29
CHAPTER 3
A LOW FREQUENCY PULSE WIDTH MODULATION
STRATEGY FOR CYCLOCONVERTERS
3.1
INTRODUCTION
Power quality problems are of increasing concern in contemporary
industry particularly with progress in the application of ac-ac conversion
systems involving power electronic converters (Lander 1987). It augurs the
need of proper harmonic management methods to acquire a complete
understanding of harmonic source behaviors and necessitate the need to
explore the interactions between the converter and other system components
(Syam et al 2004 a, Wang et al 2000, Syam et al 2004 b).
The ac-ac
conversion systems and their control strategies are thus under intense research
to meet the requirements of the utilities.
A control strategy refers to a set of firing signals that trigger the
devices in a cycloconverter to achieve a specific target output voltage and
frequency (Miyazawa et al 1989, Ishiguro et al 1991). The use of a particular
control strategy may help to improve the shape of the input current and/or
output voltage (Basirifar et al 2011). However the available approaches do not
appear to offer the desired performance. It is in this prelude that modifications
in the existing PWM techniques are contemplated to contribute in this
perspective.
There is a growing interest in the use of cycloconverters for bulk
power transfer in a good number of applications (Das et al 1997).
30
Cycloconverters inherit the ability to provide simultaneous voltage and
frequency transformation without the help of intermediate stage reactive
component by synthesizing a low frequency waveform from appropriate
sections of a higher frequency source (Pelly 1971, Lander 1987, Chu et al
1989, Shicheng Zheng et al 2009). However it assuages the dent of harmonics
and consequent effects of increased losses, deliberate de-rating and lower
efficiency. It therefore continues to urge the development of new measures to
build methodologies in order to refine the shape of the specified output
voltage (Agrawal et al 1992, Jeevananthan et al 2007 a).
3.2
PROBLEM FORMULATION
The main aim is to design a low frequency, programmed PWM
(PPWM) strategy in order to minimize the distortion in the output voltage of
single phase cycloconverters. The proposed EAC (Grahame Holmes et al
2003) attempts to reduce the lower order harmonic content especially in the
low output voltage range. The performance of this scheme is evaluated
through MATLAB based simulation and validated using a prototype. The
approach involves deriving mathematical relations for a host of techniques
and ventures to highlight the merits of this methodology.
3.3
EXISTING TECHNIQUES
The cycloconverter, being a primordial converter imbibes a facility
to work with a family of control schemes. A number of firing schemes appear
to revolve around to suit specific applications. Jeevananthan et al (2007 a)
proposed a host of most commonly used methods (a) constant firing angle
method
(CFAM),
(b)
cosine
wave
crossing
method
(CWCM),
(c) HWSVFAM and (d) QWSVFAM. The standard single phase 230V, 50 Hz
AC waveform seen in Figure 3.1 (a) is considered as the input and a
frequency transformation (Kf) ratio of three is chosen to explain the schemes.
31
3.3.1
Constant Firing Angle Method
Figure 3.1(b) shows the output voltage waveform for CFAM, where
the delay angle
is allowed to vary according to the voltage requirements of
the load. This is a simple straightforward method, which gives a crude output
waveform with considerable harmonic content resulting in its poor THD.
3.3.2
Cosine Wave Crossing Method
Figure 3.1(c) shows the output voltage waveform for CWCM in
which the comparison of a sinusoidal reference of output frequency with two
cosine carriers of input frequency (cosine and inverted cosine) generates the
gating pulses.
3.3.3
Half Wave Symmetry Variable Firing Angle Method
The HWSVFAM is a closer approximation to a sine wave that is
synthesized by phase delaying the firing of the switches as shown in
Figure 3.1(d). It is a modified version of CFAM and operates with a different
firing angle for different input half cycles. However, the firing angle of the
first and Kf th (third for this case) pulses are same, and hence the name half
wave symmetry. The load receives a full input half cycle at its peak period
Kf 1
2
th
and progressively reduces its conduction for other sections owing
to fact that the output fundamental component requires controlled phase area
proportional to its instantaneous values.
