 = 0.2, KL = 2, KF = 1
0.004 rip,RMS,T
0.012
0.000 0.0
2
SV
0.003
0.000
I'
DMAX - DMIN
0.015
SV
I'
I'
2
rip,RMS,T
0.004
OPT
0.001
0.012 0.009
S
0.006 0.003
0.000
OPT
0.000
0.0
0.1
0.2
M1
0.3
0.4
(c) Case 11. M3 = 0.2, KL = 1, KF = 3
0.5
0.0
0.1
0.2
M1
0.3
0.4
0.5
(d) Case 12. M3 = 0.2, KL = 2, KF = 3
Fig. 7. Square RMS value (p.u.) of the current ripple in T as function of M1 (ϕ3 = π/5). The control algorithm has been implemented on a Digital Signal Processor (DSP) TMS320F2812 with a switching frequency of 2.5 kHz. The experimental tests have been carried out with M1 = 0.33, M3 = 0.165, ω1 = 2 π 10 rad/s, ω3 = 2 π 30 rad/s (KF = 3) and ϕ3 = π/5. The results are presented in Figs. 8 and 9. In these figures, the waveforms of the modulating signal m1, of the zero-sequence component m0 and of the current in phase 1 are shown. As expected, being the reference voltage space vectors v1,ref and v3,ref different from zero, the proposed OPT-PWM is better than S-PWM. This result is clearly recognizable analyzing the zoomed current waveforms.
Conclusion A new optimal carrier-based PWM technique for five-phase inverters, having the best performance in terms of output current ripple for any output voltage waveform, has been derived in this paper. The output current ripple has been minimized by optimizing the zero-sequence component of the modulating signals in each switching period. A quantitative comparison in terms of output current ripple among several continuous and discontinuous modulation techniques has been presented. The comparison has emphasized the nearlyoptimal behavior of the SV-PWM and the poor behavior of DMIN-PWM and DMAX-PWM, in each operating condition. Furthermore, the analysis has shown that S-PWM yields minimum current ripple only in some specific operating conditions. The results of experimental tests, which have been carried out using a five-phase inverter prototype available in laboratory, confirm the effectiveness of the proposed optimal PWM strategy.
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m1 m0
m0 i1
i1 (zoomed)
Fig. 8. Experimental results. S-PWM. Waveforms of the modulating signal m1 (0.2/div, 20 ms/div), of m0 (0.2/div, 20 ms/div) and of the current in phase 1 (1 A/div, 20 ms/div and 0.1 A/div, 200 µs/div).
m1 i1
i1 (zoomed)
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