Microprocessor-Based Random PWM Schemes for DC-AC Power ...

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 12, NO. 2, MARCH 1997

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Microprocessor-Based Random PWM Schemes for DC–AC Power Conversion S. Y. R. Hui, Senior Member, IEEE, I. Oppermann, and S. Sathiakumar

Abstract—Two classes of microprocessor-based random PWM (RPWM) real-time schemes for dc–ac power conversion are compared and evaluated. Performance of the RPWM schemes based on the mathematical and logical approaches is examined. The proposed schemes exhibit excellent harmonic content with all low and high-order harmonics suppressed and are suitable for both MOSFET and IGBT inverters.

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Index Terms— DC–AC power conversion, power harmonics, power inverters, random pulse width modulation, randomized switching.

I. INTRODUCTION

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ULSE-WIDTH modulation (PWM) schemes for dc–ac power conversion have received much attention in the last two decades. Many PWM schemes have been developed and implemented successfully for different applications. The most standard scheme is the sinusoidal PWM (SPWM) scheme in which the PWM signal is generated from the comparison of a fundamental sinusoidal waveform and a highfrequency carrier waveform. Many recent PWM schemes are microprocessor-based and can be optimized for the minimization of harmonics and/or motor torque in drive applications. In general, traditional PWM schemes provide a PWM waveform with a large fundamental voltage component with low-order harmonics suppressed. However, the harmonic power is usually concentrated in the high-frequency range due to the high-frequency switching of the power inverter. These highfrequency harmonics can have adverse effects, such as acoustic noise, harmonic heating in electric machines, and radio interference. In principle, acoustic noise can be reduced if the switching frequency is above 18 kHz. However, such highfrequency switching results in high switching losses in power inverters. Recently, new PWM schemes based on the use of random number generation have been proposed for comparison with the fundamental sinusoidal waveform in order to generate RPWM waveform. It has been shown [1]–[11] that the randomness added into the PWM waveform can cause the harmonic power to spread over the harmonic spectrum so that no harmonic component has a significant magnitude. The resulting RPWM spectrum effectively consists of a large fundamental components with both low- and high-

Manuscript received July 25, 1995; revised July 29, 1996. S. Y. R. Hui is with the Department of Electrical Engineering, University of Sydney, N.S.W. 2006, Australia, and also with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong. I. Oppermann and S. Sathiakumar are with the Department of Electrical Engineering, University of Sydney, N.S.W. 2006, Australia. Publisher Item Identifier S 0885-8993(97)01838-3.

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(d) Fig. 1. Logical arrangements for pseudo random number generation for (a) 8-b, (b) 10-b, (c) 12-b, and (d) 14-b systems.

order harmonics suppressed. Recent reports have confirmed that the RPWM approach offers advantageous features such as reduced radio interference from converter equipment and improved acoustic and vibration effects in electronic drive systems [8], [10]–[12]. Consequently, this RPWM approach is a valuable method for many applications in which interference between neighbor systems must be avoided. A comprehensive review of existing of RPWM technique can be found in [11]. In the literature, most of the RPWM schemes have been implemented in logical circuits with the aid of microprocessors [1]–[3], [11], although a pure mathematical approach generated by a microprocessor without external logical circuits has also been reported [5], [7]. Most of the early RPWM schemes [1]–[3], [6], [7] reported employ a very high frequency (ranging from 60–480 kHz) for the random number generation, resulting in corresponding high inverter switching frequency (15–25 kHz), which is only suitable for MOSFETbased inverters and not for IGBT inverters. Recent reports [5],

0885–8993/97$10.00  1997 IEEE

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 12, NO. 2, MARCH 1997

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Fig. 2. Probability density functions of the logical RPWM schemes. (a) 8-b waveform spectrum ( = 12). (d) 14-b waveform spectrum ( = 14). (c) 12-b waveform spectrum (

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TABLE I A SUMMARY OF THE PDF’S OF THE LOGICAL RPWM SCHEMES (SIMULATION)

