Multilevel PWM Waveform Decomposition and Phase-Shifted Carrier Technique

Transcription

1 Multilevel PWM Waveform Deomposition and Phase-Shifted Carrier Tehnique R. Naderi* and A. Rahmati* Abstrat: Multilevel PWM waveforms an be deomposed into several multilevel PWM omponents. Phase-shifted arrier (PSC) is an effiient deomposition tehnique. In this paper, we have first demonstrated the equality of PSC and alternative phase opposition disposition tehniques. Seond, we have modified PSC to aommodate other disposition tehniques. Third, we have investigated the effets of using asymmetrial arriers on the spetrum of the resulting PWM waveform. Fourth, we have proposed a logial algorithm for deomposing all types of multilevel PWM waveforms. Keywords: Carrier-Based PWM, Phase-Shifted Carrier, Multilevel inverters. 1 Introdution 1 Multilevel PWM inversion is an effetive and pratial solution for inreasing power and reduing harmonis of AC waveforms. A multilevel inverter has four main advantages over the onventional bipolar inverter: First, the voltage stress on eah swith is dereased, ompared to existing topologies where swithes are onneted in series. Therefore, the rated voltage, and onsequently, the total power of the inverter ould be safely inreased. Seond, the rate of hange of voltage (dv/dt) is dereased due to the lower voltage swing of eah swithing yle. Third, harmoni distortion is redued due to more output levels. Fourth, lower aousti noise and EMI is obtained: [1]. Based on these advantages, various iruit topologies and modulation strategies have been reported for better utilization of multilevel inverters. Multilevel topologies are lassified into three ategories: Diode Clamped inverters, Flying Capaitor inverters, and Casaded inverters (also alled hain inverters). The topologies have an equal number of main swithes. A 5- level inverter with different topologies is shown in Fig. 1. Diode Clamped inverter has the least number of apaitors among these three topologies but requires additional lamping diodes. Flying Capaitor inverter requires the most number of apaitors. Casaded inverter has simple struture but it needs various separated DC soures. The main drawbak of Diode Iranian Journal of Eletrial & Eletroni Engineering, Paper first reeived 4 th July 2007 and in revised form 17 th November ** The Authors are with the Department of Eletrial Engineering, Iran University of Siene and Tehnology, Narmak, Tehran, Iran. rnaderi@ee.iust.a.ir, rahmati@iust.a.ir. Clamped inverter is the unbalaned DC link apaitor. This limits the appliation of Diode Clamped inverter to appliations with five or more levels. This problem may be mitigated using ombinational topologies. For example, ombining a 3-level Diode Clamped with H- bridge topology results in a 5-level mixed struture alled neutral point lamped (NPC), mitigating the unbalaned DC link problem of the Diode Clamped onfiguration and also eliminating the need for separate DC soures: [2] to [5]. Various swithing strategies have been developed to generate swithing angles, to fit diverse topologies and appliations. Carrier based strategies are of major interest due to their simpliity and flexibility. The most popular arrier-based tehniques are: 1. Alternative phase opposition (APO) disposition 2. Phase opposition (PO) disposition 3. Phase (PH) disposition. The main differene between these swithing strategies is the shape of the undesired harmonis spetrum. While other phase displaements ould also be used for ontiguous triangular arriers, we fous on these three types beause of their effiieny and popularity: [6] and [7]. In Setion 2 of this paper, we first present PSC as an equivalent for APO disposition tehnique using a 5- level example. Further, we modify the PSC tehnique to aommodate with PO and PH disposition tehniques. In Setion 3, we will generalize the PSC tehnique for the 5-level example to N-level. In Setion 4, we will explain the effets of using asymmetrial arriers on the spetrum of the multilevel PWM waveform. We then propose a logial deomposition tehnique for effiient 150 Iranian Journal of Eletrial & Eletroni Engineering, Vol. 4, No. 4, Otober 2008

