Sinu Pwm1

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 51, NO. 8, AUGUST 2004

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Compact Analytical Solutions for Determining the Spectral Characteristics of Multicarrier-Based Multilevel PWM W. H. Lau, Member, IEEE, Bin Zhou, and Henry S. H. Chung, Senior Member, IEEE

Abstract—Multilevel pulsewidth modulation (PWM) has become very important in high-power applications in recent years. Many different types of topology and control strategy have been developed to cater for the increasing need. One of the major concerns of the control strategy is to reduce the undesirable harmonics in order to improve the performance and efficiency. This paper presents the derivation of analytical solutions for determining the spectral characteristics of the multicarrier-based multilevel sinusoidal PWM. By decomposing the multilevel PWM into a series of sub-PWM and followed by using double Fourier series analysis, compact analytical solutions for any level with any carrier phase disposition have been obtained. Closed-form solutions have also been derived for double-sided multilevel PWM under certain specific carrier phase settings. The analytical solutions have been cross verified extensively with simulations and the results match with each other very well. It is shown that the proposed analytical solutions are by far the most compact reported in the literature. Index Terms—Double Fourier series, multilevel pulsewidth modulation (PWM) inverter, spectral characteristics.

I. INTRODUCTION

P

ULSEWIDTH modulation (PWM), in particular the two-level inverters and converter, has found many applications in power electronic applications. However, two-level PWM inverters cannot be used for high-power applications due to the limited power rating of the devices. In recent years, various topologies including the neutral-point-clamped inverter [1], the capacitor-clamed inverter [2], and the cascaded inverter [3], have been developed for multilevel inverter implementation. Various control strategies associated with these topologies have also been developed. The classic multicarrier-based sinusoidal PWM (SPWM) [4] is one of the most common strategies. Other common approaches include space-vector control (SVC) [5], selective harmonic elimination [6], and space-vector modulation (SVM) [7]. Readers may refer to a recent special issue of the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS for a comprehensive coverage in multilevel inverters [8]. One of the major concerns of using multilevel PWM is how to reduce the undesirable harmonics in order to improve the Manuscript received September 25, 2003; revised December 12, 2003. This work was supported by the RGC of the HKSAR, China, under Research Grant CityU 1192/99E. This paper was recommended by Associate Editor M. K. Kazimierczuk. W. H. Lau and B. Zhou are with the Department of Computer Engineering and Information Technology, City University of Hong Kong, Kowloon, Hong Kong. (e-mail: [email protected]; [email protected]). H. S. H. Chung is with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong. (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSI.2004.832790

performance and efficiency. An analytical study on the spectral characteristics of various control strategies has been reported elsewhere, and this paper will only concentrate on the study of multicarrier-based sinusoidal PWM. As indicated in the analysis of the two-level PWM by Bennett [9], double Fourier series is proven to be the most convenient tool for analyzing the nonlinear PWM problem. The first attempt to develop analytical expression for the multilevel PWM spectrum has been reported by Carrara et al. [4]. However, his derivation involves a fairly complicated decomposition of the PWM waveform, and the analytical solution is not easy to use. In the recent work reported by Holms [10]–[12], new sets of analytical solutions for naturally and regularly sampled multilevel PWM have been derived. Though the analysis is simple enough to follow, the solution is still rather complicated and the solution is only applied to odd-level PWM. The authors have also used the double Fourier series to analyze the multilevel SPWM and reported the results in [13]. Since the analysis was carried out in a way that the symmetrical properties of signal interaction has not yet been exploited, the analytical solution is still a bit cumbersome. After carefully reexamining this approach, it is found that simple and compact analytical solutions can be developed for various multicarrier-based multilevel PWM strategies. In this paper, the detailed derivation of the analytical approach will be presented. In the following discussion, the multilevel PWM inverter is referred to the multicarrier-based multilevel PWM inverter for convenient purpose. The analysis is based on naturally sampled multilevel PWM with arbitrary carrier phase dispositions and a solution for overmodulation is also derived. The double-sided odd-level PWM will be used to illustrate the derivation followed by the double-sided even-level PWM. The analytical solution for single-sided odd-level PWM will also be given. By comparing the results given in [4] and [10]–[13], the proposed analytical solutions are by far the most compact, and closed-form solutions can also be obtained under certain specific carrier phase dispositions for various strategies. One of the most attractive features of the proposed approach is that the spectral characteristics can be examined in detail for any carrier phase setting for an arbitrary level of PWM. II. SPECTRAL CHARACTERISTICS OF DOUBLE-SIDED ODD-LEVEL PWM In this section, the detailed derivation of the multilevel PWM spectral characteristics will be presented. Similar to the analysis of a two-level SPWM inverter reported in [9], this technique is extended to derive an analytical solution for a double-sided oddlevel PWM inverter (DOPWM). In order to produce an odd-level

