Bi32385391

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, Brazil

ABSTRACT
The pulse density modulation technique switches or gate drives. The DC blocking capacitor
has 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 by
any level of output power. The sub harmonics of low-frequency components present in the PDM
the switching frequency are in general undesired signal, as illustrated in Figure 6. If the magnetizing
and are seen as a drawback. In this work are inductance saturates the frequency in which the
developed analytical equations to express the series resonance occurs can change, leading the
harmonics and sub harmonics components of a circuit (transformer plus ozone cell) to an unknown
PDM signal. It is still analyzed the PDM and variable operation point which should be
parameters 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 the
Keywords - Double Fourier Series, Pulse Density ozone generating cell. In such application it is useful
Modulation, Ozone Generation, Resonant to investigate whether there are or not harmonic and
Converters sub harmonic components in the low frequency
range which could lead to a malfunctioning of the
I. INTRODUCTION power converter.
In many applications in which it is In [9] it is shown a methodology to estimate
sufficient 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 approach
output power by using the PDM - Pulse Density to accurately calculate the PDM spectra based only
Modulation. This modulation technique allows on the voltage of the PDM pulse train by means of
operation with constant switching frequency and analytical equations based on the PDM signal. These
pulse width, controlling the power by varying the equations can give to the designer a fast way to
pulses in a given time base, while keeping soft check whether the system against resonances which
switching capability in the whole power range for may be excited by the sub harmonics of the PDM
which the converter was designed [1]. It is still signal. It is also possible to determine for what range
possible to simultaneously control the PDM of PDM densities or pulse widths the system
modulation 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 of
delivered 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 to
extended range of high frequency high power any other application related to the use of PDM
resonant converters. The main applications where modulation in electronics power converters.
square wave PDM control are commonly applied to
are power converters intended to be used for ozone
generation [3][4][5], high power welding machines, CDC
high frequency induction heating for plasma Vdc S1 S3
applications [6] and power supplies with inherent Load
soft switching capability. S2
S4
As an example of application of PDM
modulation one can take the a typical ozone
generation converter, as shown in Fig. 1 where the
load has a capacitive behavior allowing resonant Fig. 1 - Typical resonant converter used for ozone
operation since the transformer has significant generation.
inductance., as shown in Fig. 2 [7]. The transformer
can became saturated if the inverter produces a non
zero average voltage. Such unbalance can be
originated with small differences in the power

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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 it
Fig. 2 – Transformer and load model, including DC blocking becomes possible to obtain the analytical expression
capacitor. 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

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 0
Fig. 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

 
1
C 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) -1
Carrier 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) 0.1

Carrier 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 synthesized
or reconstructed from the coefficients calculated FFT Algorithm
above by means of (10). 0.6 Analytical DFS
Fig. 7 shows an example of Unipolar PDM signal 0.5
which 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 0.3

corresponding to   2  0.5 . Fig. 3 also shows the 0.2

spectrum of this PDM obtained by the FFT of the 0.1
PDM signal. Fig. 8 compares the spectrum obtained
0
via FFT and the spectrum analytically calculated 0 500 1000 1500 2000 2500 3000 3500 4000
using 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

expressions previously developed for the Unipolar the coefficients is not shown, since they can be
PDM signal. promptly extracted from (12).
The next section deal with the calculation of the In order to validating the decomposition of the
DFS coefficients for the Bipolar PDM signal which Bipolar PDM signal the following figures shown the
is an extension of the analysis presented in the frequency and time domain plots when varying the
current section. modulating index of the reference signal  and the

IV. 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, the
The 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 proportion
obtained 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 of
part of the width of high frequency pulses.
the carrier. However, are of great importance for the
sidebands to the carrier, and especially for the
Therefore:, 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 to
when analyzing how the PDM waveform is control both the pulse density of the modulating
generated in an single-phase full-bridge converter.
signal and the pulse width of the carrier signal. This
The first arm of the inverter generates the PDM results shown that using carrier duty cycles different
f x, y  c 0
pulses 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 kind
generates 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 any
connected load.
The signal generated by the inverter is given by (12)
which express the Bipolar PDM signal in time
domain. 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


Fig . 11 – PDM spectrum and waveform for    e
  0,35  2

Fig. 9 – PDM spectrum and waveform for  
V. CONCLUSIONS
and   . 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 model
Fig. 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

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[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
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control method for high-frequency
applications," Power Electronics and Drive
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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
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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–
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[9] H. Calleja, J.Pacheco, Frequency Spectra of
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México, 2000.
[10] D. G. Holmes and T. A. Lipo, Pulse width
modulation for power converters: Principles
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724 p.

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