A Novel Hysteresis Random Spread-Spectrum Method in Random PWM Selective Harmonic Elimination for Single-Phase Inverter

Li Guohua Liu Chunwu Wang Yufeng

A Novel Hysteresis Random Spread-Spectrum Method in Random PWM Selective Harmonic Elimination for Single-Phase Inverter

1,211

（1. College of Electrical and Control Engineering Liaoning Technical University Huludao 125105 China 2. College of Mechanical Engineering Liaoning Technical University Fuxin 123000 China）

In random pulse width modulation (RPWM), the instantaneous switching frequency of inverter is required to be distributed in a wide frequency range for better randomness. However, when the switching frequency continues to be high, the switching loss will be increased. When the switching frequency continues to be low, the increase of output current ripples will be introduced. The existing selective harmonic elimination method in RPWM cannot effectively control the switching frequency of inverter as needed in the process of random spread-spectrum. In this paper, a novel hysteresis random spread-spectrum method in RPWM selective harmonic elimination for single-phase inverter is proposed. With the reasonably chosen random number () according to the relationship betweenand switching frequency in RPWM, the proposed method can guarantee that the instantaneous switching frequency is randomly distributed over a wide range and the average switching frequency can be controlled within a preset range. When the current ripples are large due to low switching frequency in RPWM selective harmonic elimination method, the average switching frequency can be increased to reduce it.

Hysteresis random spread-spectrum, random pulse width modulation (RPWM), selective harmonic elimination, single-phase inverter

0 Introduction

In the conventional sinusoidal pulse width modulation (SPWM)[1-4], the switching frequency of the inverter is usually fixed and high-order harmonics are generated at the switching frequency and its multiples. This is the main reason for electromagnetic interference (EMI) and electromagnetic noise[5-7]. Increasing the switching frequency can reduce the noise, but it does not work well on occasions where the switching frequency must be limited to low values, such as high-power traction drive systems[8-9]. Random pulse width modulation (RPWM) can distribute the noise power which is concentrated at the switching frequency and its multiples evenly within a certain frequency range[10]. It is an effective method to reduce the electromagnetic noise and EMI at present[11].

RPWM can basically be classified as: random switching frequency PWM[12-13], random pulse position PWM[14], random switching PWM[15], random zero vector PWM[16]and hybrid random PWM[17]. At present, the study focuses of RPWM mainly include: how to generate random numbers and how to calculate the switching period or place the pulse position randomly[18-19]. Thereby, the aforementioned RPWM methods couldn’t eliminate the specific harmonics selectively such as the resonance frequency of the load. The selective harmonic elimination pulse width modulation (SHEPWM)[20]can eliminate specific harmonics, but it is only aimed at low-order harmonics such as 6±1[21-22]and has little effect on eliminating the high-order harmonics.

In order to shape the noise spectrum of the inverter output voltage, the low-pass filter and the band-pass filter were used respectively to reduce harmonics in the specific frequency range[23-25]. However, the digital filter brings large computation costs. In contrast, a novel idea was suggested in Ref.[26] to calculate the switching period by using the duty ratio to reduce the noise at the specific frequency. This algorithm is simple and the calculation is small. But, with this method, harmonics below 20kHz is usually difficult to eliminate. A method of selective voltage harmonic elimination is proposed in Ref.[27]. The problem in Ref.[26] was better solved and the harmonics whose frequencies are lower than 10kHz can be eliminated. However, the general formulas of the random numberand its corresponding switching frequency extreme value are not given in Ref.[27]. The method of random modulation in Ref.[28] for selectively reducing the noise at one or more frequency points was proposed, but this method led to an increase in the peak of power spectral density (PSD). The noise at the specific frequency can also be reduced in Ref.[29], but the switching frequency of the inverter must be less than the resonance frequency. When the resonance frequency is low, the switching frequency of the inverter will be over low. However, the aforementioned methods can not effectively control the switching frequencies of inverters. Namely, the average switching frequencies of the inverter in Ref.[23-29] can not be increased or decreased according to need.