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Fig. 1.

Double-sided odd multilevel PWM.

Fig. 2.

Double-sided even multilevel PWM.

PWM, an even number of carriers is required, whereas an odd number of carriers is required to generate an even-level PWM, as shown in Figs. 1 and 2. For the analysis given in the following, it is assumed that all of the carriers are of the same carrier freand same magnitude , which is normalized to 2 V quency peak to peak. Carriers located at the positive band of the modu, . Similating signal are called positive carriers , , are called negative carriers. larly, the carriers The phase shifts of the positive and negative th carriers are repand , respectively. The modulating signal is resented by , where is the peak amdefined as is the angular frequency, and is the phase of the siplitude, nusoidal modulating signal, respectively. Since all of the phase terms for both the carriers and modulating signal can be controlled in an arbitrary manner, their effects on the output spectrum can be evaluated for various phase dispositions. A. Three-Dimensional PWM Model The main idea of the proposed approach is to determine the sub-PWM signals generated by individual positive and negative carriers and then to obtain the multilevel PWM by summing all individual sub-PWM signals together. Each sub-PWM can be considered as a result of a carrier modulated by a dc biased sinusoidal signal. Similar to the traditional two-level siis defined as nusoidal PWM inverter, the modulation index for a -level DOPWM. The time axis can and be presented by two different angular abscissas, . Within 0 to , can be divided into several sections according to the intersection points between the carrier envelope and modulating signal. The envelope of the carrier is given by and , and may or may not intwo straight lines, tersect with the modulating signal. Fig. 3 shows three different

Fig. 3. Interaction between the ith positive carrier signal V with a dc bias of (1 2i).

0

Fig. 4.

C

and the modulating

Definition of pulsewidth for a single cycle of a positive carrier.

kinds of interaction between the th positive carrier and the ). Obviously, there modulating signal with a dc bias of ( will be no intersections if the magnitude of the dc biased modulating signal is less than that of the carrier as shown in Fig. 3. and are the projections of the intersection points on and , respectively. is equal to 0 if the axis for the carrier signal and modulating signal do not intersect. Conand the value of is given as sequently, (1) By comparing the amplitude of the carrier and the modulating signal, a series of waveforms will be generated as the output. If the carrier is smaller than the modulating signal, the output will be 1, otherwise 0. Fig. 4 shows the PWM output for a single cycle of a positive carrier, and we can write

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(2) (3)

LAU et al.: COMPACT ANALYTICAL SOLUTIONS FOR DETERMINING SPECTRAL CHARACTERISTICS

a single cycle of a negative carrier and parameters given by

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and

are (5) (6)

For the negative carrier, the 3-D PWM model function is given by

(7) can be obtained The final multilevel PWM function and , by summing up all the 3-D functions, generated by individual positive as well as negative carriers. B. Spectral Characteristics is a periodic function in terms of both and Since with a period of , it can be represented using double Fourier series as follows:

Fig. 5. Interaction between the ith negative carrier signal V with a dc bias of (2i 1).