2 deomposition of multilevel PWM waveforms. Using PSC tehnique, we will explain how to generate the ontrol signals for swithes for bipolar and unipolar onfigurations in Setion 5. 2 Phase-Shifted Carrier Tehnique 2.1 General Definitions To simplify our disussion about different dispositions and PSC tehnique, we have used a speial notation for arrier and modulating signals. We have also defined some useful parameters: 1-Modulating signal: 2-Carrier signal: y (A, ω, φ ) = A sin( ω t +φ ) m m m m m m m As shown in Fig. 2 y is a periodi funtion with the period (T = 2π/ω ). w is a parameter between 0 and 1 whih represents the triangle width. For w = 0.5, y is a symmetrial triangular waveform and for w = 0 (1), y is a negative (positive) ramp sawtooth waveform. 3-Modulation index: = 2A M m (N 1)A 4-Modulation frequeny ratio (Pulse number): ω P= ω m y (A, ω, φ, w) = A ω t+φ 1 2πw 2 ω t+φ π(1 w) 1 w 2 0 ω t+φ < 2πw + 2πw ω t+φ < 2π 5-The number of stritly positive levels: N N = 2 Fig. 1 Different iruit topologies for a 5-level inverter (a) Diode Clamped, (b) Flying Capaitor, () Casaded. Fig. 2 Triangular arrier. Naderi & Rahmati: Multilevel PWM Waveform Deomposition and Phase-Shifted Carrier Tehnique 151

3 2.2 APO, PO, and PH Dispositions at a Glane Carrier based N-level PWM operations onsist of N- 1 different arriers. The arriers have the same frequeny ω, the same peak-to-peak amplitude A, and are disposed so that the bands they oupy are ontiguous. They are defined as: C = y (A, ω, φ, w) + (n N 2)A, n= 1,..., N 1 n n The zero referene is plaed in the middle of the arrier set. And we hoose the arrier C 1 as the referene arrier. For APO disposition, all arriers are alternatively in phase opposition. The arriers are defined as: eliminate the harmoni at P. In this ase, PH disposition is more onvenient due to the very little values of other harmonis. 2.3 PSC as an Equivalent for APO We start our disussion onsidering the 5-level PWM example and then generalize it to N levels. For the APO ase, the four arriers are alternatively in phase opposition. Using Eq. (1), the arriers are defined as: C = y (A, ω,0,0.5) 1.5A 1 C = y (A, ω, π,0.5) 0.5A 2 C = y (A, ω,0,0.5) + 0.5A 3 C = y (A, ω, π,0.5) + 1.5A 4 C = ( 1) y (A, ω, φ,0.5) + (n N 2)A (1) n 1 n Fig. 3(a) shows the arriers and the modulating signal for a 5-level PWM using APO disposition tehnique with M = 0.75 and P = 80. For PO disposition all the arriers above the zero referene are in phase among them but in opposition with those below. They are defined as: 2n N = ω φ + n C ( 1) y (A,,,0.5) (n N 2)A, N odd Note that the PO disposition is undefined for even numbers of N. The Modulating signal and the arriers for the 5-level example using PO disposition is shown in Fig. 3(b). For PH disposition, all arriers are in phase. They are defined as: C = y (A, ω, φ,0.5) + (n N 2)A n Fig. 3() shows the arrangement of arriers in the 5- level example for PH disposition. The harmoni ontent of the PWM waveforms using APO, PO, and PH dispositions are shown in Fig. 3(d,e,f) respetively. It is lear that the main differene is about the first set of undesired harmonis (those aggregated near P). For APO and PO dispositions, no harmoni exists at P due to odd symmetry of their PWM waveforms. For the PH ase, however, the waveform is asymmetri and the harmoni at P is relatively high. Thus, APO and PO dispositions are more onvenient to be used in single-phase inverters. For three-phase inverters, however, the triplen harmonis of voltage (urrent) will be eliminated due to Y ( ) onnetion of the load. Choosing P as a multiple of three, we an Fig. 3 A 5-level PWM operation with M = 0.75 and P = 80. Modulating signal and arriers for (a) APO, (b) PO, () PH. Waveform s spetrum for (d) APO, (e) PO, (f) PH dispositions. 152 Iranian Journal of Eletrial & Eletroni Engineering, Vol. 4, No. 4, Otober 2008