In RPWM, the instantaneous switching frequency of the inverter is required to be distributed in a wide frequency range[30]. When the switching frequency continues to be high, the switching loss of the IGBT will be increased. Usually, if the switching frequency of the inverter rises to a certain value, the inverter should reduce the rated power by about 5% for every 1kHz increase. When the switching frequency continues to be low, the issues (such as the increase of output current ripples) will be introduced. These all put forward high requirements for inverter switching frequency control. How to control the instantaneous switching frequency and average switching frequency of the inverter optimally are not considered in the aforementioned methods[23-29]when shaping the noise spectrum of the output voltage for the inverter.

This paper proposed a novel hysteresis random spread-spectrum method in RPWM selective harmonic elimination forsingle-phaseinverter. Firstly, the mechanism underlying RPWM selective harmonic elimination is analyzed and two ideas of eliminating specific harmonics in RPWM are derived. Based on the two ideas, the general formulas to calculate the switching period, the random numberand its upper and lower limits (maxandmin) of the switching frequencies are obtained as well. While realizing selective voltage harmonic elimination in RPWM, the average switching frequency can be controlled to increase or decrease according to the need by the frequency hysteresis comparator and reasonably chosen random number. It is simple and easy to implement.

1 Selective HarmonicElimination in RPWM

Topology of single-phase voltage inverter as shown in Fig.1, the bipolar PWM has been adopted in the single-phase voltage inverter. The output voltage (ab) is equal todcor-dc.

Fig.1 Topology of single-phase voltage inverter

The formulas of the pulse sequence forth cycle are respectively shown in

In equation (1) and equation (2),is a constant, Tis period ofth switching cycle,Dis duty ratio ofth switching cycle,tis start time ofth switching cycle andg() indicates the pulse ofth switching cycle.

The PWM pulses are put in the front of the switching period and in the center of the switching period, respectively, as shown in Fig.2a and Fig.2b. A sequence of pulses in Fig.2a and Fig.2b can be regarded as the sum of output voltage (ab) and DC component (dc) of Fig.1.

Fig.2 Pulse sequence diagram

If the pulse functiong() does not contain a specific harmonic, then the waveform ofabdoes notcontain the harmonic. Equation (3) expresses the pulse train. The Fourier transform of equation (3) is given at equation (4). The real and imaginary parts of equation (4) are given at equation (5) and equation (6).

Idea 1 is that the first summation of the (+)th term in equation (7) to equation (9) is removed by the second summation of theth term. The first summation of the (++1)th term is removed by the second summation of the (+1)th term, etc. Con- sequently, the summations in each term can be removed with each other.

Idea 2 is that the second summation of the (+)th term in equation (7) to equation (9) is removed by the first summation of theth term. The second summation of the (++1)th term is removed by the first summation of the (+1)th term, etc. Consequently, the summations in each term can be removed with each other. Wheretis the start time of (+)th switching period andis a positive integer number.

2 MechanismAnalysis of Selective Har-monic Elimination Based on Pulse Posi- tion at Front of Switching Period

2.1 General Formula for Switching Period and Duty Ratio

Equation (10) are available according to idea 1 on the basis of equation (8).

Simultaneously, equation (12) is a general formula of the relation between the switching period and the duty ratio. Equation (13) and equation (14) are obtained whenare equal to 2and 3, respectively. Consequently, the cases canalso be obtained whereare equal to other positive integers. Thein equations are positive random numbers.

Equation (15) is given according to the idea 2.

Simultaneously, equation (16) is a general formula of the relation between the switching period and the duty ratio. Equation (17) and equation (18) are obtained whenare equal to 1 and 2, respectively. Consequently, the cases can be obtained whereare equal to other positive integers.

2.2 General Formula for Random Number k and Its Switching Frequency Extreme Value

In the idea 1, the general formula (takingT1as an example) of the upper and lower limits (maxandmin) for the random number () can be obtained by using equation (13) on the basis of the specific frequency (0), the duty ratio and the limit value of switching period, as shown in Tab.1. Similarly, the cases can also be obtained whereare equal to other positive integers.

Tab.1 The ,, and in pulse position at front of switching period

In the idea 2, the general formula of themaxandmincan be obtained according to equation (17), as shown in Tab.1. Similarly, the cases can also be obtained whereare equal to other positive integers.is a positive integer in the range of extreme values.