0

C

and the modulating

(8) Substituting (2)–(7) into

(9) Fig. 6. Definition of pulsewidth for a single cycle of a negative carrier.

for positive and negative carriers are given, respectively, as

Since the PWM output voltage is a function of two variables and , it can be represented by a three-dimensional (3-D) model [14]. Considering both Figs. 3 and 4, a 3-D PWM model funccan be defined for the th positive carrier and can tion be written as (10) (4)

Similarly, Fig. 5 shows three different kinds of interaction and the modulating signal between the th negative carrier with a dc bias of ( ). Fig. 6 shows the PWM output for Authorized licensed use limited to: GOVERNMENT COLLEGE OF ENGINEERING. Downloaded on January 22, 2009 at 04:05 from IEEE Xplore. Restrictions apply.

(11)

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The three possible and combinations for (10) and (11), ,( ) and ( ), i.e., will be derived in detail. Since the analysis is carried out using even functions, i.e., symmetrical triangular carriers and cosine becomes zero and hence . modulating signal, Equations (12)–(17) can be obtained (12) Fig. 7. Definition of pulsewidth for DEPWM.

(13)

(14)

(15)

(16)

(17) where (18) and The final carriers, i.e., have

. is a summation of the PWM outputs from all . Finally, we and

the angular frequency of the modulating signal and carriers, i.e., and . The general solution for a -level PWM with carriers and double-sided is then given by (20), shown at the bottom of the page. , is given For an overmodulation situation, i.e., in (21), shown at the bottom of the page, and the second term in . (20) is replaced by Equation (20) is further examined under certain specific phase combinations, and it is found that, if the carrier phases and , respectively, i.e., are set to be alternative phase opposition disposition (APOD), it can further be simplified to a compact closed form given in (22), shown is the th-order at the bottom of the page, in which Bessel function of the first kind. Hence, the spectrum of a DOPWM inverter with APOD strategy can be directly computed using (22) without involving any numerical approximation techniques. III. SPECTRAL CHARACTERISTICS OF OTHER MULTICARRIER-BASED PWM STRATEGIES The derivation of the spectral characteristic for the doublesided even-level PWM (DEPWM) is similar to that given in Section II, but with an additional 0th carrier, as shown in Fig. 2. is defined as The modulation index for a double-sided -level PWM. The definition of pulsewidth of DEPWM is shown in Fig. 7, where

(19) In order to have a complete view on the effects to the spectrum due to various phase dispositions, phase shift is introduced to

(23a)

(20) (21)

(22)

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(23b) The 3-D PWM function

is given as follows:

(24a)

Fig. 8. Multicarrier cascaded multilevel PWM inverter topology used for the analysis.

(24b)

(24c) where

level PWM The general solution for a double-sided can be obtained by substituting (26)–(32) into with , is (8), whereas the solution for overmodulation, i.e., obtained by replacing , given in (33), shown at the bottom of the next page. , a compact Similar to the case for DOPWM and closed-form solution for DEPWM can also be obtained for and , as shown in (34) at the bottom of the next page. For the case of single-sided odd-level PWM, its derivation and general analytical solution is rather complicated. However, a compact analytical solution has also been found for and , and is given in (35), shown at the bottom of the next page. IV. SIMULATION RESULTS AND DISCUSSIONS

(25) and , the coefficients are given in For (26)–(31), shown at the bottom of the page, where (32)

A. Inverter Topology The analytical solution derived is for analyzing the spectrum of a multicarrier-based cascaded multilevel PWM inverter, and the topology is shown in Fig. 8. The only inconvenient part of the proposed general solution is that it contains an integral term which does not have a direct solution and requires numerical integration technique (e.g., adaptive Simpson

(26) (27) (28) (29) (30) (31)