4 Fig. 4 Carrier and PWM waveforms for (a) C 1, (b) C 2, () C 3, (d) C 4 and (e) resulting PWM waveform generated by adding up the last parts. And the amplitude of the modulating signal is: A m = 2MA The arriers and the spetrum of the resulting waveform are shown in Fig. 3(a,d). Sine the swithing frequeny is equal to ω, the undesired harmonis are aggregated near the multiples of P. Note that no even harmoni exists due to odd symmetry of the PWM waveform. Instead of the tehnique ited for the APO disposition, we an use PSC tehnique to generate the same swithing angles. In this tehnique the 5-level operation splits into four different bipolar PWM operations. The frequeny of eah arrier is four times smaller than the frequeny of arriers for APO ase, and the arriers are phase shifted by π/2. They are defined as: C = y (A, ω,0,0.5) 1 C = y (A, ω, π 2,0.5) 2 C = y (A, ω, π,0.5) 3 C = y (A, ω,3π 2,0.5) 4 Fig. 4(a,b,,d) show the four bipolar operations using these arriers and the resulting bipolar PWM waveforms. The amplitude of the sinusoidal wave is: A = MA 2 m As depited in Fig. 4(e), summing up the four resulting bipolar PWM waveforms gives the same 5- level PWM wave as we had in the APO disposition tehnique. Note that the frequeny of the resulting 5- level PWM is four times greater than that of eah bipolar PWM waveform: ω = 4 ω, P= 4P Naderi & Rahmati: Multilevel PWM Waveform Deomposition and Phase-Shifted Carrier Tehnique 153

5 Fig. 5 Amplitude and phase in terms of harmonis for the PWM waveforms in Fig. 4 respetively. Fig. 5(a,b,,d) show the amplitude and phase of bipolar PWM waveforms generated by the arriers C 1, C 2, C 3, and C 4 in terms of harmonis, respetively. Note that for better illustration, the phase angle of the harmonis with the amplitude more than 0.01A are shown and the others are set to zero. As depited, for any amount of phase shifting the amplitude diagrams are the same and the differene is in phase angles. The undesired harmonis aggregated near the P, 3P, 5P,, (2m-1)P harmonis are in phase opposition between C 1 and C 3, and between C 2 and C 4. Furthermore, the aggregated harmonis near the 2P, 6P,, (2m-1)2P harmonis are in phase opposition between C 1 (C 3 ) and C 2 (C 4 ). Therefore, these harmonis are eliminated when we sum up the four bipolar PWM waveforms and only the harmonis aggregated near 4mP (mp) harmonis remain. The amplitude of the resulting 5- level PWM wave in terms of harmonis is shown in Fig. 5(e). The spetrum of an N-level APO PWM or its PSC equivalent onsists of the harmonis aggregated near m(n-1)p in the spetrum of bipolar sub-operations for PSC. The other sets of harmonis in the spetrum of bipolar sub-operations are eliminated due to the opposition of their phase in different sub-operations. 2.4 Demonstration of Equality To demonstrate the equality of the resulting waveforms, we an use mathematial relations. However, summing up four expressions to obtain the results that we already have for APO disposition is not quite interesting. Instead, we use a method originally developed by Bennett to obtain the mathematial relation whih defines the PWM waveforms in the form of Fourier series. Using this approah gives a deeper understanding about different aspets of the onept and enables us to develop the equivalent PSC tehniques for PO and PH dispositions: [8] and [9]. This method hanges the problem of interseting two periodi funtions by the equivalent problem of interseting a non-periodi funtion with a two orientation periodi funtion. It states that by referring to a bipolar modulation in whih only one arrier signal is involved, to build the mathematial model of the 154 Iranian Journal of Eletrial & Eletroni Engineering, Vol. 4, No. 4, Otober 2008