3.3.4
Quarter Wave Symmetry Variable Firing Angle Method
The QWSVFAM is basically a modified phase controlled
cycloconversion, which makes use of hybrid commutation (both natural and
forced commutation) for enabling quarter wave symmetry in the output. Each
32
input half cycle is phase controlled in such a way that the position of each
output pulse (phase controlled half cycle) is as close as possible to the desired
sinusoidal output voltage in addition to their proportionality average value as
seen in the output voltage waveform in Figure 3.1(e). It is similar to
HWSVFAM in the first quarter of any output half cycle while the second
quarter differs, which is the exact mirror image of the first quarter. In this
method, the delay angles of segments vary in such a way that in addition to
average value of each segment, the pulse orientations also lie as closely as
possible to the variations of desired sinusoidal output voltage and renders
reduced the harmonics.
Though the symmetric firing schemes viz., HWSVFAM and
QWSVFAM support a marginal improvement, still there is a definite need to
introduce refinements in the available strategies to harness a much lower
distortion in the output voltage.
Figure 3.1 Input/ output voltage waveforms (a) input (b) output-CFAM
(c)
output-CWCM
(d)
output-HWSVFAM
and
(e) output-QWSVFAM
33
3.4
MATHEMATICAL EXTRACTION OF THE PROPOSED
EAC STRATEGY
The existing cycloconversion strategies introduce lower order
harmonics and increase the THD. The mission is to develop a low frequency,
PPWM strategy that provides an improved performance in terms of minimal
lower order harmonics and reduced THD. The EAC is a refined PWM
strategy suitable for single-phase to single-phase cycloconverter. The width of
the PWM pulses in this approach is determined by making the area of the
PWM signal equal to that under the sampled target waveform, irrespective of
the number of samples. It orients to compute the pulse widths for a wide
range of input and output conditions. The basic principle of the EAC method
is illustrated in Figure 3.2.
Figure 3.2 EAC for cycloconverter
34
The area enfolded by the target output waveform in the kth sampling
period is represented by,
k
m
i
AkT
M sin
0t
dt
k 1
m i
AkT
M
cos
0
k 1
cos
mK f
k
mK f
(3.1)
The pulse width at each sampling period is calculated by equating
the area under the target in every sampling period with the area of the actual
output. The area enfolded by the actual output waveform in the corresponding
kth sampling duration is given by
k 1
m
2m 2
i
AkO
sin K f
0
tdt
k 1
m
2m 2
i
AkO
2
Kf
sin
k
0.5
0
m
sin
(3.2)
2
where, m is the number of samples per half cycle of the input waveform,
M
2
is the modulation depth,
o
is the output frequency,
Kf
o
is the
i
frequency transformation ratio and
i
is the input frequency.
The pulse width, ‘ ’ for kth sampling period is calculated by
equating the area under the target waveform represented by Equation (3.1) to
the area under the actual output waveform given by Equation (3.2).
35
M
k 1
cos
mK f
0
2 sin
cos
1
k
mK f
2
Kf
sin
0
X
Y
k
0.5
m
sin
2
(3.3)
where
X
k 1
K f M cos
Y
2
j 1
1
cos
mK f
sin k
k
mK f
0.5
m
The Fourier constants and harmonic analysis of output voltage are
determined as follows.