Generally, more than onein Tab.1 are satisfied. The general formula for eachand its corresponding upper and lower limits of the switching frequencies can be calculated. As shown in Tab.1, the general formulas for eachand its upper and lower limits (fmaxandfmin) of the switching frequency in the idea 1 and idea 2 can be calculated according to equation (13) and equation (17), respectively. Similarly, the cases can also be obtained whereare equal to other positive integers.

In the Tab.1,maxandminare the upper and lower limits of the duty ratio.maxandminwhich are usually preseted are the upper and lower limits of the instantaneous switching frequency.is a positive integer.fmaxandfminare the upper and lower limits of the switching frequency which are derived by each. Under the same conditions, the largeris, the smaller thefmaxandfminare and the smalleris, the largerfmaxandfminare.

It can be seen from the equations of themaxandminin Tab.1 that the range of selection foris reduced with the decrease of0.The higher0is, the larger the selection range ofis.

3 Mechanism Analysis of Selective Har-monic Elimination Based on Pulse Posi- tion at Center of Switching Period

3.1 General Formula for Switching Period and Duty Ratio

Equation (19) are available according to idea 1 on the basis of equation (9).

Simultaneously, equation (20) is a general formula of the relation between the switching period and the duty ratio. Equation (21) and equation (22) are obtained whenare equal to 1 and 2, respectively. Consequently, the cases can be obtained whereare equal to other positive integers.

Equation (23) is available according to the idea 2.

Simultaneously, Equation (24) is a general formula of the relation between the switching period and the duty ratio. Equation (25) and equation (26) are obtained whenare equal to 1 and 2, respectively. Consequently, the cases can also be obtained whereare equal to other positive integers.

3.2 General Formula for Random Number k and its Switching Frequency Extreme Value

As shown in line 2 and line 3 of Tab.2, the extreme values (maxandmin) of the random numbercan be obtained by using equation (21) and equation (25) according to the idea 1 and idea 2, respectively.is a positive integer in the range of extreme values. By similar calculation above, the cases whereare equal to other positive integers can be obtained.

Tab.2 The ,, and in pulse position at center of switching period

Eachand its upper and lower limits of the switching frequencies in line 4 and line 5 of Tab.2 are calculated according to equation (21) and equation (25). In the same way, the cases whereare equal to other positive integers can also be obtained.

It should be noted that the denominator may have a negative number when using the above equations to calculatefmaxandfmin. It is shown that the frequency () and duty ratio () can make the denominator of equations be zero without taking the limit value. In this case, the corresponding upper limit frequency is equivalent to +∞.

4 Hysteresis Random Spread-Spectrum inRPWM Selective Harmonic Elimination Method

It can be observed from Tab.1 and Tab.2 that the larger random number () is, the smallerfmaxandfminare. On the contrary, the limit value of the cor- responding instantaneous switching frequency is larger with a smaller. This makes it possible to control the average switching frequency of the inverter by reasonably selecting the random number (). So a smalleris selected to increase the average switching frequency when the average switching frequency is low. This can prevent the current ripples from increasing caused by the low switching frequency. Similarly, a largerischosen to reduce the average switching frequency and switching loss when the average switching frequency is high.

Equation (27) is the formula of the average switching frequency (An) at the start time of theth sampling cycle according to Fig.2a and the idea 1. In order to avoid repeating the accumulation of the first1 switching frequencies in the process of calculating the average, it only needs to multiply the average switching frequencyA(n–1)at the beginning of the previous cycle and the number of cycles, then add it to the previous cycle frequencyf1to calculate the average frequency.

As shown in Fig.3, the average switching frequency ( fAn) of the inverter was calculated by equation (27) and the upper and lower limits of the frequency hysteresis comparator were set to f1 and f2. The average switching frequency error (Df) is determined according to the result of hysteresis comparison, as shown in equation (28).

The next sampling cycle (T+1) is calculated by using equation (13) after selecting the random number () and then the compare registers in DSP is assigned a value. Finally, the PWM driving signal of the single-phase inverter is generated.

The effectiveis determined by the upper and lower limits (maxandmin) of the instantaneous switching frequency of the inverter. Among them, themaxcan not exceed the allowable switching frequency of the IGBT and a certain margin is kept. In addition, for those occasions where the carrier frequency must be low, themaxshall be selected flexibly according to the actual requirements. In this paper,maxis set to 8kHz andminis set to 1.5kHz.