6 modulation proess we an exploit the fat that the arrier is made of straight segments (it is a triangular wave). Eah straight segment has an intersetion point with the sinusoidal wave, but these straight segments are alternatively positive and negative ramps. Thus, we an onsider the intersetion of a negative ramp segment and the sinusoidal wave as the intersetion of extending the last positive segment and a sinusoid biased by A and in phase opposition with the sinusoid of last segment. Putting it together, we an modify the problem by defining a surfae (z = F(x,y)), whih onsists of different sinusoids alternatively in phase opposition toward y axis. The sinusoids subdivide the xy plane into different zones. The amount of z omponent in eah zone is alternatively +A/2 and A/2. Interseting this surfae with the plane (y = Px), gives us the PWM waveform. Fig. 6(a) shows the surfae, the plane, and their intersetion. To obtain the expression for the PWM waveform, the double Fourier series of the surfae is alulated. The intersetion is mathematially obtained introduing the relation: y = Px To find the time expression of the PWM waveform whih is the intersetion line between the surfae and the plane we must introdue the following relations: x = ωm t, y = ω t In the ase of over-modulation (M > 1) the sinusoids generated by the proess desribed above partially overlap, delimiting new zones that must not be taken into aount, the value of F inside these zones being the same as outside. To demonstrate the equality of APO and PSC tehniques, we just need to show that the resulting surfaes for these two tehniques are equal. The surfae presented, is periodi in both x and y. As depited in Fig. 6(b) whih shows the top view of the surfae, φ identifies the position of the Referene point (R). Thus, aording to periodiity of the surfae and by assuming R as the zero point, onsidering one period of the surfae gives us all we need to know about it: 0 x < 2π, π y < π, R = (0, 0) Fig. 6 Three-dimensional model for a bipolar modulation (a) the surfae z = F(x,y), the plane y = Px, and their intersetion line. (b) top view of the surfae and the plane with the arrier phase shifted by φ. Naderi & Rahmati: Multilevel PWM Waveform Deomposition and Phase-Shifted Carrier Tehnique 155

7 Fig. 7(a,b,,d) show the top view of equivalent surfaes for C1, C2, C3, and C4, respetively. Note that here we subdivide the proess into four bipolar proesses for eah arrier. Eah bipolar proess gives a surfae whih an assume only two values. These values are 0 and A for arriers above the zero referene, 0 and -A for the arriers below. Superimposing these surfaes gives the resulting surfae for the APO disposition tehnique. The resulting surfae and its top view are shown in Fig. 7(e,f). The top view of the equivalent surfaes for C 1, C 2, C 3, and C 4 is shown in Fig. 8(a,b,,d) respetively. Note that in this ase the only differene between surfaes is about their referene point. Thus, eah surfae is shifted by π/2 toward y axis. Adding up these surfaes gives the resulting surfae for the PSC tehnique whih is shown in Fig. 8(e,f). It is lear that the surfae for APO disposition and the one for PSC tehnique are the same. The only differene that we see in the figures is that four periods of the surfae for PSC is shown, omparing to one period that is shown for APO. It is beause the y axis in PSC is four times greater in sale than the y axis in APO disposition (Remember that ω = 4ω ). As we illustrated, for our 5-level example, the surfaes and onsequently the resulting PWM waveforms for APO disposition and PSC tehniques are equal. 2.5 PSC Equivalents for PO and PH In this setion, we modify the presented PSC tehnique to aommodate with PO and PH dispositions. Considering our 5-level example, the surfae F for PO disposition is obtained using the same proess as APO. The surfae and its top view are shown in Fig. 9(b,). As depited in Fig. 9(a), we an subdivide the resulting surfae for APO disposition into six different parts toward x axis. The parts are hosen based on the boundaries in whih the sinusoidal wave travels from the region of one arrier into another. In our example the sinusoidal wave passes from the region of C3 to that of C4 at: y = A 0.75sin(θ) = 0.5 θ = So, the six parts are defined as: B1 : 0< x<θ B4 : π < x < π+θ B2 : θ < x < π θ B5 : π + θ < x < 2 π θ B3 : π θ < x < π B6 : 2π θ < x < 2π Fig. 7 The surfae F orresponding to bipolar operations using (a) C1, (b) C2, () C3, (d) C4, (e), (f) orresponding surfae and its top view for APO disposition tehnique. 156 Iranian Journal of Eletrial & Eletroni Engineering, Vol. 4, No. 4, Otober 2008

8 Fig. 8 The surfae F orresponding to bipolar sub-operations using (a) C 1, (b) C 2, () C 3, (d) C 4, (e), (f) orresponding surfae and its top view for PSC tehnique. Fig. 9 PO disposition (a) parts that must be shifted, (b), () the orresponding surfae and its top view. Naderi & Rahmati: Multilevel PWM Waveform Deomposition and Phase-Shifted Carrier Tehnique 157

