Double Tangent Sampling Method for Sinusoidal Pulse Width Modulation

Transcription

1 Compuaional and Applied Mahemaics Journal 2018; 4(1): 8-14 hp:// ISS: (Prin); ISS: (Online) Double Tangen Sampling Mehod for Sinusoidal Pulse Widh Modulaion Ligeng Shao, Yanming Wu School of Elecrical and Informaion Engineering, Dalian Jiaoong Universiy, Dalian, China address (Ligeng Shao) Ciaion Ligeng Shao, Yanming Wu. Double Tangen Sampling Mehod for Sinusoidal Pulse Widh Modulaion. Compuaional and Applied Mahemaics Journal. Vol. 4, o. 1, 2018, pp Received: January 11, 2018; Acceped: January 26, 2018; Published: February 27, 2018 Absrac: The conrol mehod of inverer for low power single-phase moor mainly uses pulse widh modulaion echnology. Based upon he radiional naural mehod, he angen mehod and he secan mehod, he swiching ime poin calculaion formulae are derived for he double angen mehod. Compared he swiching ime wih above hree mehods and analyzing he, he SPWM wave generaed by he double angen mehod are closes o he wave generaed by he naural mehod. The double angen mehod can improve he accuracy of SPWM wave. The harmonic analysis indicaes ha he SPWM wave obained by he double angen mehod has greaer fundamenal componen, and is low order harmonic componens are closer o he naural mehod compared wih he SPWM wave obained by he secan mehod. The double angen mehod realizes easily in digial sysems. I is feasible o obain an ideal SPWM wave by he double angen mehod. Keywords: Sinusoidal Pulse Widh Modulaion (SPWM), Tangen Sampling Mehod, Secan Sampling Mehod, Double Tangen Sampling Mehod 1. Inroducion Sinusoidal pulse widh modulaion (SPWM) is widely used in power elecronics echnology recenly [1]-[4]. SPWM waves are characerized by consan ampliude pulses wih differen duy cycle for each period. A sequence of volage pulses can be generaed by swiching of he power elecronic devices. The modulaion of pulse widh can conrol he oupu volage of inverer, and reduce is harmonic componen [5]-[7]. In variable frequency speed conrol sysem, SPWM is one of he mos imporan modulaion echnique applied o moor conrol and inverer [8]-[10]. A presen, he naural mehod is an ideal SPWM echnology since is waveform is closes o he sinusoidal wave [11]. The mehod can ge good qualiy of waveform. The conrolling precision of he fundamenal wave is high. However, he naural mehod needs o solve ranscendenal equaions. This is difficul o realize by digial conrol devices in engineering applicaion [12, 13]. In order o avoid handling he ranscendenal equaions, some echniques such as symmeric regular, asymmeric regular and modified regular have been presened. These mehods can be designed flexible and easier o implemen. However, heir modulaion waveforms conain higher harmonic conens o some exen. Therefore, researchers devoe o develop he mehods which SPWM waves conain low harmonic conen. The effec of modulaion waves is more close o he naural mehod of SPWM as possible. And hese mehods are easy digial realizaion. In his sudy, a double angen mehod o generae SPWM wave was explored on he basis of he naural mehod, he angen mehod and he secan mehod. A firs, he principle of he double angen mehod was analyzed, Then he swiching ime calculaion formula were derived. Finally, he precision of swiching ime and he characerisics of harmonics for he double angen mehod was analyzed, and compared hem wih hose for he naural mehod and he secan mehod. 2. Several Sampling Mehods 2.1. aural Sampling Mehod The naural mehod is depiced in Figure 1. The

2 Compuaional and Applied Mahemaics Journal 2018; 4(1): pulse widh of modulaion wave varies wih he frequency and ampliude of sinusoidal wave. Figure 1 indicaes ha he inersecion of riangle waveform and sinusoidal waveform is asymmerical o he median of riangular carrier waveform. Suppose ha he maximum ampliude of he sinusoidal volage is U rm. The sinusoidal volage can be expressed as u r = U rm sinω. Given he maximum ampliude of bipolar riangular carrier wave U m = 1, he modulaion deph is defined M Urm/ Um 0 < M < 1. Then he sinusoidal wave ur = Msinω. The urn-on ime is denoed by on. The urn-off ime is denoed by off. And he period of riangular carrier wave is denoed by T c. In a period of riangular carrier wave, he coordinaes of = ( ) poin C is (, -1); he coordinaes of poin D is 0.5 ( + 0.5, 1); and he coordinaes of poin E is (, 1). The linear equaions of CD and CE can be obained, respecively. 4 lcd: y = x (1) 4 lce : y = x 4 1 (2) In order o obain he accurae SPWM waveform, he urn-on ime and he urn-off ime can be calculaed by Equaion (3) and (4). Msinω 4 / = 0 (4) Where is he poin number of sinusoidal wave in half period; is he sequence number of poins. The naural mehod conrols power elecronic devices on-off a he naural inersecion poins of he sinusoidal modulaion wave and high-frequency riangular carrier wave. The swiching poin ime is solved by numerical ieraion since Equaion (1) and (2) are ranscendenal equaion. Therefore, his mehod is no suiable for real-ime digial conrol [10]. Among pulse widh modulaion mehods, alhough he regular mehod is simple o calculae, he harmonic conens are relaively greaer. Tangen mehod and secan mehod can obain higher accurae SPWM waveform, and effecively reduce harmonic conens [14] Tangen Sampling Mehod Figure 2 shows angen mehod. The mehod samples a he negaive pea of riangle wave. A angen of sinusoidal wave is drawn a he poin F. The swiching ime is deermined by he inersecion poins of he angen and riangular carrier wave. The coordinaes of poin F are (, M sin ). Le ω = θ( ), he angen equaion is lab: u = θ( ) Mcos( ) M cos Msin + The formula of he swiching ime can be obained by solving he angen equaion and wo linear equaions of riangular carrier wave simulaneously. (5) Figure 1. aural mehod. Msinω + 4 / = 0 (3) Figure 2. Tangen mehod.

