Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
1
Chapter 4DC to AC Conversion
(INVERTER)
• General concept• Single-phase inverter• Harmonics• Modulation• Three-phase inverter
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
2
DC to AC Converter (Inverter)
• DEFINITION: Converts DC to AC power by switching the DC input voltage (or current) in a pre-determined sequence so as to generate AC voltage (or current) output.
• General block diagram
IDC Iac
�
−
VDC Vac
�
−
• TYPICAL APPLICATIONS:– Un-interruptible power supply (UPS), Industrial
(induction motor) drives, Traction, HVDC
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
3
Simple square-wave inverter (1)
• To illustrate the concept of AC waveform generation
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Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
4
AC Waveform Generation
VDC
S1
S4
S3
+ vO −
VDC
S1
S4
S3
S2
+ vO −
VDC
vO
t1 t2
t
S1,S2 ON; S3,S4 OFF for t1 < t < t2
t2 t3
vO
-VDC
t
S3,S4 ON ; S1,S2 OFF for t2 < t < t3
S2
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
5
AC Waveforms
FUNDAMENTAL COMPONENT
3RD HARMONIC
5RD HARMONIC
πDCV4
Vdc
-Vdc
V1
31V
51V
INVERTER OUTPUT VOLTAGE
π 2π
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
6
Harmonics Filtering
• Output of the inverter is “chopped AC voltage with zero DC component”. It contain harmonics.
• An LC section low-pass filter is normally fitted at the inverter output to reduce the high frequency harmonics.
• In some applications such as UPS, “high purity” sine wave output is required. Good filtering is a must.
• In some applications such as AC motor drive, filtering is not required.
vO 1
+
−
L
CvO 2
(LOW PASS) FILTER
+
−
vO 1vO 2
BEFORE FILTERING AFTER FILTERING
INVERTER LOADDC SUPPLY
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
7
Variable Voltage Variable Frequency Capability
�
� t
Vdc1
Vdc2 Higher input voltageHigher frequency
Lower input voltageLower frequency
• Output voltage frequency can be varied by “period” of the square-wave pulse.
• Output voltage amplitude can be varied by varying the “magnitude” of the DC input voltage.
• Very useful: e.g. variable speed induction motor drive
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
8
Output voltage harmonics/ distortion
• Harmonics cause distortion on the output voltage.
• Lower order harmonics (3rd, 5th etc) are very difficult to filter, due to the filter size and high filter order. They can cause serious voltage distortion.
• Why need to consider harmonics?– Sinusoidal waveform quality must match TNB
supply. – “Power Quality” issue.– Harmonics may cause degradation of
equipment. Equipment need to be “de-rated”.
• Total Harmonic Distortion (THD) is a measure to determine the “quality” of a given waveform.
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
9
Total Harmonics Distortion (THD)
( )
( ) ( )
( )
frequency. harmonicat impedance theis
:THDCurrent
known, is vaveformfor the voltagerms theIf
....
voltage,harmonicth theis If :THD Voltage
,1
2
2,
,1
2
2,1
2
,1
2,2
2,3
2,2
,1
2
2,
n
n
nn
RMS
nRMSn
RMS
nRMSRMS
RMS
RMSRMSRMS
RMS
nRMSn
n
Z
ZV
I
I
I
THDi
V
VV
THDv
V
VVV
V
V
THDv
nV
=
=
−=
+++=
=
�
�
�
∞
=
∞
=
∞
=
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
10
Fourier Series
• Study of harmonics requires understanding of wave shapes. Fourier Series is a tool to analyse wave shapes.