Equation (29) for switching frequency of the inverter is obtained according to the reciprocal of equation (13). The effective random number () which satisfies the upper and lower limits of the instantaneous switching can be calculated by equation (29), wheref1is the (+1)th cycle switching frequency of the inverter.

The upper and lower limits (1and2) of the average switching frequency hysteresis are con- strained by the upper and lower limits (maxandmin) of the instantaneous switching frequency. In addition, a certain hysteresis width shall be ensured, as shown in equation (30).

In summary, under the premise of ensuring that the instantaneous switching frequency does not exceed the range frommintomax, the average frequ- ency is controlled within the set range from1to2.

5 Simulation and Experiment

The parameters of the simulation and experi- mental system are as follows: inverter was connected to RL load with=15Wand=5mH, also the DC voltage is equal to 48V. The instantaneous switching frequency of the inverter is distributed from 1.5kHz to 8kHz.=0.3, 0.5 and 0.7, also0=6kHz, 8kHz and 10kHz are preseted, respectively. Duty ratio is achieved from (1+sin())/2, whereis angular frequency of main harmonic. For instance, the upper and lower limits of duty ratio are equal to 0.85 and 0.15, respectively if=0.7. The fundamental frequency of the output voltage is 50Hz.

The experimental system is shown in Fig.4. TMS320F2812 32-bit DSP for the master chip of system and the IGBT BSM50GB120DN2 for the main circuit power switches have been adopted. A dead time has been set to 4.27ms. In the experiment, the oscilloscope (DS1052E) and power quality analyzer (HIOKIPW3198) have been used.

Fig.4 Experimental system

5.1 Realization of Selective Harmonic Elimination in RPWM

The proposed method is verified by the software of Matlab/Simulink. Fig.5 shows the waveforms of fixed switching frequency 3kHz for SPWM and the conventional RPWM.

Fig.5 Simulated waveforms of PSD

It can be seen that the harmonics are mainly concentrated at 3kHz and its multiples in Fig.5a. Compared with the fixed switching frequency SPWM, it can be seen from Fig.5b that there is no obvious peak in the PSD of the conventional RPWM and the harmonics are distributed evenly, but it can not selectively eliminate the specific frequency.

The waveforms of the proposed method under the same parameters are shown in Fig.6. Among these figures, the upper and lower limits (1and2) of the average switching frequency are set to 2 100Hz and 4 100Hz, respectively. The0is set to 6kHz.

Compared with the conventional RPWM, the proposed method can significantly reduce the frequency component of0and its multiples, whenare equal to 0.3, 0.5 and 0.7, respectively.

5.2 Analysis of Experimental Results of Different f0

The experimental waveforms of the PSD for the proposed method (0=6kHz, 8kHz and 10kHz) are shown in Fig.7. It can be seen that the harmonic power at0and its multiples can be reduced significantly when0are equal to three frequencies, respectively. It is basically consistent with the simulation results.

As seen in Fig.7, the PSD waveforms have a significant gap with the range of about hundreds hertz around0and its multiples. The reason is that the frequency close to0is reduced by a certain amplitude when the harmonic power at0is completely eliminated. The closer to0is, the greater the amplitude is decreased. The impact of phase error in system can be overcome by the characteristics, such as the error in time-delay, etc.

The experimental waveforms of driving voltage for IGBT, the output voltage and current for the inverter are shown in Fig.8 when0is equal to 10kHz.

Fig.8 Experimental waveforms of voltage and current

5.3 Comparison with existing method

The distribution of instantaneous switching frequency percentage and average switching frequency are shown in Fig.9. Waveforms 1, 2 and 3 are for the proposed method in this paper. The upper and lower limits1and2of the frequency hysteresis comparator are: 2 100～2 200Hz, 3 000～3 100Hz, 4 000～4 100Hz, also all the hysteresis width is set to 100Hz. Waveform 4 is for the existing RPWM selective harmonic elimination method, taking Ref.[27] as an example. All0of aforementioned are 10kHz.

It can be seen from Fig.9 that the average switching frequency can be controlled within the set of three smaller frequency ranges1～2by reasonable selection of the random number. And the instantaneous switching frequency can be ensured to be randomly distributed within the allowable rangemin～max. The randomness is well.