[10], [12], however, suggest that a random number generation rate of around 50 kHz would be sufficient so that the averaged switching frequency can be kept below 10 kHz, which is a frequency suitable for IGBT inverters. There has not been any comprehensive study on comparing the performance of different RPWM generation methods and

Ns = 8). (b) 10-b waveform spectrum (Ns = 10).

their performance. As the microprocessor has become an essential component in most of the modern power electronic equipment for dc–ac power conversion, this project aims at studying two classes of real-time RPWM schemes that can be implemented directly in microprocessors without using external logical random number generator circuits. These schemes are based on the mathematical and logical approaches. The mathematical approach has been discussed before in [5] and [7]. This paper would put more emphasis on the characteristics of the logical approach. However, spectral characteristics of both approaches are included and compared. Both experimental and theoretical results are presented and discussed.

II. RANDOM PWM GENERATION The RPWM schemes examined in this project can be classified into two categories, namely 1) the mathematical

HUI et al.: MICROPROCESSOR-BASED PWM SCHEMES FOR DC–AC POWER CONVERSION

Fig. 3. Harmonic (Rn+1 Rn

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mathematical

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Fig. 5. Harmonic spectra of 8-b logical RPWM scheme.

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(b) Fig. 6. Harmonic spectra of 10-b logical RPWM scheme.

(b) Fig. 4. Harmonic Rn (Rn+1

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approach and 2) the logical approach. In each RPWM scheme, the random number generated is compared with a sinusoidal reference signal at a certain sampling frequency. The result of this comparison forms the digital RPWM signal.

A. Mathematical RPWM (MRPWM) The mathematical approach is based on a mathematical equation that can generate random number. The general form of the random number equation is as follows: (1)

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 12, NO. 2, MARCH 1997

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Fig. 9. Measured RPWM waveform and the corresponding reference sinusoidal waveform from mathematical scheme 2.

(b) Fig. 7. Harmonic spectra of 12-b logical RPWM scheme.

(a) Fig. 10. Measured RPWM waveform and the corresponding reference sinu12). soidal waveform from logical scheme (

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MRPWM schemes examined in 16-b calculation are Scheme 1: Scheme 2: B. Logical RPWM (LRPWM)

(b) Fig. 8. Harmonic spectra of 14-b logical RPWM scheme.

where and are the random numbers at the th and th step, respectively, and are prime numbers, and is the number of bits for number representation. This approach is very suitable for microprocessor implementation because it involves only one multiplication and one addition, together with the modulus operation. Various pairs of prime numbers and can be employed [5], [7]. The two

This approach is based on the logical operation of several bits of a digital binary number and is commonly known as pseudo-PWM code generator in communications. By first performing certain logical operations on several bits, the modular-2 operation of these bits forms a new bit value. By shifting the binary number by 1 b with the new bit forming the least significant bit, a new binary number is then generated. The number generation is known as pseudo-random number generation because the random number pattern is repetitive. This approach simply requires XOR and shift operations, and is therefore suitable for real-time microprocessor implementation. The LRPWM generation, in principle, can be developed

HUI et al.: MICROPROCESSOR-BASED PWM SCHEMES FOR DC–AC POWER CONVERSION

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(b) Fig. 11.

(a) Measure probability density function for the 8-b LRPWM scheme, (b) measure probability density function for the 10-b LRPWM scheme.

by different number of bits. In this paper, LRPWM schemes for 8, 10, 12, and 14 b are investigated. These schemes generate numbers in a repetitive random manner. The only number that cannot be used as initial value is all zeros, which is also the number absent in the number generation. III. SIMULATION STUDY OF LOGICAL RPWM SCHEMES In order to obtain real time random number generators that offer good randomness, various logical arrangements have been conducted for the LRPWM scheme. The probability density function (PDF) is used to examine the quality of randomness. A system with good randomness should have a PDF in which the harmonic components should have low amplitude. The -axis of the PDF shows the probability or

the rate of occurrence of harmonics with a certain amplitude. The scale of the -axis is the normalized amplitude of the harmonic components with the fundamental voltage as the based value. Thus, a good random scheme should have 1) harmonic components concentrated in the low-amplitude range along the -axis and 2) no dominant harmonics of significant amplitude. A computer simulation program has been developed for simulation of the switching behavior of the logical RPWM method for various bit numbers. The PDF is obtained from the harmonic spectrum of the simulated RPWM waveform over 20 fundamental cycles. Several logical arrangements (Fig. 1) have been tested and are suitable for generating pseudo-random numbers. They are first examined with computer simulation before implementation in a digital signal processor.

