Current Harmonics of Voltage-Source Active Rectifier with random Switching Frequency Vaclav Kus, Tereza Josefova Department of Electromechanics and Power Electronics University of West Bohemia Pilsen, Czech Republic [email protected], [email protected] Abstract—This paper presents the issue of current harmonics drawn from the grid by three-phase voltage-source active rectifiers when the switching frequency of the devices is random; it is not an integer multiple of the supply frequency (50 Hz). The first part compares the harmonic analysis calculations derived for the piecewise linearized simplified current waveform and the simulations in the random frequency case. The second part presents a comparison of the current harmonic analysis when the switching frequency is the integer multiple of the supply frequency and when it is not. The second part also shows a dependence of the current spectrums on the random switching frequencies. This paper verifies, that the same conclusions of the previous publications apply to both the switching frequency that is the integer multiple of the supply frequency and the random switching frequency. Keywords—voltage-source active rectifier, frequency, Fourier transform, frequency analysis.

I.

the authors have decided to present the issue of current harmonics values when the switching frequency of voltagesource active rectifier is random; it is not the integer multiple of the supply frequency.

II.

CURRENT DRAWN FROM THE GRID BY THREE-PHASE VOLTAGE-SOURCE ACTIVE RECTIFIER

A. Control of the Voltage-Source Active Rectifier The voltage-source active rectifier is composed of six switches; each switch is formed by the IGBT and antiparallel diode. The voltage-source active rectifier has the DC side connected with the capacitor (Fig. 1). Constant voltage is maintained on this capacitor by the proper control [1], [3], [11]-[13].

INTRODUCTION

The use of voltage-source active rectifiers is described in many papers, because they draw current with very low distortion and have a very good power factor [1]-[3]. The harmonics in the current spectrum that do not have negligible values are measured many times. In addition, the frequency of the current harmonics depends on the switching frequency of voltage-source active rectifiers. Therefore, several papers have been written about these problems. They describe the issue of the current harmonics of voltage-source active rectifiers using simulations, theoretical calculations and measurements [4]-[7]. These papers state that the values of these current harmonics are very low in absolute values. If the harmonics are shown in the percentage values, they should be presented relative to the nominal values of the devices [8]. It is also shown that the absolute values of the current harmonics are almost independent of the load of voltage-source active rectifier [9]. The literature also describes the frequency spectrums as a function of the voltage-source active rectifier switching frequency [10]. All these papers show the switching frequency as an integer multiple of the supply frequency (50 Hz). With regard to the discussions at conferences and to the review of the paper [9], This work was supported by the European Regional Development Fund and Ministry of Education, Youth and Sports of the Czech Republic under project No. ED 2.1.00/03.0094: Regional Innovation Centre for Electrical Engineering (RICE)

978-1-4799-3807-0/14/$31.00 ©2014 IEEE

Fig. 1. The three-phase voltage-source active rectifier

Proper functioning of the active rectifier is depicted in the Figure 2, which also shows the current drawn from the grid in phase with the supply voltage. The current waveform is composed of individual sections, which approximate an increasing or a decreasing exponential function. Simulation of the voltage-source active rectifier with the voltage type control was created in C language.

The same conclusions for the total harmonic distortion of current THDI apply to both the switching frequency that is the integer multiple of the supply frequency and the random switching frequency.

Fig. 2. The correct operation of voltage-source active rectifier, fsw=525Hz

B. Harmonic Analysis of the Current Drawn From The Grid by Voltage-Source Active Rectifier Formulas for the current harmonics calculation are derived in the previous papers [8], [9]. Derivation of formulas is calculated using the simplified current. This current was divided into individual parts and the parts were linearized. The next section shows only the resulting formulas.

A. The Comparison of the Harmonic Analysis from the Simulations and the Harmonic Analysis Calculated by the Formulas, the Switching Frequency Equal to 525 Hz Harmonic analysis is calculated using the already derived formulas [9]. The low frequency is selected as the fundamental frequency (current harmonics equal larger values for these frequencies, this follows from the previous papers and it is also shown in the last part of the present paper). First, the difference is verified between the analytical calculations derived for the linearized current waveform and the simulations of voltagesource active rectifier. The results are in Figure 3-5.

