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Bi32385391

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Bi32385391

  1. 1. Jakson Paulo Bonaldo, José Antenor Pomilio / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.385-391 Spectral Analysis of Square-wave Pulse Density Modulation: An Analytical Approach Jakson Paulo Bonaldo* and José Antenor Pomilio* *School of Electrical and Computer Engineering, Campinas, BrazilABSTRACT The pulse density modulation technique switches or gate drives. The DC blocking capacitorhas been used in resonant power converters adds an additional low frequency series resonance,because its ability to maintains soft-switching for as shown in Fig. 3.Such resonance can be excited byany level of output power. The sub harmonics of low-frequency components present in the PDMthe switching frequency are in general undesired signal, as illustrated in Figure 6. If the magnetizingand are seen as a drawback. In this work are inductance saturates the frequency in which thedeveloped analytical equations to express the series resonance occurs can change, leading theharmonics and sub harmonics components of a circuit (transformer plus ozone cell) to an unknownPDM signal. It is still analyzed the PDM and variable operation point which should beparameters variation and their effects in the avoided for safe operation of the circuit [8].frequency spectra. Fig.4 shows a typical PDM pulse train and the resulting current when these pulses are applied to theKeywords - Double Fourier Series, Pulse Density ozone generating cell. In such application it is usefulModulation, Ozone Generation, Resonant to investigate whether there are or not harmonic andConverters sub harmonic components in the low frequency range which could lead to a malfunctioning of theI. INTRODUCTION power converter. In many applications in which it is In [9] it is shown a methodology to estimatesufficient to control the average power in a relatively the PDM spectra observing the modulated current.long time interval it is possible to modulate the However, in this work is proposed a new approachoutput power by using the PDM - Pulse Density to accurately calculate the PDM spectra based onlyModulation. This modulation technique allows on the voltage of the PDM pulse train by means ofoperation with constant switching frequency and analytical equations based on the PDM signal. Thesepulse width, controlling the power by varying the equations can give to the designer a fast way topulses in a given time base, while keeping soft check whether the system against resonances whichswitching capability in the whole power range for may be excited by the sub harmonics of the PDMwhich the converter was designed [1]. It is still signal. It is also possible to determine for what rangepossible to simultaneously control the PDM of PDM densities or pulse widths the systemmodulation ratio and the duty cycle allowing a finer operation is sage.and more accurate adjust of the output power It is important state that the application ofdelivered to a given load [2]. the proposed analysis is not limited to ozone The PDM control may be applied on an generation devices, but it can be fully applicable toextended range of high frequency high power any other application related to the use of PDMresonant converters. The main applications where modulation in electronics power converters.square wave PDM control are commonly applied toare power converters intended to be used for ozonegeneration [3][4][5], high power welding machines, CDChigh frequency induction heating for plasma Vdc S1 S3applications [6] and power supplies with inherent Loadsoft switching capability. S2 S4 As an example of application of PDMmodulation one can take the a typical ozonegeneration converter, as shown in Fig. 1 where theload has a capacitive behavior allowing resonant Fig. 1 - Typical resonant converter used for ozoneoperation since the transformer has significant generation.inductance., as shown in Fig. 2 [7]. The transformercan became saturated if the inverter produces a nonzero average voltage. Such unbalance can beoriginated with small differences in the power 385 | P a g e