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quadrature) to compute its value. In practice, due to the inconvenience involved in generating the carriers with specific phase setting, three control strategies, namely the alternative phase opposition disposition (APOD), phase opposition disposition (POD), and phase disposition (PD), are of common concern since the carriers are all either in phase or 180 out of phase, which can easily be generated. In particular, compact closed-form solutions given in (22), (34), and (35) are derived for carrier phases in either in phase or 180 out of phase. However, the proposed general solution is not limited to the above carrier phase dispositions since other phase relationships can also be generated using frequency synthesizing techniques with the current digital technology. B. Simulation Results Pspice and Matlab have been used to verify these general solutions extensively, and their validity has been confirmed. Despite the fact that it may not be practical to set the phases of the carriers and modulating signal arbitrarily, the proposed solution is able to evaluate the phase effects on the spectral components. Figs. 9 and 10 show the simulated and analytical results for fourand five-level double-sided PWM spectra, and the phase dispositions are purposely made to be arbitrary, which is uncommon for verifying the validity of the proposed analytical solutions. Overmodulation is also one of the concerns for multilevel PWM, and the proposed solution allows us to have a close examination of the output spectrum when overmodulation occurs. Fig. 11 shows the spectra obtained from simulation and analytical solu. The tion of a five-level PWM with overmodulation, above examples demonstrate that the simulated and analytical results match very well. Although hardware implementation has not been carried out, the proposed solution has been cross verified by comparing the experimental result of a five-level cascaded PWM inverter reported in [12] and the analytical spectrum matches that of the experimental spectrum very well.

Fig. 9. Spectrum of a double-sided four-level PWM with f = 50 Hz, f = 2100 Hz, M = 0:9,  = =4,  = 0, and  = =3. (a) Simulation result. (b) Analytical result.

(33)

(34)

(35)

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Fig. 10. Spectrum of a double-sided five-level PWM with f = 50 Hz, f = 2100 Hz, M = 0:9,  = =3,  =  ,  = =4, and  = 6=5. (a) Simulation result. (b) Analytical result.

C. Discussions With the analytical solutions provided in this paper, one can perform thorough examination of the spectral characteristics of any level multicarrier-based PWM strategies. With some uncommon carrier phase dispositions, it is shown in Figs. 9 and 10 that the spectra are rather unclean and are not suitable for real applications. However, if the carrier phases are set appropriately and the corresponding spectra are described in (22), (34), and (35), the spectrum will be very clean and ready for real applications. Fig. 12 shows the spectra of the double-sided four-, five-, six-, and seven–level PWM. It can be seen that the energy of the spectral components shrinks as the number of levels increases, but more spectral components with significant energy appear at the same time. This phenomenon has also been observed for the single-sided odd-level PWM with specific carrier phase dispositions as shown in Fig. 13, except that the spectral components are denser than that of the double-sided PWM due to the asymmetrical properties of the carriers. With this tool, one can choose the most appropriate number of levels and control strategy by observing the spectral characteristics. The solution derived is based on ideal situations such as ideal switching devices and ideal voltage supplies with perfect bal-

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Fig. 11. Spectrum of a double-sided five-level PWM with f = 50 Hz, f = 2100 Hz, and M = 1:2. (a) Simulation result. (b) Analytical result.

ance. In practice, the voltage unbalance problem is an important issue in a multilevel inverter and also is not an easy task to tackle. Since the proposed technique is based on sub-PWM analysis, it is easy to modify the dc voltage for each carrier. The dc voltage can be represented by a nominal value plus an offset voltage and, by solely considering the contributions from the nominal voltages, the proposed solution can be derived. However, it is not an easy task or, rather, it is quite impossible to obtain a simple representation to generalize the voltage unbalance effect to the spectrum unless the offset voltages are exactly the same for dc supplies. Nevertheless, such a condition is unlikely to exist in practice. Another concern of applying the analytical solution to real applications is that the effects of output filter have not been considered. Since multilevel inverter is one type of voltage source inverters, the output current is dependent on the output voltage waveform. It is thus true that the output filter or its variant (such as the machine) is used to filter out undesirable harmonics. Nevertheless, the source of the harmonic generation is originated from the PWM generation process. This is the major reason why power electronics researchers spend a great deal of effort studying the modulation strategies to understand their spectral characteristics. However, incorporating the filter parameters