9 Comparing the surfaes for APO and PO dispositions, it is evident that if we shift the parts B 2 and B 5 by π toward y axis in the APO surfae, the PO surfae will be obtained. To generate the arriers needed for the PSC equivalent, we should impose the shifting operation to the orresponding parts of arriers. In other words, we should shift the parts B 2 and B 5 in eah arrier by π/4. Note that the y axis is four times greater in sale than the y axis. That is why the amount of phase shifting has been divided by 4. For example, the PO equivalent for C 3 is shown in Fig. 10(a) and is defined as: y (A, ω t, π,0.5) ω t : B, B,B,B = π y (A, ω t,,0.5) ωmt : B 2, B5 4 m C3(PO) 5 For the PH disposition, all we need to do is to shift the parts B 2, B 4, and B 6 by π toward y axis. Like the PO ase, the amount of phase shifting toward y axis is π/4. Parts that must be shifted, the surfae and its top view are shown in Fig. 11. The PH equivalent for C 3 is shown in Fig. 10(b) whih is: y (A, ω t, π,0.5) ω t : B,B,B = π y (A, ω t,,0.5) ω t : B,B, B m m C3(PH) 5 3 Generalization to N Levels The proedure we have introdued ould be generalized to other onditions like N-level inversion or even over-modulation. The surfae F orresponding to N-level PWM using APO disposition tehnique, has three publi shapes. Fig. 12(a,b) show the surfaes for N odd (N even and odd), respetively. And the surfae for N even is shown in Fig. 12(). The surfaes onsist of different zone separators whih are a part of a sine wave. In fat the part of modulating signal whih overlaps the region of eah arrier onfigures a set of zone-separators in the orresponding surfae whih are plaed due to the phase angle of that arrier and separate the zones that have two different z values orresponding to the boundary levels of that arrier. For N even the zone-separator starts from ±π/2 where as in the ase of N odd, it starts from the zero point. It is beause unlike the odd ase for N even the modulating signal starts from the middle point of C N arrier. The absissas of the points in whih the sinusoidal wave intersets the predefined levels between 0 and π/2 are denoted by x 1, x 2,, x N. The number of strit positive levels required by the modulation proess, the influene of the parameters N and M on the shape of the figures is onsidered by defining N whih is defined as: [ + ] MN 1 N odd N = 1 M MN + N even 2 Aording to the symmetry of sinusoidal wave and uniform distribution of levels, the other angles are defined as: 0 x<π 2 x,, x 1 N π 2 x<π π x,, π x N 2 π x< 3π 2 π+ x,, π+ x 1 N 3π 2 x< 2π 2π x,,2π x N 2 Fig. 10 (a) The modified arrier for PO disposition (C 3(PO) ), (b) The modified arrier for PH disposition (C 3(PH) ). 158 Iranian Journal of Eletrial & Eletroni Engineering, Vol. 4, No. 4, Otober 2008

10 Fig. 11 PH disposition (a) parts that must be shifted, (b), () the surfae F and its top view. Fig. 12 General surfaes for APO disposition (a) N odd (N even), (b) N odd (N odd), () N even. Naderi & Rahmati: Multilevel PWM Waveform Deomposition and Phase-Shifted Carrier Tehnique 159