  2. 2. Jakson Paulo Bonaldo, José Antenor Pomilio / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.385-391 Cb Rs Ls II. PDM – BASIS OF PULSE DENSITY MODULATION SPECTRAL ANALYSIS Rp Cp Lm Ca Holmes and Lipo [10] show a methodology Vinv Vo for determining the spectrum of a Sinusoidal Pulse Vz Cv Width Modulation – SPWM. Based on the concept of sinusoidal switching cell, this paper considers a Transformer O. G. Cell switching cell for a rectangular wave, whereby itFig. 2 – Transformer and load model, including DC blocking becomes possible to obtain the analytical expressioncapacitor. of the frequency spectrum for rectangular pulse density modulation. This work is organized in more three The concept of switching cell can be used sections. In the second section it is explained the to find the Double Fourier Series (DFS) of a signal general theory behind the PDM spectral analysis. given by f(t), which is composed of a carrier and a Section three deals with the unipolar PDM pulses modulating waveform. The modulating is also which could be applicable on DC-DC power known as the reference signal. The unit switching converters when they are operating with PDM cell identifies edges within for which the function modulation. Finally, section four brings the analysis f(t) is constant for periodic variation of the carrier of the bipolar PDM pulses which are the main issue x(t) and modulating y(t) signals. of this work. The function f t   f xt , yt  , within each region boundary of the unit cell, represents the resulting voltage in the arms of a single-phase H- bridge inverter. For a low frequency reference, or modulating, rectangular-type unit cell is proposed as shown in Figure 1. It can happen variations in both, the pulse width of the low frequency reference signal and in the high frequency carrier wave, that is the converter switching frequency. Therefore, the cell inner contour is determined by  and  , where:   duty-cycle of the reference low frequency   duty-cycle of the high frequency carrier  ot 2 Fig. 3 – Input impedance and voltage gain. 0   ct Vdc 0  0 2 Fig. 1 – Rectangular Switching Cell. The range of  and  are between 0 and 2 . The variation of  modify the PDM pulse density, what means change the duty cycle of the modulating signal increasing the high frequency Fig. 4 - PDM pulses applied on an ozone generating pulses inside a low frequency period, leading to cell. changes in the average power delivered to a given 386 | P a g e
  3. 3. Jakson Paulo Bonaldo, José Antenor Pomilio / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.385-391 load. Variations in  change the width of the high side band harmonics. For m=n=0 it is defined the first term of the DFS, which corresponds to the frequency pulses, affecting the voltage value. Fig. 6.a shows a grid consisting of the unit cell average value of the pulse width modulated replication and Fig. 6.b shows the resulting signal. For m=0, the harmonic frequencies are modulated signal. While the line with slope defined by n, being named as base band harmonics. For n=0 the harmonic frequencies are o / c is over the gray area of the unit cell, the due to the carrier and defined by m, named carrier resulting voltage value is equal to VDC. The voltage harmonics. The sidebands harmonics are created returns to zero when the line crosses the boundary when m≠0 and n≠0. and reaches the white region of the unit cell, thus The DFS coefficients, in the trigonometric generating the pulse density modulated signal. form, can be calculated as follows, where the limits o of integration must be used according to the switching cell. c 2 2    f x, y cosmx  nydxdy 1 Amn  2 2 0 0 a) (3)Vdc 2 2    f x, y sinmx  ny dxdy 1 0 Bmn  b) t 2 2 0 0Fig. 6 – Arrangement of unit cells which originates the PDM (4)signal. Or, in the complex form:    f x, y e 1 j  mx ny  The general form of the Double Fourier C mn  Amn  jBmn  dxdy Series is given in (1), in which the first term refers to 2 2 0 0 the average value. The second term refers to the (5) harmonics of the base band. The third term consists of harmonics of the carrier and the fourth consists of The DFS analysis is initially developed for the sideband harmonic of the carrier. an unipolar PDM waveform, as shown in Fig. 7,    A0 n cosny   B0 n sin ny   A00 f  x, y   which is expressed in terms of the DFS coefficients. 2 The unipolar DFS is then expanded to the bipolar n 0 DFS by means of the sum of two unipolar     Am 0 cosmx   Bm 0 sin mx   waveforms shifted in time by the angle corresponding to the complementary carrier duty- m 1   cycle.   