Calculation of Fourier coefficients ahl and bhl for l-th straight line of simplified linearized current waveform: 2

2

sin

cos

(1)

cos

sin

(2)

Where members cl and dl derived from l-th straight line il are: ·

,

Fig. 3. The comparison of the simulated current waveform ia and the piecewise linearized current waveform ia, fsw=525Hz

(3)

The resulting members ah and bh for all linearized simplified current waveform: ∑ ∑ , (4) The amplitude of the h-th harmonic of the current: (5) Current harmonics drawn from the grid by voltage-source active rectifier appear around the switching frequency fsw in accordance with the theory of frequency modulation [10]:

Fig. 4. The comparison of harmonic analysis of the simulated current waveform ia and the piecewise linearized current waveform ia, fsw=525Hz

(6) ·

1

The spectrum of the PWM signal is composed of groups of harmonics around integer multiples of the switching frequency fsw. There are sidebands of the harmonics around each hmultiple. Spacing is equal to the k-multiples of the supply frequency fm. 2

∆

III.

1

(7)

THE CURRENT DRAWN FROM THE GRID, FREQUENCY IS NOT MULTIPLE OF THE SUPPLY FREQUENCY

This part presents a comparison of the simulations and the calculations using the mentioned formulas. Furthermore, it shows comparisons of the current waveforms and its harmonic analysis in the case of a totally random switching frequency.

Fig. 5. The comparison from Figure 4 in the presentage values

The figures show, that results of harmonic analysis calculated in accordance linearized simplified current waveform are accurate. B. The Comparison of the Harmonic Analysis of the Current when the Switching Freqency is 500Hz and 525 Hz Now, we focus on the comparison of the current waveforms during using two different switching frequencies. Figure 6

shows the comparison of current waveforms, Figure 7-8 show harmonic analysis of these current waveforms.

Fig. 9. The comparison of the current waveforms ia, fsw=510Hz and fsw=540Hz) Fig. 6. The comparison of the current waveforms ia, fsw=500Hz and fsw=525Hz

Fig. 10. The comparison of harmonic analysis in the precentage values of the current waveforms ia, fsw=510Hz and fsw=540Hz Fig. 7. The comparison of harmonic analysis in the absolute values of the current waveforms ia, fsw=500Hz and fsw=525Hz

Fig. 11. The comparison of harmonic analysis in the absolute values of the current waveforms ia, fsw=510Hz and fsw=540Hz Fig. 8. The comparison of harmonic analysis in the precentage values of the current waveforms ia, fsw=500Hz and fsw=525Hz

Results confirm that current harmonics do not increase even if the switching frequency is not integer multiple of the supply frequency. The current harmonics appear around the switching frequency in accordance with the frequency modulation Eq. 6 and Eq. 7. It is also confirmed that the current harmonics decrease with the increasing switching frequency. C. The Comparison of the Harmonic Analysis of the Current when the Switching Freqency is 510Hz and 540 Hz This part presents the comparison of current waveforms and their harmonic analysis in the case two switching frequencies, that both frequencies are not the integer multiple of the supply frequency. Figures 9-11 confirm the earlier conclusions. The current harmonics appear around the switching frequency and they decrease with the increasing switching frequency.

D. The Total Harmonic Distortion of Current, when the Switching Freqouency Is Random The last part presents the dependence of the total harmonic distortion of current on the random switching frequencies Fig. 13. It verifies that the same conclusions for the total harmonic distortion of current apply to the random switching frequencies as well as for the switching frequencies that are integer multiples of the supply frequency. Figure 12 shows the harmonic analysis of the currents in the percentage values in the case of five totally random switching frequencies. Harmonic analysis again demonstrated that the current harmonics appear around the switching frequency and its multiples and that they decrease with the increasing switching frequency.