11 Based on this, we an subdivide the surfae into parts between two onseutive angles whih are alled bases. There are four families of bases in APO disposition. Although to keep ompatibility of this definition with those works whih have done in this field we should add two more angles to our list, here we avoid doing this and we hoose the name part instead of base to avoid any kind of misunderstanding. The parts are defined as: B = [x, x ),,B = [2π x,2 π) N odd N 4 2 B = [0, x ),, B = [2π x,2 π) N even N 3 2 The equality of the APO and PSC tehniques is generally, beause of the same onfiguration (same shape and positioning) of these parts when onsidering one period of APO and 1/(N-1) period of PSC. In fat, the phase shifted bipolar surfaes in the PSC tehnique partially ooperate to form the same surfae as the one for APO. In general, an N-level PWM using APO disposition tehnique onsists of N-1 different arriers with the frequeny ω whih are defined by Eq. (1). The equivalent PSC tehnique onsists of N-1 different arriers with the frequeny ω /(N-1) whih are defined as: ω (n 1)2π = = N 1 N 1 C y (A,,, w), n 1,, N 1 n In a more general ase an N-level APO disposition PWM operation ould be deomposed into m different N (m) -level APO, PO, or PH PWM operations in a way that: is: (m) m(n 1) = N 1, m N The referene arrier for eah N (m) -level operation ω (n 1)2π N m m 2 n= 1,,m (m) (m) C = y (A,,, w) + (1 ) n (2) The arrangement of the other arriers in eah N (m) - level operation depends on the type of operation hosen (APO, PO, or PH) and must set aording to their referene arrier. Although deomposing an APO operation into several PO or PH operations seems strange, but it was preditable. As depited in Fig. 3 the first set of undesired harmonis differs among three dispositions, however the other sets are equal. Thus, for example if we add two N (2) -level PH operations whih are in phase opposition aording to Eq. (2), the first set of undesired harmonis will be eliminated and the seond set remains whih is the same for APO, PO and PH dispositions. The advantage of this generalization show itself speially, in the ase of asaded inverters, where we need to split an N-level operation into several 3-level operations to ontrol the swithes in eah unipolar ell. Then, eah 3-level operation uses one arrier but opposite phase sinusoids to generate the swithing angles needed for eah bipolar leg. For PO and PH dispositions, the strategy is different. Fig. 13(a) shows the surfae for PO disposition (definitely for N odd) in its general form. Comparing to the general surfaes illustrated for APO disposition, to aommodate the surfae of PSC with PO we should shift the parts B 2, B 4,, B N-3, B N, B N+2,, B 2N-3 by π toward y axis. In other words we must shift the even parts between 0 and π, and the odd parts between π and 2π by π/(n-1) toward y axis. Obviously, in the ase of deomposing into m different N (m) -level operations, the partly phase shifting operation must be applied to all the arriers but the amount of phase shifting is π/m. Thus, the referene arriers for eah N (m) -level operation are defined as: (m) (m) N C = (1 ) + n (PO) 2 ω (2n 1) π B [0, π) even y (A,,, w) ω t : m m m B odd [, 2 ) π π ω (n 1)2π y (A,,, w) t : otherwise ω m m m For PH disposition, all even parts must be shifted by π/m. The general surfaes for PH disposition with N odd and even are shown in Fig. 13(b,). The referene arriers are: (m) (m) N C n (PH) = (1 ) + 2 ω (2n 1) π y (A,,, w) ω t : B m m ω (n 1)2π y (A,,, w) ω t : B m m m even m odd Here we developed the PSC tehnique to aommodate with APO, PO, and PH dispositions. Modifying the PSC tehnique for other possible phase displaements is out of sope of this paper but the surfae represented here is a good means for handling those situations. Moreover, orret sign hoie of the dynami phase shift at eah interval results in a more uniform distribution of swithing angles. This effet is investigated in [10]. 160 Iranian Journal of Eletrial & Eletroni Engineering, Vol. 4, No. 4, Otober 2008

12 4 Speial Features 4.1 Asymmetrial Carriers Yet, our disussion was restrited to symmetrial triangular arriers. For the asymmetrial ases the situation is quite different. The effet of triangle width (w) on harmoni ontent of a bipolar PWM waveform is shown in Fig. 14. For 0.5 < w < 1 the spetrum only differs in the phase angles. Obviously, the harmoni pollution in the spetrum of a PWM waveform using sawtooth arrier (w = 0 or 1) is more than that of a symmetrial triangular arrier (w = 0.5). In other words, the more w approahes from 0.5 to 0 or 1 the more harmoni distortion we will have at the inverter output. Thus using symmetrial triangular arrier is more onvenient: [11]. Moreover, the more w approahes to 0 or 1 the more unwanted harmonis are likely to onentrate on the multiples of P. On the other hand, the first set of undesired harmonis for an N-level PO or PH dispositions are similar to (N-1)th set of undesired harmonis for a bipolar modulation using w = 0.25(0.75) or w = 0(1) respetively. Therefore, we an use a regular PSC using sawtooth arriers instead of PH and a regular PSC using w = 0.25(0.75) arriers instead of PO disposition. Although, from the harmoni pollution point of view this replaement is not effiient, but in some ases it is useful due to omplexity of the arriers needed in the PSC equivalents of PO and PH tehniques. Fig. 13 General surfaes (a) PO (N odd), (b) PH (N odd), () PH (N even). Naderi & Rahmati: Multilevel PWM Waveform Deomposition and Phase-Shifted Carrier Tehnique 161