In contrast, the existing method in Ref.[23-29] can not control the average switching frequency of the inverter flexibly. The average switching frequency in Ref.[21] is around 3 150Hz in Fig.9a under the same conditions, which is uncontrollable. In this paper, the average switching frequency can be increased to about 4 050Hz and controlled to increase or decrease in the range of 2 100～4 100Hz. So the flexibility is better. The current waveforms for the above two methods are presented in Fig.10 and Fig.11. It can be seen from the comparison that switching frequency of the method in Ref.[27] is lower than the proposed method as a whole and the current ripples are significantly larger.

Fig.9 The waveforms of switching frequency

Fig.10 The waveforms of uab and iab when the existing method is adopted

In the method of this paper, the upper and lower limits of the average switching frequency can be obtained if the range of random selection is limited to the maximum three random numbers or the minimum three random numbers. Moreover, the larger0is, the more effective random numbers are and the higher the degree and precision of average switching frequency to control is.

Fig.11 The waveforms of uab and iab when frequency hysteresis is 4 000～4 100Hz

6 Conclusion

In this paper, a novel hysteresis random spread- spectrum method in RPWM selective harmonic elimination forsingle-phaseinverter has been proposed. Compared with the fixed switching frequency SPWM and the traditional RPWM, the harmonics can be distributed uniformly in certain frequency range and some unwanted harmonics can be eliminated successfully with proposed method. Based on the realization of selective harmonic elimination in RPWM, the flexible increase and decrease of the average switching frequencies have been achieved by the hysteresis random spread-spectrum method. When the current ripple is large due to the low switching frequency, the average switching frequency can be increased to reduce it, while the distribution range of instantaneous switching frequency is unchanged. The proposed method is characterized by flexibility, simple algorithm and easy implementation.

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單相逆變器隨機(jī)PWM選擇性消諧滯環(huán)隨機(jī)擴(kuò)頻方法

李國華1,2劉春武1汪玉鳳1

（1. 遼寧工程技術(shù)大學(xué)電氣與控制工程學(xué)院 葫蘆島 125105 2. 遼寧工程技術(shù)大學(xué)機(jī)械工程學(xué)院 阜新 123000）

在隨機(jī)脈沖寬度調(diào)制（RPWM）中，要求逆變器的瞬時(shí)開關(guān)頻率分布在較寬的頻率范圍內(nèi)，以獲得較好的隨機(jī)性。然而，當(dāng)開關(guān)頻率持續(xù)偏高時(shí)，逆變器的開關(guān)損耗將會(huì)增加；當(dāng)開關(guān)頻率持續(xù)較低時(shí)，會(huì)引起輸出電流紋波的增加。現(xiàn)有的RPWM選擇性諧波消除方法在隨機(jī)擴(kuò)頻的過程中不能有效地控制逆變器的開關(guān)頻率。該文提出一種單相逆變器RPWM選擇性消諧滯環(huán)隨機(jī)擴(kuò)頻方法。根據(jù)RPWM策略中有效隨機(jī)數(shù)與開關(guān)頻率的關(guān)系，合理選擇隨機(jī)數(shù)，既可以保證逆變器瞬時(shí)開關(guān)頻率在較寬的范圍內(nèi)隨機(jī)分布，又可以將平均開關(guān)頻率控制在預(yù)先設(shè)定的頻率范圍內(nèi)。在RPWM選擇性消諧過程中，當(dāng)開關(guān)頻率較低而引起電流紋波增加時(shí)，該方法可以通過提高平均開關(guān)頻率來減小紋波。

滯環(huán)隨機(jī)擴(kuò)頻 隨機(jī)脈寬調(diào)制（RPWM） 選擇性諧波消除 單相逆變器

TM46

10.19595/j.cnki.1000-6753.tces.200018

This work is partially supported by National Natural Science Foundation of China (51307076) and Natural Science Foundation of Liaoning Province, China (20180550268).

January 7, 2020;

March 3, 2020.

Li Guohua male, born in 1981, PhD. Major research interests include power electronics, pulse-width modulation schemes and harmonic compensation in power system.E-mail: dkliguohua@163.com

Liu Chunwu male, born in 1995, Postgraduate. Major research interests include power electronics and pulse-width modulation schemes.E-mail: lcw19950204@163.com (Corresponding author)

（編輯 陳 誠）