REFERENCES [1]

[2]

[3]

[4] Fig. 12. The comparison of the harmonic analysis, fsw=833Hz, fsw=1478Hz, fsw=2009Hz and fsw=2486Hz

fsw=525Hz, [5]

Figure 13 is used as a recapitulation of simulations in case of random switching frequencies (switching frequencies from Figure 12). The total harmonic distortion of current (THDI) decreases with increasing switching frequency. This also applies to the random switching frequency.

[6]

[7]

[8]

[9]

[10] Fig. 13. The dependence of total harmonic distortion of current THDI on the switching frequency

IV.

CONCLUSION

The analytical calculations and the simulations prove that the voltage-source active rectifier draws the current from the grid with very low distortion. The paper shows that the switching frequency can be arbitrary with regard to the current harmonics. The values of the current harmonics and the total harmonic distortion of current decrease with the increasing switching frequency.

[11]

[12]

[12]

[13]

Kus, V.; Skála, J.; Hammerbauer, J.: Electromagnetic compatibility of the power electronic systems. Book, edited by BEN, Praha, Czech Republic ISBN: 978-80-7300-476-7, 11/2013. (in Czech) Akagi, H.: New trends in active filters for power conditioning,, IEEE Transactions on Industry Applications, vol.32, no.6, pp.1312-1322, Nov/Dec 1996. Kus, V.: Impact of power electronic converters on power distribution network. Book edited in BEN, Praha, Czech Republic, ISBN: 978-807300-062-8, 2002. (in Czech) Filote, C.; Ciufudean, C.; Alaei, S.; Cozgarea, A.: "Harmonic elimination and power factor improvement of three-phase rectifier using RNSIC variant," Clean Electrical Power (ICCEP), 2011 International Conference on , vol., no., pp.40-45, 14-16 June 2011. Singh, B.; Singh, B.; Chandra, A.; Al-Haddad, K.; Pandey, A.; Kothari, D.: A review of single-phase improved power quality AC-DC converters, Industrial Electronics, IEEE Transactions on , vol.50, no.5, pp. 962- 981, Oct. 2003. Bilik, P.; Zidek J.; Kus, V.; Josefova, T.: Harmonic Currents of Semiconductor Pulse Switching Converters, Energy and Power Engineering, Vol. 5 No. 4B, 2013, pp. 1120-1125. Bilik, P.; Koval, L.: Modular system for distributed power quality monitoring, Electrical Power Quality and Utilisation, 2007. EPQU 2007. 9th International Conference on , vol., no., pp.1,5, 9-11 Oct. 2007. Kus, V.; Josefova, T.; Bilik, P.: Harmonic currents generated by the voltage-source active rectifier, Power Engineering, Energy and Electrical Drives (POWERENG), 2013 Fourth International Conference on , vol., no., pp.373,378, 13-17 May 2013. Josefova, T.; Kus, V.: New perspectives on the analysis of current harmonics of the voltage source active rectifier. In: Proceedings of the 14th international scientific conference Electric Power Engineering 2013. Ostrava: VSB - Technical University, 2013. s. 319-324. ISBN 978-80-248-2988-3. (in Czech) Kus, V.; Josefova, T.,: The use of theory of frequency modulation for the calculation of current harmonics of the voltage-source active rectifier. Applied Electronics (AE), 2013 International Conference on , vol., no., pp.1,4, 10-12 Sept. 2013. Josefova, T.; Kus, V.: Analysis of current drawn by the voltage-source active rectifier from the electricity network, Energy (IYCE), 2013 4th International Youth Conference on , vol., no., pp.1,7, 6-8 June 2013. Malinowski, M.; Jasinski, M.; Kazmierkowski, M.P.: Simple direct power control of three-phase PWM rec-tifier using space-vector modulation (DPC-SVM). Industrial Electronics, IEEE Transactions, vol.51, no.2, pp. 447- 454, April 2004 on , vol., no., pp.1-6, 26-29 Sept. 2010. Balazs, G.G.; Kiss, P.; Horvath, M.; Schmidt, I.: Application of the double domain simulation in compari-son of new current control methods for PWM controlled AC electric locomotives, Harmonics and Quality of Power (ICHQP), 2012 IEEE 15th International Conference , vol., no., pp.232-237, 17-20 June 2012. Otčenášová, A.; Altus, J.; Braciník, P.: Power Quality in Practice, Analysis of Power Quality and Available Solutions, International Review on Modelling and Simulation (IREMOS), Vol. 5, N. 4, Italy, 2012, 08, ADE, str.: 1672-1678, ISSN 1974-9821.