13 Fig. 14 The effet of triangle width (w) on the spetrum of orresponding bipolar PWM waveform. 4.2 Logial Deomposition In the ase of PSC equivalent for PO and PH dispositions, the major onern is about the implementation. Comparing to a regular PO or PH disposition, implementing a PSC equivalent needs twie more arriers. Moreover, the angles in whih we should swith between arriers depend on the modulation index (M) and vary due to different amplitudes needed. A somewhat easier way is to generate the resulting PWM waveform using onventional PO or PH methods and then deompose it to bipolar PWM sub-waveforms using a logial strategy. The strategy is based on preparing the most possible amount of time between two onseutive edges in the bipolar PWM subwaveforms. To do this a stak is defined whih saves the privilege level of eah arrier to show whih one is the best to be omplemented. The stak onsists of N-1 ells. Eah ell an save a number from 1 to N-1 whih refers to a speified sub-waveform. The privilege level is defined aording to the distane of a ell from the end of stak. Thus, the ell at the start of stak has the most privilege where as the one at the end has the least. The algorithm is: 1. Choose an initial state for eah bipolar subwaveform in a way that their total sum is equal to the starting value of the main PWM waveform. 2. Load the stak with arbitrary initial sequene (e.g., the sequene 1,, N-1) 3. Keep the states of the arriers until an edge on the main wave is deteted. 4. For rising (falling) edge, look up the stak from start to the end to find the most privileged ell whih its orresponding arrier is in 0 (1) state. 5. Complement the state of orresponding arrier and transfer the ell to the end of stak (a irular shift using the stak from the seleted ell to the end). 6. Go to step 3. The results for a 5-level deomposition is shown in Fig. 15. The resulting bipolar PWM waveforms are the same as those in PSC tehnique. Implementing suh an algorithm needs a sequential logial iruit. At least two sequenes are needed. One for read the stak and one for modify it. On the other hand, the pulses are generated by an analog proess thus they might have very small amount of pulse widths. In this ase, we should use a tiny pulse arrester or modifier iruit to prevent abnormal operation of our sequential iruit. 5 Applying to Parallel and Series Cells The deomposition tehniques that we have developed have two main advantages. First, a multilevel PWM waveform is deomposed into several subwaveforms with lower frequeny. Seond, the main harmonis of sub-waveforms are equal. Thus, they are well suited to be used with parallel or series onneted ells: [12]. The ells are usually bipolar (Fig. 16(a)) or unipolar (Fig. 16(b)). Applying the deomposition method to bipolar ells is quite simple. We should deompose the N-level PWM waveform into N-1 bipolar subwaveforms and apply them to eah bipolar leg (Q 1 must be fed by the bipolar waveform and Q 2 must be fed by its negation). For the unipolar ells the situation is slightly different. The unipolar ells are naturally 3-level inverters. Thus we should deompose our N-level PWM waveform into (N-1)/2 3-level sub-waveforms. Then eah ell must generate one of these 3-level PWM subwaveforms. Again, the 3-level waveform should be deomposed into two bipolar operations. The positive leg (Q 1 and Q 2 ) must be fed straightly and the negative leg (Q 3 and Q4) must be fed inversely. In Brief, we have two strategies for unipolar ells: 162 Iranian Journal of Eletrial & Eletroni Engineering, Vol. 4, No. 4, Otober 2008