f (x)
a0
2
a n cos
n 1
nx
3
bn sin
nx
3
T
2
an
4
nx
f (x) cos dx
T0
3
an
2Vm
3
3
sin x cos
0
nx
dx
3
k 1
an
2Vm
3
m
2m 2
sin x cos
k 1
m
2m 2
nx
dx
3
(3.4)
36
k 1
m
Vm
3
an
2m 2
sin(
k 1
m
2m 2
1
an
( 1) j
1
Vm
n
n
1)x sin(
1)x dx
3
3
n 3
1
n 3
cos
n
1
3
k 1
cos
n
1
3
k 1
m
m
2m
2m
2
2
cos
n
1
3
k 1
cos
n
1
3
k 1
m
m
2m
2
2m
2
(3.5)
bn
3
2Vm
3
sin x sin
0
nx
dx
3
k 1
bn
Vm
3
m
2m 2
cos
k 1
m
2m 2
1
bn
( 1) j
1
Vm
n
n
1 x cos
1 x dx
3
3
n 3
1
n 3
sin
n
1
3
k 1
sin
n
1
3
k 1
m
m
2m
2m
2
2
sin
n
1
3
k 1
sin
n
1
3
k 1
m
m
2m
2
2m
2
(3.6)
where j=1 for odd chopping and j=2 for even chopping for a particular value
of Kf.
The third order Fourier constants and harmonic analysis of output
voltage are determined as follows.
b3
2Vm
3
3
sin x sin xdx
0
37
k 1
b3
m
Vm
3
1 cos 2x dx
k 1
m
b3
1
2m 2
j 1
2m 2
Vm
3
cos
2k 1
sin
m
(3.7)
k 1
a3
a3
2Vm
3
1
j 1
m
2m 2
sin x cos xdx
k 1
m
2m 2
Vm
sin
3
2k 1
m
sin
(3.8)
The advent of PWM appears to revelutionise the performance of
switched converters in the sense they inherit the facility to be programmed
and suit precise design requirements. The firing pulse generated using some
form of PWM based approaches hold merit and influence the operation of the
circuit in a satisfactory manner. It is precisely the fact as to why a pulse width
modulation technique is contemplated as a prerogative in the perspective of
triggering a power device.
3.5
MULTIPLE PULSE MODULATION METHOD
The multiple pulse modulation (MPM) where several equidistant
pulses per half cycle are generated at the points of intersection of the carrier
and reference signal waves is explored as a firing mechanism suitable for
cycloconverters. A triangular carrier and a square reference wave are chosen
and the pulse width is allowed to change in accordance with the requirements
38
by varying the amplitude of the square wave. The Figure 3.3 shows the output
voltage waveform for MPM.
Figure 3.3 Output voltage waveform
The value of the actual output voltage (rms) in the cycloconverter
is given by,
V0
V0
2
1
3
Vm
2
2
Vm sin
2
td t
3
2
Vm sin
2
Vm 2 sin 2 td t
td t
2
sin 2
2
(3.9)
2
The Fourier constants and harmonic analysis of output voltage are
determined as follows.
f (x)
a0
2
T
2
an
a n cos
n 1
nx
3
4
nx
f (x) cos dx
T0
3
b n sin
nx
3
(3.10)
39
an
an
3
4Vm
6
Vm
3
0
(2 Vc )
2m
(k 1)
m
sin(
Vc
2m
(k 1)
m
1
an
( 1) j
nx
dx
3
sin x cos
Vm n 3
1
1
n 3
n
n
1)x sin(
1)x dx
3
3
cos
n
1
3
(k 1)
m
(2 Vc )
2m
cos
n
1
3
(k 1)
m
(2 Vc )
2m
cos
cos
n
1
3
n
1
3
(k 1)
m
(k 1)
m
Vc )
2m
Vc )
2m
(3.11)
bn
bn
bn
2Vm
3
sin x sin
0
nx
dx
3
(2 Vc )
2m
(k 1)
m
Vm
3
( 1) j
3
cos(
Vc
2m
(k 1)
m
1
Vm
1
sin
n 3
1
n 3
sin
n
n
1)x cos(
1)x dx
3
3
n
1
3
(k 1)
m
(2 Vc )
2m
n
1
3
(k 1)
m
(2 Vc )
2m
sin
sin
n
1
3
n
1
3
(k 1)
m
(k 1)
m
Vc )
2m
Vc )
2m
(3.12)
40
The third order Fourier constants and harmonic analysis of output
voltage are determined as follows.