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Fig. 12. Spectra of double-sided PWM with M = 0:9, f = 50 Hz, f = 2100 Hz,  = (i 1) , and  = i for odd-level PWM and  =  = i and  = 0 for even-level PWM. (a) Four-level. (b) Five-level. (c) Six-level. (d) Seven-level.

0

into the analysis is inevitably complicated, and it is unlikely that a compact solution will be obtained. Finally, it should be noted that the proposed solution is obtained based on double Fourier analysis which has a basic

Spectra of single-sided PWM with M = 0:9, f = 50 Hz, f =  =  = 0. (a) Three-level. (b) Five-level. (c) Seven-level. (d) Nine-level.

Fig. 13.

2100 Hz, and

assumption that the signal must be periodic. If the ratio of the carrier and modulating signal frequencies is not a rational number, the PWM signal is not periodic and the analytical solution cannot be applied. It is also of interest to point out that

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LAU et al.: COMPACT ANALYTICAL SOLUTIONS FOR DETERMINING SPECTRAL CHARACTERISTICS

all even harmonics will be eliminated due to the odd half-wave symmetry if the ratio is an odd integer, and this property may also help control the harmonics. V. CONCLUSION This paper presents an elegant approach to derive the general solutions for various multicarrier-based multilevel PWM strategies. The main idea is to sum up all the PWM waveforms obtained by decomposing the multilevel PWM into a number of sub-PWM waveforms. The general solutions have been verified using Pspice and Matlab simulations extensively. The proposed approach offers the following advantages: 1) easy understanding of the derivation; 2) adaptability for arbitrary carrier phase and modulating signal phase disposition; 3) only a simple numerical integration technique is required to compute the analytical solution; 4) the overmodulation situation is also taken into account; and 5) a compact closed-form solution can be obtained for specific carrier phase disposition. These general solutions provide a very convenient tool for researchers who would like to have an in-depth insight into the spectrum of multicarrier-based multilevel PWM with arbitrary phase disposition.

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[12]

, “Multicarrier PWM strategies for multilevel inverters,” IEEE Trans. Ind. Electron., vol. 49, pp. 858–867, Aug. 2002. [13] C. M. Wu, W. H. Lau, and H. Chung, “Generic analytical solution for calculating the harmonic characteristics of multilevel sinusoidal PWM inverter,” in Proc. 1999 IEEE ISCAS, vol. 5, 1999, pp. 184–187. , “Analytical technique for calculating the output harmonics of an [14] H-bridge inverter with dead time,” IEEE Trans. Circuits Syst. I, vol. 46, pp. 617–627, May 1999.

W. H. Lau (M’88) received the B.Sc. and Ph.D. degrees in electrical and electronic engineering from University of Portsmouth, Portsmouth, U.K., in 1985 and 1989, respectively. In 1990, he joined the City University of Hong Kong, where he is currently an Associate Professor in the Department of Computer Engineering and Information Technology. His current research interests are in the area of digital signal processing, digital audio engineering, visual speech signal processing. Dr. Lau is currently the Vice-Chairman of the IEEE Hong Kong Section. He is the Financial Chair of the TENCON 2006. He was the recipient of the IEEE Third Millennium Medal. He was the Registration Co-Chair of the ICASSP 2003 and ISCAS 1997. He was the Chairman of the IEEE Hong Kong Joint Chapter on CAS/COM for 1997 and 1998.