3 10 Ligeng Shao and Yanming Wu: Double Tangen Sampling Mehod for Sinusoidal Pulse Widh Modulaion on 4 + Mcos( ) Msin = Mcos + 4 (6) 2 1 M(sin sin ) 2 u = θ + Msin (8) off 4 + Mcos Msin = Mcos 4 (7) 2.3. Secan Sampling Mehod As shown in Figure 3, a he poin C of he negaive pea of he riangular carrier wave, a line perpendicular is drawn o abscissa axis. The line inersecs he sinusoidal wave a poin F. Through he poin D of he posiive pea of he riangular carrier wave, a line perpendicular is done o abscissa axis. The line inersecs sinusoidal wave a poin G. Similarly, he inersecion poin H is goen. The poin F is respecively conneced wih wo inersecion poins G and H. There are wo secans wihin a riangle carrier period. The swiching ime is deermined by he inersecion poins of riangular carrier wave and he wo secans. The poins G and F are conneced. The secan line GF and he riangular carrier wave inersec a poin A. The linear l GF equaion is on Figure 3. Secan mehod. Solving Equaion (1) and (8), he urn-on ime can be obained. ( ) ( ) ( ) + ( ) ( ) Msin / + 2Msin / 2Msin 2 1 /2 = 2 { 2 Msin / Msin 2 1 /2} (9) off ( ) ( ) ( ) + ( ) ( + ) Msin / + 2Msin / 2Msin /2 = 2 { 2 Msin / Msin 2 1 /2} (10) 3. Double Tangen Sampling Mehod Figure 4 shows double angen mehod. The mehod samples a he posiive pea of he riangle wave. Two angens for sinusoidal wave are done a wo poin G and H, respecively. The swiching ime is deermined by he inersecion poins of he angens and riangular carrier wave. The formula of he swiching ime can be obained by solving he wo angen equaions and linear equaions of riangular carrier wave. There are wo angens wihin a period of he riangle carrier wave. Through he poins D and E of he posiive pea of he riangular carrier wave, wo doed lines are done perpendicular o abscissa axis. The lines inersec sinusoidal wave a poin G and H, respecively. A he poin G and H, angens are done for he sinusoidal wave, respecively. The wo angens and he riangular carrier wave inersec a poin A and B. The swiching ime is deermined by he inersecion A and B. Figure 4. Double angen mehod.

4 Compuaional and Applied Mahemaics Journal 2018; 4(1): The coordinaes of poin G are (, Msin ) The coordinaes of poin H are (, Msin + ). The wo angen equaions are as following: lga: u = θmcos( ) M cos Msin lhb: u = θmcos( ) M cos Msin + (11) (12) Solving equaion (1) and (11), (2) and (12), he swiching ime can be obained. on M ( 0.5) cos Msin = 0.5 Mcos + 4 (13) off M ( + 0.5) cos Msin = Mcos 4 (14) The formula (13) and (14) are deduced by he double angen mehod. They are algebraic and easy o implemen digially. Seing he carrier raio 2, 0.5 sin, 0.5 cos, 0.5 sin + and 0.5 cos + can be calculaed by he microprocessor. While he modulaion deph M changes, he swiching ime can be calculaed o obain SPWM wave. 4. Accuracy and Harmonic Analysis 4.1. Accuracy Analysis Supposing, he carrier raio is 18. The modulaion deph is M=0.8. is aen 1 o 9. Table 1 gives he swiching ime of angen mehod, secan mehod and double angen mehod. The accuracy of hese naural angen Table 1. Swiching ime and error accuracy. mehods is compared wih ha of naural mehod. As saed in Table 1, he angen mehod and he secan mehod can conrol he error in a small range. While is aen 1 o 9, he angen mehod can conrol he error wihin ~0.062%; he secan mehod can mae he error wihin %~0.025%. In comparison he angen mehod wih he secan mehod, i indicaes ha he accuracy of secan mehod is higher han ha of angen mehod. The error of double angen mehod can be conrolled beween 0.013%~0.012%. This verifies ha he precision of he double angen mehod is higher han ha of he secan mehod and angen mehod. The double angen mehod is beer han he angen and secan mehod. The swiching ime of he double angen mehod is closes o ha of he naural mehod. The accuracy of he double angen mehod can mee he requiremens of higher conrol. secan double angen on off on off on off on off on off