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 12, NO. 2, MARCH 1997

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(d) Fig. 11. (Continued.) (c) Measure probability density function for the 12-b LRPWM scheme, and (d) measure probability density function for the 14-b LRPWM scheme.

The probability density functions for the 8-, 10-, 12-, and 14-b LRPWM systems under the simulation study are shown in Fig. 2(a)–(d), respectively. Examination of these four PDF’s shows that the 8-b system has most of its voltage harmonics concentrated on very low per-unit amplitude. The probability or occurrence rate peaks at 9% at 0.01 per unit value. However, the 8-b system also contains harmonics that are of relatively high amplitude when compared with other higher bit systems. The highest per-unit harmonic amplitude is 0.187 in the 8-b system whilst those for the 10-, 12-, and 14-b systems are 0.127, 0.099, and 0.095, respectively. The results are summarized in Table I. This table shows that the peak , but harmonic amplitude decreases with the number of bits . the mean value of the harmonics increases slightly with

The reason for this trend is that the RPWM scheme serves to spread the harmonic power over a wide frequency range. If the harmonic power concentrates in a few frequencies (i.e., a poor RPWM scheme), the average harmonic amplitude is low for the nondominant harmonics but there will be a few dominant harmonics of high amplitude in the spectrum. However, if the RPWM scheme is good in spreading the harmonic power over a wide frequency range, then the peak harmonic amplitude should decrease and the mean harmonic amplitude should slightly increase, as predicted in the simulation. IV. EXPERIMENTAL VERIFICATION In order to evaluate the performance of the mathematical and logical approaches, both schemes have been implemented

HUI et al.: MICROPROCESSOR-BASED PWM SCHEMES FOR DC–AC POWER CONVERSION

in a TMS320C25 processor. The rate of random number generation is set at 50 kHz. The RPWM waveform is obtained by comparing a sinusoidal reference at a rate of 6.4 kHz. The resultant averaged switching frequency is less than 3 kHz. Since the performance of PWM schemes usually improve with the increase of the switching frequency, PWM schemes that can generate high-quality waveforms at low switching frequency are of great significance, especially for high-power applications. Harmonic spectra of 2- and 25-kHz range are recorded for each scheme so that both low- and high-order harmonics can be observed. The fundamental frequency is set at 50 Hz and the modulation index is one. Two sets of prime number pairs have been tested for the mathematical RPWM approach. These combinations are and Figs. 3 and 4 show the harmonic spectra of the two MRPWM schemes. It can be seen that both MRPWM schemes are successful in spreading the harmonic power over the spectrum because there is no dominant switching harmonic component. The 2kHz spectra show that these two RPWM waveforms contain some low-order harmonics that are of low amplitude. For the logical RPWM approach, the harmonic spectra for of 8-, 10-, 12-, and 14-b systems are included in Figs. 5–8, respectively. For 8-b representation, there are only 255 possibilities. For a 50-Hz fundamental cycle and a sampling rate of 50 kHz, there are 1000 sampled intervals within the fundamental cycle and the 8-b random number sequence repeats itself almost four times per 50 Hz cycle. Fig. 5 shows that the logical RPWM scheme with contains high-order harmonics of significant amplitude at well-defined sampling frequencies. The dominant harmonics are due to the low resolution of the 8-b system. The 2-kHz spectrum [Fig. 5(a)] shows that it also contains some low-order harmonics of significant amplitude. However, besides the few dominant harmonics, the amplitude of other nondominant harmonics remains at very low level. This characteristic agrees with the prediction that poor RPWM scheme has some dominant harmonics of high amplitude and nondominant harmonics of low amplitude. Marked improvement in the harmonic content can be achieved by increasing the number of bits and thus the number of random numbers. Harmonic spectra of RPWM schemes with equal to or higher than ten (Figs. 6–8) indicate that these schemes are successful in spreading the harmonic power over than entire spectrum because no dominant harmonic component can be observed. Each of these schemes has over 1000 numbers in the random number sequence which does not repeat itself within the 50-Hz cycle. The spread of the harmonic power leads to a slight increase in the amplitude of all the nondominant harmonics, as predicted and explained in Section III. Measured RPWM waveforms obtained from the mathematical scheme 2 and the logical scheme with are included in Figs. 9 and 10, respectively. These waveforms confirm that RPWM schemes with low switching frequency ( 3 kHz) can provide PWM waveforms of high quality. In general, the two mathematical schemes and the logical schemes with provide comparable PWM waveforms and similar harmonic content. For the logical schemes, the