b3
b3
a3
2Vm
3
Vm
3
0
(2 Vc )
2m
1 cos 2x dx
( 1) j
a3
3.6
Vm
3
sin x sin xdx
(k 1)
m
b3
a3
3
2Vm
3
1
(k 1)
m
Vc
2m
Vm
3
(1 Vc )
2(k 1)
cos
m
m
(Vc 1)
m
(3.13)
3
sin x cos xdx
0
(k 1)
m
( 1) j
m
sin
(2 Vc )
2m
sin 2xdx
(k 1)
m
1
Vc
2m
Vm
2(k 1)
sin
3
m
m
sin
(Vc 1)
m
(3.14)
SIMULATION
The power circuit of a cycloconverter for a single-phase output
with a single-phase input is shown in Figure 3.4. Basically it consists of two
back-to-back connected full converters (P-converter and N-converter).
A series of positive (rectified) input half cycles appears across the resistive
load when the P-converter is gated, while series of negative input half cycles
result when the N-converter is operated in an open blanking mode of
approach. The configuration sets the load frequency to be a fraction of the
input frequency.
41
Figure 3.4 Power circuit of single phase cycloconverter
The single-phase cycloconverter with different control strategies is
simulated using MATLAB. The circuit specifications are intuitively chosen to
acquire a target output voltage of close to 100V (rms) fundamental at one
third the input frequency across a 110
resistive load.
An intuitive investigation reveals that the viable number of samples
(m) lies in the range of eight and two for a modulation depth (M) that varies
from 0.1 and 0.636. The suitable value for m in the low output voltage range
is eight where the M varies from 0.1 to 0.18 while the same is two in the
higher output voltage range for which M is between 0.54 and 0.636.
Therefore, the study echoes the appropriate choice of the modulation depth
and the number of samples to land at the desired output voltage. The trail and
error procedure yields a modulation depth of 0.46 and 3 samples to yield the
specified target voltage in all the schemes. The output voltage obtained for
M=0.46 and m=3and their respective harmonic spectra are depicted in
Figures 3.5 to 3.10. It is to be noted that the performance of EAC is obtained
for the same number of samples as observed from Figures 3.9 and 3.10.
42
Figure 3.5 Output voltage and spectrum – CFAM
Figure 3.6 Output voltage and spectrum – CWCM
Figure 3.7 Output voltage and spectrum –HWSVFAM
Figure 3.8 Output voltage and spectrum –QWSVFAM
43
Figure 3.9 Output voltage and spectrum – MPM
Figure 3.10 Output voltage and spectrum – EAC
The entries in Table 3.1 contain the amplitudes of the target
fundamental and harmonic components 3, 5 and 7 in terms of percentage of
fundamental component for the same modulation depth and number of
samples to illustrate the lower value of THD for EAC. It is interesting to note
from Table 3.2 that EAC offers an exhilarating performance even when
compared with MPM scheme on a similar platform. The results re-incarnate
the creation of the new firing strategy so as to accomplish an improved
performance for cycloconverters.
44
Table 3.1
Comparison of lower order harmonics and THD using
existing phase angle controlled and proposed EAC methods
V3
V5
V7
CFAM
Vo1
(V)
104.2
46.56
80.37
64.24
THD
(%)
124.29
CWCM
104.3
46.6
80.39
64.33
124.16
HWSVFAM
104.5
45.48
71.45
50.18
119.71
QWSVFAM
104.4
6.01
8.58
77.74
99.49
EAC
104.2
1.45
12.18
17.25
75.38
Strategy
Table 3.2 Comparison of lower order harmonics and THD using MPM
and proposed EAC methods
Strategy
Vo1
(V)
V3
V5
V7
THD
(%)
MPM
105.0
39.11
44.96
14.88
114.58
EAC
104.2
1.45
12.18
17.25
75.38
The variation of amplitudes of harmonic orders over a specific
output voltage range for a modulation depth ranging from 0.44 to 0.52 with
three samples for the different control schemes seen in Figure 3.11 go to
highlight the supremacy of EAC.