5 12 Ligeng Shao and Yanming Wu: Double Tangen Sampling Mehod for Sinusoidal Pulse Widh Modulaion naural angen secan double angen on off on off on off on off Harmonic Analysis The bipolar SPWM wave is a sequence of periodic pulses. The funcion which he waveform expresses is periodic and odd. Is period is 2. The posiive and negaive half period of sinusoidal wave is equal divisions ( usually aes muliples of 3). Then he funcion can be defined on he inerval [0, ]. f ( θ ) Ud on( ) θ < of( ) f = 1 Ud θ < on( ) or off( ) θ < (15) where = 1,2,3.... Funcion (15) can be expanded in Fourier series. ( ) f ( θ ) = a0 + ancos( nθ) + bnsin( nθ) (16) n= 1 f θ is due o an odd funcion on [, ], SPWM wave does no conain even harmonics. Thus, a 0 = 0, a = 0. where ( θ ) f = b sin( nθ) (17) n n= 1,3,5... n 2 bn = f ( θ ) sin ( nθ ) dθ 0 2U θon θoff d = 1 sin ( ) sin nθ dθ ( nθ ) dθ sin ( nθ ) dθ + θon θoff = 1 2Ud = 2 cos( non( )) cos( noff( )) + cos( n) 1 n = 1 (18) Under he same condiions, he bipolar SPWM wave is generaed by he secan mehod and double angen mehod, respecively. The harmonic characerisics of he bipolar SPWM wave can be analyzed using Fourier series (18). Table 2 shows he fundamenal and harmonic coefficiens of SPWM waves achieved by he secan mehod and he double angen mehod, respecively. The harmonics are wihin 40h. The fundamenal coefficien of SPWM wave obained by he double angen mehod is closes o ha of SPWM wave obained by he naural mehod. The harmonic coefficien of SPWM wave obained by he double angen mehod is lower han ha of SPWM wave obained by he secan mehod. The volage disorion of SPWM wave obained by he double angen mehod is ligher han ha obained by he secan mehod. Table 2. Harmonic coefficien. aural mehod Secan mehod Double angen

6 Compuaional and Applied Mahemaics Journal 2018; 4(1): aural mehod Secan mehod Double angen The harmonic coefficiens generaed by he naural mehod, secan mehod and double angen mehod is expressed as b n, b Sn, b Dn, respecively. Taing n = 1, 3, 5, , he relaive value of harmonic coefficien σ is defined as ( ) bsn bdn bn σ = 100% bn (19) Figure 5 depics he relaive value of harmonic coefficiens of SPWM wave achieved by he secan mehod and double angen mehod. To low order harmonics (order less han 20), he relaive value of harmonic coefficiens of he double angen mehod is lower han hose of he secan mehod obviously. To high order harmonics, he relaive value of harmonic coefficiens of he double angen mehod is also lower for mos of harmonic orders. The harmonic characerisics of SPWM wave achieved by he double angen mehod is prior o hose of SPWM wave achieved by secan mehod wihin 40 orders. While he demand of oupu volage is higher, i is a good choice ha he SPWM wave is generaed by he double angen mehod. (b) Double angen mehod Figure 5. Relaive value of harmonic coefficiens. 5. Conclusions Compared wih he angen mehod and he secan mehod, he double angen mehod can generae he SPWM wave which error is smalles. Harmonic analysis shows ha he fundamenal componen of SPWM wave generaed by double angen mehod is high. The harmonic componens of SPWM wave obained by double angen mehod are lower han hose of he secan mehod and he angen mehod. The double angen mehod can be implemened in digial conrol sysems, and improve he accuracy of sinusoidal pulse widh modulaion. This mehod can be applied o indusrial conrol and engineering applicaion. Acnowledgemens This wor was suppored by he aional aural Science Foundaion of China under projec [gran number ]; he aural Science Foundaion of Liaoning Province under projec [gran number ]. References [1] H. F. Xiao, K. Lan and L. Zhang, A quasi-unipolar SPWM full-bridge ransformerless PV grid-conneced inverer wih consan common-mode volage, IEEE Transacions on Power Elecronics, 30 (6): , [2] M. Laa, E. Kouroulis and A. Dollas, Developmen of an FPGA-based SPWM generaor for high swiching frequency DC/AC inverers, IEEE Transacions on Power Elecronics, 29 (1): , [3] I. Cola and E. Kabalci, Developing a novel sinusoidal pulse widh modulaion (SPWM) echnique o eliminae side band harmonics, Inernaional Journal of Elecrical Power & Energy Sysems, 44 (1): , (a) Secan mehod [4] M. M. Renge and H. M. Suryawanshi, Mulilevel inverer o reduce common mode volage in AC moor drives using SPWM echnique, Journal of Power Elecronics, 11 (1): 21-27, 2011.

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