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TABLE II A SUMMARY OF THE PDF’S OF THE LOGICAL RPWM SCHEMES (MEASUREMENT)

Fig. 12. Weighted harmonic index of logical RPWM schemes.

harmonic content improves slightly with increasing number of bits because of the increasing resolution. Both approaches are suitable for real time implementation due to their low computing requirements. The logical approach, which provides random number in a repetitive manner (thus named pseudo random number generation), is also suitable for implementation in digital circuitry because the random numbers can easily be stored in EPROM’s. The normalized probability density functions of the four logical RPWM schemes are included in Fig. 11(a)–(d). The mean and peak magnitude of the measured harmonics are shown in Table II. It can been seen that the general trends in these measurements agree well with predictions presented in Section III. To further compare the spectral performance of the LRPWM schemes, a weighted harmonic index can be used. The weighted harmonic index is defined as the sum of the product of number of occurrence and the corresponding harmonic amplitude. A good RPWM scheme should have a low value of weighted index because most of the harmonics lie at the low-amplitude end of the probability density function. Fig. 12 shows the bar chart for the weighted indexes for the 10-, 12-, and 14-b systems. It confirms that spectral performance improves with increasing number of bits. In order to check whether the fundamental frequency will affect the spectral characteristic, an experiment has been carried out using the 14-b logical RPWM scheme with a reference voltage signal of 4 Hz (instead of 50 Hz). Fig. 13 shows the corresponding harmonic spectrum. It can been seen that the harmonic power is successfully spread over the spectrum at a very low fundamental frequency.

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[7] V. G. Agelidis and D. Vincenti, “Optimum nondeterministic pulsewidth modulation for three-phase inverters,” in IEEE IECON 1993, pp. 1234–1239. [8] J. T. Boys and P. G. Handley, “Spread spectrum switching: low noise modulation technique for PWM inverter drives,” Proc. Inst. Elect. Eng., vol. 139, no. 3, pt. B, pp. 252–260, May 1992. [9] J. T. Boys, “Theoretical spectra for narrow-band random PWM waveforms,” Proc. Inst. Elect. Eng., vol. 140, no. 6, pt. B, pp. 393–400, Nov. 1993. [10] T. G. Habetler and D. M. Divan, “Acoustic noise reduction in sinusoidal PWM drives using a randomly modulated carrier,” IEEE Trans. Power Electron., vol. 6, pp. 356–363, July 1991. [11] A. M. Trzynadlowski, F. Blaabjerg, J. K. Pedersen, R. L. Kirlin, and S. Legowski, “Random pulse width modulation techniques for converterfed drive systems—A review,“ IEEE Trans. Ind. Applicat., vol. 30, pp. 1166–1175, Sept./Oct. 1994. [12] R. L. Kirlin, S. Kwok, S. Legowski, and A. M. Trzynadlowski, “Power spectra of a pwm inverter with randomized pulse position,“ in PESC 1993, pp. 1041–1047. Fig. 13. Harmonic spectrum of 14-b LRPWM scheme with a fundamental frequency of 4 Hz.