I.

the authors have decided to present the issue of current harmonics values when the switching frequency of voltagesource active rectifier is random; it is not the integer multiple of the supply frequency.

II.

CURRENT DRAWN FROM THE GRID BY THREE-PHASE VOLTAGE-SOURCE ACTIVE RECTIFIER

A. Control of the Voltage-Source Active Rectifier The voltage-source active rectifier is composed of six switches; each switch is formed by the IGBT and antiparallel diode. The voltage-source active rectifier has the DC side connected with the capacitor (Fig. 1). Constant voltage is maintained on this capacitor by the proper control [1], [3], [11]-[13].

INTRODUCTION

The use of voltage-source active rectifiers is described in many papers, because they draw current with very low distortion and have a very good power factor [1]-[3]. The harmonics in the current spectrum that do not have negligible values are measured many times. In addition, the frequency of the current harmonics depends on the switching frequency of voltage-source active rectifiers. Therefore, several papers have been written about these problems. They describe the issue of the current harmonics of voltage-source active rectifiers using simulations, theoretical calculations and measurements [4]-[7]. These papers state that the values of these current harmonics are very low in absolute values. If the harmonics are shown in the percentage values, they should be presented relative to the nominal values of the devices [8]. It is also shown that the absolute values of the current harmonics are almost independent of the load of voltage-source active rectifier [9]. The literature also describes the frequency spectrums as a function of the voltage-source active rectifier switching frequency [10]. All these papers show the switching frequency as an integer multiple of the supply frequency (50 Hz). With regard to the discussions at conferences and to the review of the paper [9], This work was supported by the European Regional Development Fund and Ministry of Education, Youth and Sports of the Czech Republic under project No. ED 2.1.00/03.0094: Regional Innovation Centre for Electrical Engineering (RICE)

978-1-4799-3807-0/14/$31.00 ©2014 IEEE

Fig. 1. The three-phase voltage-source active rectifier

Proper functioning of the active rectifier is depicted in the Figure 2, which also shows the current drawn from the grid in phase with the supply voltage. The current waveform is composed of individual sections, which approximate an increasing or a decreasing exponential function. Simulation of the voltage-source active rectifier with the voltage type control was created in C language.

The same conclusions for the total harmonic distortion of current THDI apply to both the switching frequency that is the integer multiple of the supply frequency and the random switching frequency.

Fig. 2. The correct operation of voltage-source active rectifier, fsw=525Hz

B. Harmonic Analysis of the Current Drawn From The Grid by Voltage-Source Active Rectifier Formulas for the current harmonics calculation are derived in the previous papers [8], [9]. Derivation of formulas is calculated using the simplified current. This current was divided into individual parts and the parts were linearized. The next section shows only the resulting formulas.

A. The Comparison of the Harmonic Analysis from the Simulations and the Harmonic Analysis Calculated by the Formulas, the Switching Frequency Equal to 525 Hz Harmonic analysis is calculated using the already derived formulas [9]. The low frequency is selected as the fundamental frequency (current harmonics equal larger values for these frequencies, this follows from the previous papers and it is also shown in the last part of the present paper). First, the difference is verified between the analytical calculations derived for the linearized current waveform and the simulations of voltagesource active rectifier. The results are in Figure 3-5.