14 amount of time between two onseutive angles in eah sub-waveform. It is useful for deomposing all types of multilevel PWM waveforms. Fig. 15 Logial Deomposition of a 5-level PWM waveform into bipolar sub-waveforms. 1. Deompose the N-level operation into N-1 bipolar sub-operations. Use the PWM waveforms generated by arriers with phase 0 φ < π to ontrol the positive legs. For negative legs, use the negation of the PWM waveforms generated by the arriers with phase π φ < 2π. 2. Deompose the N-level operation into N-1 bipolar sub-operations. Use the PWM waveforms generated by arriers with phase 0 φ < π to ontrol the positive legs. For negative legs, use the same arriers but the negation of modulating signal (opposite phase sine wave) to generate the PWM waveforms. It is lear that the seond strategy needs twie less number of arriers, while the results of using these strategies are the same. Fig. 16 (a) Bipolar ell, (b) Unipolar (H-bridge) ell. 5 Conlusion In this paper we have presented PSC tehnique as an equivalent for APO disposition and have modified it to aommodate PO and PH disposition tehniques. While the PSC tehnique and APO disposition generate the same waveform, PSC is more onvenient to be used with asaded and parallel multilevel topologies. We have developed the equivalent PSC tehniques for PO and PH dispositions based on partly shifted arriers. These tehniques are also useful for asaded and parallel three-phase topologies. We have shown that using asymmetrial arriers in PSC tehnique an also help us to simply approah to the properties of PO and PH waveforms. We have proposed logial deomposition based on preparing the most possible Referenes [1] Meynard T. A. and Foh H., Multi-level onversion: high voltage hoppers and voltagesoure inverter, Pro. IEEE Power Eletron. Spe. Conf., Vol. 1, pp , Jun./Jul [2] Rodriguez J., Lai J. S. and Peng F. Z., Multilevel inverter: a survey of topologies, ontrols, and appliations, IEEE Trans. Ind. Eletron., Vol. 49, No. 4, pp , Aug [3] Jin B. S., Lee W. K., Kim T. J., Kang D. W. and Hyun D. S., A study on the multi-arrier PWM methods for voltage balaning of flying apaitor in the flying apaitor multi-level inverter, IEEE Ind. Eletron. Conf., pp , Nov [4] Lee Y. H., Kim R. Y. and Hyun D. S., A novel SVPWM strategy onsidering DC-link balaning for a multi-level voltage soure inverter, Applied Power Eletron. Conf. (APEC), Vol. 1, pp , Mar [5] Marhesoni M., High-performane urrent ontrol tehniques for appliations to multilevel high-power voltage soure inverters, IEEE Trans. Power Eletron., Vol. 7, No. 1, pp , Jan [6] Carrara G., Gardella S., Marhesoni M., Salutari R. and Siutto G., A new multilevel PWM method: a theoretial analysis, IEEE Trans. Power Eletron., Vol. 7, No. 3, pp , Jul [7] Kang D. W. and Hyun D. S., Simple harmoni analysis method for multi-arrier PWM tehniques using output phase voltage in multilevel inverter, Pro. IEE Eletri Power Appl., Vol. 152, No. 2, pp , Mar [8] Blak H. S., Modulation Theory. New York, NY, USA: D. Van Nostrand, [9] Bowes S. R., New sinusoidal pulsewidthmodulated inverter, Pro. IEE, Vol. 122, No. 11, pp , Nov [10] Naderi R. and Rahmati A., Phase-shifted arrier PWM tehnique for general asaded inverters, IEEE Trans. Power Eletron., Vol. 23, No. 3, pp , May [11] Hamman J. and Van Der Merwe F. S., Voltage harmonis generated by voltage-fed inverters using PWM natural sampling, IEEE Trans. Power Eletron., Vol. 3, No. 3, pp , Jul [12] Ueda F., Asao M. and Tsuboi K., Parallelonnetions of pulsewidth modulated inverters using urrent sharing reators, IEEE Trans. Power Eletron., Vol. 10, No. 6, pp , Nov Naderi & Rahmati: Multilevel PWM Waveform Deomposition and Phase-Shifted Carrier Tehnique 163

15 Roozbeh Naderi was born in Amol, Iran, in He reeived the B.S. degree in Eletronis Engineering from Mazandaran University, Babolsar, Iran, in 2005, and the M.S. degree in Eletronis Engineering from Iran University of Siene and Tehnology (IUST), Tehran, in His researh interests are in modulation strategies for power eletroni systems, multilevel inverters, and resonant inverters. Abdolreza Rahmati was born in Abadeh, Iran, in He reeived the B.S. degree in Eletronis Engineering from Iran University of Siene and Tehnology (IUST), Tehran, in 1979, and the M.S. and Ph.D. degrees in Power Eletronis from Bradford University, West Yorkshire, U.K. in 1987 and 1990, respetively. He has been a member of the faulty at IUST where he is urrently an Assoiate Professor. He was Deputy for Eduational Affairs and Postgraduate Studies in the Department (1992 to 2003), and a Visiting Professor at the Illinois Institute of Tehnology (IIT), Chiago (September 2004 to May 2005). His fields of interest are Miroproessors and Miroontroller-Based System Design, Motor Drives and Control, HVDC Transmissions, Modulation Strategies for Power Eletroni Systems, Multilevel Inverters, and Power Devies. Dr. Rahmati is a member of the Institution of Engineering and Tehnology (IET) and Engineering Counil, U.K Iranian Journal of Eletrial & Eletroni Engineering, Vol. 4, No. 4, Otober 2008