V. CONCLUSIONS Two types of microprocessor-based random PWM schemes have been presented and compared. Both the mathematical and logical approaches are shown to be suitable for real time RPWM waveform generation. The RPWM schemes are compared using their probability density functions and the weighted harmonic index. For the logical approach, the number of bits is crucial factor in determining the quality of the RPWM waveforms. It has been shown that at least 10 b are required for good RPWM waveform generation in this implementation. Among the four logical schemes tested, the 14-b scheme provides the best spectral performance. One advantage of using the logical approach is that the random number sequence, which repeats itself in real-time generation, can be stored in EPROM. The RPWM schemes have been implemented successfully at a switching frequency that is much less than 10 kHz, although the switching frequency could be substantially increased if necessary. For the logical schemes, the averaged switching frequency is less than 3 kHz. Therefore, these schemes can be used in both low– and high-power dc-to-ac power conversion.

S. Y. R. Hui (M’87–SM’94) was born in Hong Kong in 1961. He received the B.Sc. degree (Hons.) from the University of Birmingham, U.K., in 1984 and the D.I.C. and Ph.D. degrees from Imperial College of Science and Technology, London, U.K., in 1987. In 1987, he was appointed as a Lecturer in Power Electronics at the University of Nottingham, U.K. In 1990, he went to Australia and took up a lectureship at the University of Technology, where he became a Senior Lecturer in 1991. In January 1993, he joined the University of Sydney, where he is a Reader of Electrical Engineering and Director of the Power Electronics and Drive Research Group. Presently, he is also a Professor of Electronic Engineering at the City University of Hong Kong, Hong Kong. His current research interests include all aspects of power electronics. Dr. Hui is a Fellow of the IEAust.

I. Oppermann received the B.S. and B.E.E. degrees from the University of Sydney, Sydney, Australia, in 1989 and 1991, respectively. He is currently completing the Ph.D. degree in wireless mobile communications at the University of Sydney. His research interests include spread spectrum communications, CDMA techniques, wideband channel models, and CDMA receiver structures.

ACKNOWLEDGMENT The authors would like to thank F. Pasalic for developing some software programs for this project. REFERENCES [1] A. M. Trzynadlowski, S. Legowski, and R. L. Kirlin, “Random pulse width modulation technique for voltage-controlled power inverters,” in IEEE IAS Meet., 1987, pp. 863–868. [2] S. Legowski and A. M. Trzynadlowski, “Hypersonic MOSFET-based power inverter with random pulse width modulation,” in IEEE IAS Meet., 1989, pp. 901–903. [3] , “Power-MOSFET, hypersonic inverter with high-quality output current,” in IEEE APEC 1990, pp. 3–7. [4] S. Legowski, J. Bei, and A. M. Trzynadlowski, “Analysis and implementation of a grey-noise PWM technique based on voltage space vectors,” in IEEE APEC 1992, pp. 586–593. [5] J. K. Pedersen and F. Blaabjerg, “Implementation and test of a digital quasirandom modulated SFAVM PWM in a high performance drive system,” in IEEE IECON 1992, pp. 265–270. [6] V. G. Agelidis, P. D. Ziogas, and G. Joos, “Dead-band PWM switching patterns,” in IEEE PESC 1992, pp. 427–434.

S. Sathiakumar received the B.E., M.E., and Ph.D. degrees in electrical engineering from the Indian Institute of Science, Bangalore, India. His industrial experience includes graduate apprentice training for a period of one year, followed by employment, from 1978 to 1981, as an Assistant Development Engineer at the English Electric Company of India Ltd. He then worked as a Project Assistant/Research Associate at the Indian Institute of Science on a project sponsored by the Electronics Commission of India to develop medium-power inverters for different indigenous applications. He also worked at the University of Newcastle, Australia, on adaptive control of rotating machines. Currently, he is a Lecturer at the University of Sydney, Sydney, Australia. His fields of interest are adaptive control of electric machines, application of microprocessors and power converters for real-time control, and harmonic pollutionless PWM switching techniques for power conversion.