Calculation of Fourier coefficients ahl and bhl for l-th straight line of simplified linearized current waveform: 2

2

sin

cos

(1)

cos

sin

(2)

Where members cl and dl derived from l-th straight line il are: ·

,

Fig. 3. The comparison of the simulated current waveform ia and the piecewise linearized current waveform ia, fsw=525Hz

(3)

The resulting members ah and bh for all linearized simplified current waveform: ∑ ∑ , (4) The amplitude of the h-th harmonic of the current: (5) Current harmonics drawn from the grid by voltage-source active rectifier appear around the switching frequency fsw in accordance with the theory of frequency modulation [10]:

Fig. 4. The comparison of harmonic analysis of the simulated current waveform ia and the piecewise linearized current waveform ia, fsw=525Hz

(6) ·

1

The spectrum of the PWM signal is composed of groups of harmonics around integer multiples of the switching frequency fsw. There are sidebands of the harmonics around each hmultiple. Spacing is equal to the k-multiples of the supply frequency fm. 2

∆

III.

1

(7)

THE CURRENT DRAWN FROM THE GRID, FREQUENCY IS NOT MULTIPLE OF THE SUPPLY FREQUENCY

This part presents a comparison of the simulations and the calculations using the mentioned formulas. Furthermore, it shows comparisons of the current waveforms and its harmonic analysis in the case of a totally random switching frequency.

Fig. 5. The comparison from Figure 4 in the presentage values

The figures show, that results of harmonic analysis calculated in accordance linearized simplified current waveform are accurate. B. The Comparison of the Harmonic Analysis of the Current when the Switching Freqency is 500Hz and 525 Hz Now, we focus on the comparison of the current waveforms during using two different switching frequencies. Figure 6

shows the comparison of current waveforms, Figure 7-8 show harmonic analysis of these current waveforms.

Fig. 9. The comparison of the current waveforms ia, fsw=510Hz and fsw=540Hz) Fig. 6. The comparison of the current waveforms ia, fsw=500Hz and fsw=525Hz

Fig. 10. The comparison of harmonic analysis in the precentage values of the current waveforms ia, fsw=510Hz and fsw=540Hz Fig. 7. The comparison of harmonic analysis in the absolute values of the current waveforms ia, fsw=500Hz and fsw=525Hz

Fig. 11. The comparison of harmonic analysis in the absolute values of the current waveforms ia, fsw=510Hz and fsw=540Hz Fig. 8. The comparison of harmonic analysis in the precentage values of the current waveforms ia, fsw=500Hz and fsw=525Hz

Results confirm that current harmonics do not increase even if the switching frequency is not integer multiple of the supply frequency. The current harmonics appear around the switching frequency in accordance with the frequency modulation Eq. 6 and Eq. 7. It is also confirmed that the current harmonics decrease with the increasing switching frequency. C. The Comparison of the Harmonic Analysis of the Current when the Switching Freqency is 510Hz and 540 Hz This part presents the comparison of current waveforms and their harmonic analysis in the case two switching frequencies, that both frequencies are not the integer multiple of the supply frequency. Figures 9-11 confirm the earlier conclusions. The current harmonics appear around the switching frequency and they decrease with the increasing switching frequency.

D. The Total Harmonic Distortion of Current, when the Switching Freqouency Is Random The last part presents the dependence of the total harmonic distortion of current on the random switching frequencies Fig. 13. It verifies that the same conclusions for the total harmonic distortion of current apply to the random switching frequencies as well as for the switching frequencies that are integer multiples of the supply frequency. Figure 12 shows the harmonic analysis of the currents in the percentage values in the case of five totally random switching frequencies. Harmonic analysis again demonstrated that the current harmonics appear around the switching frequency and its multiples and that they decrease with the increasing switching frequency.

REFERENCES [1]

[2]

[3]

[4] Fig. 12. The comparison of the harmonic analysis, fsw=833Hz, fsw=1478Hz, fsw=2009Hz and fsw=2486Hz

fsw=525Hz, [5]

Figure 13 is used as a recapitulation of simulations in case of random switching frequencies (switching frequencies from Figure 12). The total harmonic distortion of current (THDI) decreases with increasing switching frequency. This also applies to the random switching frequency.

[6]

[7]

[8]

[9]

[10] Fig. 13. The dependence of total harmonic distortion of current THDI on the switching frequency

IV.