REFERENCES [1] A. Nabae, I. Takahashi, and H. Akagi, “A new neutral-point-clamped PWM inverter,” IEEE Trans. Ind. Applicat., vol. IA-17, pp. 518–523, Sept. 1981. [2] X. Yuang, H. Stemmler, and I. Barbi, “Self-balancing of the clampingcapacitor-voltages in the multilevel capacitor-clamping-inverter under sub-harmonic PWM modulation,” IEEE Trans. Power Electron., vol. 6, pp. 256–263, Mar. 2001. [3] F. Z. Peng and J. S. Lai, “Dynamic performance and control of a star var compensator using cascade multilevel inverters,” in Rec. Conf. IEEE-IAS Annu. Meeting, vol. 2, San Diego, CA, Oct. 1996, pp. 1009–1015. [4] G. Carrara, S. Gardella, M. Marchesoni, R. Salutari, and G. Sciutto, “A new multilevel PWM method: A theoretical analysis,” IEEE Trans. Power Electron., vol. 7, pp. 497–505, May 1992. [5] T. Nakajima, H. Suzuki, K. Sakamoto, M. Shigeta, H. Yamamoto, Y. Miyazaki, S. Tanaka, and S. Saito, “Multiple space vector control for self-commutated power converters,” IEEE Trans. Power Delivery, vol. 13, pp. 1418–1424, Oct. 1998. [6] L. Li, D. Czarkowski, Y. Liu, and P. Pillay, “Multilevel selective harmonic elimination PWM technique in series-connected voltage inverters,” IEEE Trans. Ind. Applicat., vol. 36, pp. 160–170, Jan./Feb. 2000. [7] Z. Suto and I. Nagy, “Analysis of nonlinear phenomena and design aspects of three-phase space-vector-modulated converters,” IEEE Trans. Circuits Syst. I, vol. 50, pp. 1064–1071, Aug. 2003. [8] J. Rodriguez, Ed., IEEE Transactions on Industiral Electronics, Special Issue on Multilevel Inverters, Aug. 2002, vol. 49. [9] W. R. Bennett, “New results in the calculation of modulation products,” Bell Syst. Tech. J., vol. 12, pp. 228–243, 1933. [10] D. G. Holmes, “A general analytical method for determining the theoretical harmonic components of carrier based PWM strategies,” in Rec. Conf. IEEE-IAS Annu. Meeting, vol. 2, Oct 1998, pp. 1207–1214. [11] B. P. McGrath and D. G. Holmes, “An analytical technique for the determination of spectral components of multilevel carrier-based PWM methods,” IEEE Trans. Ind. Electron., vol. 49, pp. 847–857, Aug. 2002.

Bin Zhou received the B.Eng. degree in electrical and electronic engineering from Shanghai Jiaotong University, Shanghai, China, in 2001. He is currently working toward the Ph.D. degree from the City University of Hong Kong. His research interests include digital signal processing, digital audio engineering, and fundamental control principles for power electronic converters.

Henry S. H. Chung (S’92–M’95–SM’03) received the B.Eng. (with first class honors) and Ph.D. degrees in electrical engineering from The Hong Kong Polytechnic University in 1991 and 1994, respectively. Since 1995, he has been with the City University of Hong Kong. He is currently an Associate Professor with the Department of Electronic Engineering. His research interests include time- and frequency-domain analysis of power electronic circuits, switched-capacitor-based converters, random-switching techniques, digital audio amplifiers, soft-switching converters, and electronic ballast design. He has authored four research book chapters and over 160 technical papers including 76 refereed journal papers in the current research area, and holds four U.S. patents. Dr. Chung was track chair of the technical committee on power electronics circuits and power systems of IEEE Circuits and Systems Society in 1997–1998. He was Associate Editor and Guest Editor of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS–Part I: FUNDAMENTAL THEORY AND APPLICATIONS. He was the receipient of the Grand Applied Research Excellence Award in 2001 from the City University of Hong Kong.

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