45
Figure 3.11 Variation of harmonic order with THD (%) for m=3
46
Similarly the variation of harmonic orders with respect to THD
obtained by varying M from 0.1 to 0.18, with eight samples for the different
control schemes in the defined output voltage range is shown in Figure 3.12.
Figure 3.12 Variation of harmonic order with THD (%) for m=8
3.7
HARDWARE IMPLEMEMTATION
The scheme seen in Figure 3.4 is implemented through a suitable
prototype constructed with the specific purpose to operate in the same
horizon. A real time windows target using MATLAB is used to generate
switching pulses to the constructed prototype converter based on MOSFET
(IRF840) switches. The switching angles are calculated in off-line for
different operating points of the converters. The virtual reality toolbox
seamlessly integrates with real-time workshop targets. It supports simulations
that use code generated by real-time workshop and a third-party compiler on
the desktop computer. The virtual reality toolbox also supports code executed
in real time on external target computers. The schematic pulse generation
using the real time target of MATLAB is shown in Figure 3.13.
47
Figure 3.13 Schematic of real time workshop based pulse generation scheme
The systematic digital generation of EAC pulses corresponding to
the target output voltage of close to 100V(rms) along with the output voltage
waveform is shown in Figure 3.14.
Figure 3.14 Pulse pattern and output voltage
The fabricated hardware with the work bench is illustrated as a
photograph in Figure 3.15. It is tested in the same operating states and the
output voltage waveforms along with their harmonic spectra are drawn in
Figures 3.16 to 3.21 for the different control schemes. It is noteworthy to
observe that the experimental arrangement annotates more or less similar
performance thus bringing out the efficacy of the developed control
algorithm. The tabulated readings in Table 3.3 authenticate the findings and
served to validate the simulated response.
48
Figure 3.15 Cycloconverter prototype
Figure 3.16 Output voltage and harmonic spectrum –CFAM
Figure 3.17 Output voltage and harmonic spectrum -CWCM
49
Figure 3.18 Output voltage and harmonic spectrum – HWSVFAM
Figure 3.19 Output voltage and harmonic spectrum - QWSVFAM
Figure 3.20 Output voltage and harmonic spectrum – MPM
Figure 3.21 Output voltage and harmonic spectrum - EAC
50
Table 3.3 Comparison of existing and proposed control strategies
Simulated values
Strategies
Experimental values
CFAM
Vo1
THD Vo1
THD
V3 V 5 V 7
V3
V5
V7
(%)
(%)
(v)
(v)
104.2 46.56 80.37 64.24 124.29 107.8 50.7 86.1 72.0 122.7
CWCM
104.3 46.60 80.39 64.33 124.16 108.0 50
87 67.9 122.4
HWSVFAM 104.5 45.48 71.45 50.18 119.71 107.7 51.5 76 53.3 119.5
QWSVFAM 104.4 6.01 8.58 77.74 99.49 103.4 7.8 8.5 79.5 102.4
MPM
105.0 39.11 44.96 14.88 114.58 103.6 39.7 46.1 15 115.7
EAC
104.2 1.45 12.18 17.25 75.38 106.2 2.3 14.1 18.9 75.5
The constructed prototype hardware is also tested with the firing
pulses generated in the same way shown in Figure 3.15 for an R-L load with
R=110
and L= 5mH. The output voltage and output current waveforms
shown in Figure 3.22 prove that proposed EAC also works with R-L load.
(a)
(b)
Figure 3.22 (a) Output voltage and (b) output current waveforms for
RL load
51
3.8
SUMMARY
The concept of EAC has been implemented for a single phase
cycloconverter to drive home the benefits of extracting an entire range of
output voltage. It has been found to yield appreciable lowering of the THD
values almost through the entire operating range of output voltage. A
significant contribution has been to illustrate the superiority of the proposed
technique even over a PWM based methodology typically in the lower ranges
of the output voltage. The performance of the existing firing schemes has
been elucidated to highlight the attractive features of the developed algorithm
and portray its capability in drive applications where low speed operation is
required.