CONCLUSION

The analytical calculations and the simulations prove that the voltage-source active rectifier draws the current from the grid with very low distortion. The paper shows that the switching frequency can be arbitrary with regard to the current harmonics. The values of the current harmonics and the total harmonic distortion of current decrease with the increasing switching frequency.

[11]

[12]

[12]

[13]

Kus, V.; Skála, J.; Hammerbauer, J.: Electromagnetic compatibility of the power electronic systems. Book, edited by BEN, Praha, Czech Republic ISBN: 978-80-7300-476-7, 11/2013. (in Czech) Akagi, H.: New trends in active filters for power conditioning,, IEEE Transactions on Industry Applications, vol.32, no.6, pp.1312-1322, Nov/Dec 1996. Kus, V.: Impact of power electronic converters on power distribution network. Book edited in BEN, Praha, Czech Republic, ISBN: 978-807300-062-8, 2002. (in Czech) Filote, C.; Ciufudean, C.; Alaei, S.; Cozgarea, A.: "Harmonic elimination and power factor improvement of three-phase rectifier using RNSIC variant," Clean Electrical Power (ICCEP), 2011 International Conference on , vol., no., pp.40-45, 14-16 June 2011. Singh, B.; Singh, B.; Chandra, A.; Al-Haddad, K.; Pandey, A.; Kothari, D.: A review of single-phase improved power quality AC-DC converters, Industrial Electronics, IEEE Transactions on , vol.50, no.5, pp. 962- 981, Oct. 2003. Bilik, P.; Zidek J.; Kus, V.; Josefova, T.: Harmonic Currents of Semiconductor Pulse Switching Converters, Energy and Power Engineering, Vol. 5 No. 4B, 2013, pp. 1120-1125. Bilik, P.; Koval, L.: Modular system for distributed power quality monitoring, Electrical Power Quality and Utilisation, 2007. EPQU 2007. 9th International Conference on , vol., no., pp.1,5, 9-11 Oct. 2007. Kus, V.; Josefova, T.; Bilik, P.: Harmonic currents generated by the voltage-source active rectifier, Power Engineering, Energy and Electrical Drives (POWERENG), 2013 Fourth International Conference on , vol., no., pp.373,378, 13-17 May 2013. Josefova, T.; Kus, V.: New perspectives on the analysis of current harmonics of the voltage source active rectifier. In: Proceedings of the 14th international scientific conference Electric Power Engineering 2013. Ostrava: VSB - Technical University, 2013. s. 319-324. ISBN 978-80-248-2988-3. (in Czech) Kus, V.; Josefova, T.,: The use of theory of frequency modulation for the calculation of current harmonics of the voltage-source active rectifier. Applied Electronics (AE), 2013 International Conference on , vol., no., pp.1,4, 10-12 Sept. 2013. Josefova, T.; Kus, V.: Analysis of current drawn by the voltage-source active rectifier from the electricity network, Energy (IYCE), 2013 4th International Youth Conference on , vol., no., pp.1,7, 6-8 June 2013. Malinowski, M.; Jasinski, M.; Kazmierkowski, M.P.: Simple direct power control of three-phase PWM rec-tifier using space-vector modulation (DPC-SVM). Industrial Electronics, IEEE Transactions, vol.51, no.2, pp. 447- 454, April 2004 on , vol., no., pp.1-6, 26-29 Sept. 2010. Balazs, G.G.; Kiss, P.; Horvath, M.; Schmidt, I.: Application of the double domain simulation in compari-son of new current control methods for PWM controlled AC electric locomotives, Harmonics and Quality of Power (ICHQP), 2012 IEEE 15th International Conference , vol., no., pp.232-237, 17-20 June 2012. Otčenášová, A.; Altus, J.; Braciník, P.: Power Quality in Practice, Analysis of Power Quality and Available Solutions, International Review on Modelling and Simulation (IREMOS), Vol. 5, N. 4, Italy, 2012, 08, ADE, str.: 1672-1678, ISSN 1974-9821.