A m  0 n   mn cosmx  ny   Bmn sin mx  ny  III. DFS CALCULATION OF AN UNIPOLAR (1) PDM WAVEFORM The switching cell showed on Fig. 5 is used Or, in the general form: to perform the DFS calculation of an Unipolar PDM   waveform defining a function f(x,y) which does not f  x, y     A mn cosmx  ny   Bmn sin mx  ny  need to be integrated all over the periods, but just until the  and  limits. The DFS calculation is m 0 n   (2) carried on apart for each kind of coefficient as Where: depicted bellow. xt    c t   c y t   ot   0 Average Coefficient e  1 c  C 00  A00  jB00  V e j 0 x  0 y  dxdy  2   dc carrier angular frequency 2 0 0 c  Vdc  Vdc  Vdc   dxdy   dy  2 carrier phase angle   o  2 2 0 0 2 2 0 2 reference angular frequency (6) o  Baseband Coefficients reference phase angle The indices m and n define the harmonic frequencies of the carrier and the modulating wave, respectively. A combination of these indices turn in 387 | P a g e
  4. 4. Jakson Paulo Bonaldo, José Antenor Pomilio / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.385-391   1C 0 n  A0 n  jB0 n   V e j 0 x  ny  dxdy  1 2 2 dc Voltage (V) 0 0 0.5   j  sin n     j cosn    Vdc 0 2n 2 -0.5 (7) -1Carrier Coefficients 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02   Time (s) 1 j  m  x  0 y C m 0  Am 0  jBm 0  2 2  V 0 0 dc e dxdy  0.3 Amplitude (V)  j  sin m     j cosm     V dc 0.2 2m 2 Vdc (8) 0.1Carrier Sideband Coefficients 0   1 Vdce j mx ny dxdy  2 2  Cmn  Amn  jBmn  -500 0 500 Hz 1000 1500 2000 2500 (9) Frequency (Hz) 0 0 Vdc cosn   cosm  cosn  m  1   Fig. 7 – DC biased PDM signal (bottom) and its   2mn 2  jsin n   sin m  sin n  m  spectrum (top).The function f(x,y,t), which denote the Unipolar   V      dc 2  sin n     cosno t  n 0   dc 2  1  cosn    sin no t  n 0   Vdc V f  x, y , t   4 n 0  2n 2n    V     dc 2   sin m     cosmc t  m c   dc 2 1  cosm    sin mc t  m c   V m 1  2m 2m   Vdc    cosn   cosm  cosn  m  1 cosmc t  m c  no t  n 0    2mn 2     m 0 n     Vdc  2mn 2 sin n   sin m  sin n  m sin mc t  m c  no t  n 0     (10)PDM signal in the time domain, can be synthesizedor reconstructed from the coefficients calculated FFT Algorithmabove by means of (10). 0.6 Analytical DFSFig. 7 shows an example of Unipolar PDM signal 0.5which has unitary voltage amplitude. The Amplitude (V)modulating pulse density is 0.5 corresponding to 0.4  2  0.5 and the carrier pulse density is 0.5 0.3corresponding to   2  0.5 . Fig. 3 also shows the 0.2spectrum of this PDM obtained by the FFT of the 0.1PDM signal. Fig. 8 compares the spectrum obtained 0via FFT and the spectrum analytically calculated 0 500 1000 1500 2000 2500 3000 3500 4000using the equations develop early in this section. Frequency (Hz) Fig. 8 – Comparison of spectra of the waveform shown in Fig. 3 with the spectra analytically calculated. Fig. 8 clearly shows that the FFT spectrum of the Unipolar PDM signal matches exactly the spectrum obtained when applying the mathematical expressions developed. This result validate the 388 | P a g e
  5. 5. Jakson Paulo Bonaldo, José Antenor Pomilio / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.385-391expressions previously developed for the Unipolar the coefficients is not shown, since they can bePDM signal. promptly extracted from (12).The next section deal with the calculation of the In order to validating the decomposition of theDFS coefficients for the Bipolar PDM signal which Bipolar PDM signal the following figures shown theis an extension of the analysis presented in the frequency and time domain plots when varying thecurrent section. modulating index of the reference signal  and theIV. DFS CALCULATION OF A BIPOLAR PDM carrier modulating index  which also dictates the pulse width of the carrier waveform.WAVEFORM As shown in Fig. 9, Fig. 10 and Fig. 11, theThe waveform of the bipolar PDM signal can be amplitude of the carrier varies proportionally to the f x, y  c 0 duty cycle of the modulating wave in proportionobtained by the difference between and f x, y  c  to  . Variations in the pulse width of the carrier,  , , where:   2   is complementary does not exert a major influence on the amplitude ofpart of the width of high frequency pulses. the carrier. However, are of great importance for the sidebands to the carrier, and especially for theTherefore:, harmonics in the baseband and the average f PDM x, y   f x, y  c 0  f x, y  c  level. This feature can be used to make the control of (11) the voltage average level.The physical meaning of (11) is better understood Beside it is possible and in many times desired towhen analyzing how the PDM waveform is control both the pulse density of the modulatinggenerated in an single-phase full-bridge converter. signal and the pulse width of the carrier signal. ThisThe first arm of the inverter generates the PDM results shown that using carrier duty cycles different f x, y  c 0pulses given by and the second arm from 0.5 which is equivalent to    generates low f x, y  c  frequency harmonics and, as stated before, this kindgenerates the pulses given by . So, the of harmonics are drawbacks for some applications,combination of both signal make up the Bipolar mainly when using a DC block capacitor.PDM signal which is in fact applied over anyconnected load.The signal generated by the inverter is given by (12)which express the Bipolar PDM signal in timedomain. For the sake of simplicity the final shape of   V         dc 2    sin n     cosn o t  n 0      1  cosn    sin n o t  n 0   Vdc f PDM x, y   4 n  0  2n    V sin m       sin m    cosm   1  cosm   sin m  cosm c t       dc 2     m 1  2m  1  cosm     1  cosm   cosm   sin m   sin m  sin m c t    cosn   cosm   cosn  m   1          cosn   cosm   cosn  m   1cosm    cosm c t  n o t  n 0      Vdc  sin n   sin m   sin n  m sin m              m  0 n    2mn  sin n   sin m   sin n  m     2    sin n   sin m   sin n  m cosm     sin m t  n t  n        c o 0      cosn   cosm   cosn  m   1sin m       (12) 389 | P a g e
  6. 6. Jakson Paulo Bonaldo, José Antenor Pomilio / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.385-391 Fig . 11 – PDM spectrum and waveform for    e   0,35  2Fig. 9 – PDM spectrum and waveform for   V. CONCLUSIONSand   . This work has presented a new approach to analytically determine the spectrum of an square wave PDM modulated signal. Each spectral component can be obtained and analyzed against variations in modulation parameters like carrier pulse width and reference modulating index. Therefore, this tool could be useful for analyzing an given spectral component of interest and check out its behavior when changes occur in the PDM parameters, making possible to rapidly determines the amplitude of that component. whether it exists in the considered PDM signal. REFERENCES [1] Ahmed, N.A.; , "High-Frequency Soft- Switching AC Conversion Circuit With Dual-Mode PWM/PDM Control Strategy for High-Power IH Applications," Industrial Electronics, IEEE Transactions on , vol.58, no.4, pp.1440-1448, April 2011. [2] Amjad, M.; Salam, Z.; Facta, M.; Mekhilef, S.; , "Analysis and Implementation of Transformerless LCL Resonant Power Supply for Ozone Generation," Power Electronics, IEEE Transactions on , vol.28, no.2, pp.650-660, Feb. 2013. [3] Vijayan, T.; Patil, Jagadish G.; , "A modelFig. 10 – PDM spectrum and waveform for of the generation and transport of ozone in  2  0,2 e    high-tension nozzle driven corona inside a novel diode," Review of Scientific Instruments , vol.83, no.12, pp.123304- 123304-11, Dec 2012 390 | P a g e
  7. 7. Jakson Paulo Bonaldo, José Antenor Pomilio / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.385-391[4] J. Alonso, J. Garcia, A. Calleja, J. Ribas, and J. Cardesin, “Analysis, design, and experimentation of a high-voltage power supply for ozone generation based on current-fed parallel-resonant push-pull inverter”, IEEE Transactions on Industry Applications, vol. 41, no. 5, pp. 1364–1372, Sept.-Oct. 2005.[5] Nakata, Y.; Itoh, J.; , "An experimental verification and analysis of a single-phase to three-phase matrix converter using PDM control method for high-frequency applications," Power Electronics and Drive Systems (PEDS), 2011 IEEE Ninth International Conference on , vol., no., pp.1084-1089, 5-8 Dec. 2011.[6] Vijayan, T.; Patil, J.G.; , "Temporal Development of Ozone Generation in Electron-Induced Corona-Discharge Plasma," Plasma Science, IEEE Transactions on , vol.39, no.11, pp.3168- 3172, Nov. 2011.[7] J. P. Bonaldo. J. A. Pomilio, “Control Strategies for High Frequency Voltage Source Converter for Ozone Generation”, Proc. of the IEEE International Symposium on Industrial Electronics, ISIE 2010, Bari, Italy, July 4 to 7, 2010, pp; 754-760.[8] L. Bolduc, A. Gaudreau, A. Dutil, “Saturation time of transformers under dc excitation”, Elsevier Electric Power Systems Research Vol. 56, 2000, pp. 95– 102.[9] H. Calleja, J.Pacheco, Frequency Spectra of Pulse-Density Modulated Waveforms, México, 2000.[10] D. G. Holmes and T. A. Lipo, Pulse width modulation for power converters: Principles and practice, Wiley Inter-Science, 2003. 724 p. 391 | P a g e

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