( )
( )
( )
t
nbnaavf
dnvfb
dnvfa
dvfa
nnno
n
n
o
ωθ
θθ
θθπ
θθπ
θπ
π
π
π
=
++=
=
=
=
�
�
�
�
∞
= where
sincos21
)(
Fourier Inverse
term) sin"(" sin)(1
term) cos"(" cos)(1
term) DC"(" )(1
SeriesFourier
1
2
0
2
0
2
0
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
11
Harmonics of square-wave (1)
( ) ( )
( ) ( )���
�
���
�−=
=���
�
���
�−=
=���
�
���
�−+=
��
��
��
π
π
π
π
π
π
π
π
π
θθθθπ
θθθθπ
θθπ
2
0
2
0
2
0
sinsin
0coscos
01
dndnV
b
dndnV
a
dVdVa
dcn
dcn
dcdco
Vdc
-Vdc
θ�ω�π 2π
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
12
Harmonics of square wave (2)
( ) ( )[ ][ ]
[ ]
[ ]
π
π
π
ππ
πππ
ππππ
θθπ
ππ
π
nV
b
n
b
nn
nnV
nnnV
nnnnV
nnnV
b
dcn
n
dc
dc
dc
dcn
41cos odd, isn When
exist)not do harmonicseven i.e.(
01cos even, is When
)cos1(2
)cos1()cos1(
)cos2(cos)cos0(cos
coscos
Solving,
20
=
=
−=
=
−=
−+−=
−+−=
+−=
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
13
Spectra of square wave
1 3 5 7 9 11
NormalisedFundamental
3rd (0.33)
5th (0.2)
7th (0.14)9th (0.11)
11th (0.09)
1st
n
• Spectra (harmonics) characteristics:– Harmonic decreases with a factor of (1/n). – Even harmonics are absent– Nearest harmonics is the 3rd. If fundamental is
50Hz, then nearest harmonic is 150Hz.– Due to the small separation between the
fundamental an harmonics, output low-pass filter design can be very difficult.
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
14
Quasi-square wave (QSW)
( ) [ ]( ) ( )[ ]
( ) ( )
( )[ ]
( )[ ]παπ
απαπ
απαπαπαπαπ
απαπ
θπ
θθπ
απα
απ
α
nnnV
nnnnV
b
nnnnnn
nnn
nnnV
nnV
dnVb
a
dc
dcn
dc
dcdcn
n
cos1cos2
coscoscos2
coscossinsincoscoscoscos
:Expanding
coscos2
cos2
sin1
2
symmetry) wave-half to(due .0 that Note
−=
−=
=+=−=−
−−=
−=���
�
���
�=
=
−−
�
π π2
α α αVdc
-Vdc
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
15
Harmonics control
( )
( )
n
n
b
b
Note
Vb
nnV
b
b
o
dc
dcn
n
o
3
1
1
90
:if eliminated be will harmonic general,In waveform. thefrom eliminated is harmonic
thirdor the,0then ,30 if exampleFor
:nEliminatio Harmonics
, adjustingby controlled be alsocan Harmonics
� by varying controlled is ,, lfundamenta The:
cos4
:is lfundamenta theof amplitude ,particularIn
cos4
odd, isn If
,0 even, isn If
=
==
=
=
=
α
α
α
απ
απ
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
16
Example
degrees30 with case wavesquare-quasifor (c) and (b)Repeat
harmonics zero-non efirst thre theusingby THDi thec)harmonics zero-non efirst thre theusingby THDv theb)
formula. exact"" theusing THDv thea):Calculate series.in 10mHL and
10RR is load The 100V. is gelink volta DC The signals. wavesquareby fed isinverter phase single bridge-fullA
=
==
α
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17
Half-bridge inverter (1)
Vo
RL
+−
VC1
VC2
+
-
+
-S1
S2
Vdc
2Vdc
2Vdc−
S1 ONS2 OFF
S1 OFFS2 ON
t0G
• Also known as the “inverter leg”.
• Basic building block for full bridge, three phase and higher order inverters.
• G is the “centre point”.
• Both capacitors have the same value. Thus the DC link is equally “spilt” into two.
• The top and bottom switch has to be “complementary”, i.e. If the top switch is closed (on), the bottom must be off, and vice-versa.
Power Electronics and Drives (Version 3-2003):
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18
Shoot through fault and “Dead-time”
• In practical, a dead time as shown below is required to avoid “shoot-through” faults, i.e. short circuit across the DC rail.
• Dead time creates “low frequency envelope”. Low frequency harmonics emerged.
• This is the main source of distortion for high-quality sine wave inverter.
td td
"Dead time' = td
S1signal(gate)
S2signal(gate)
S1
S2
+
−−−−
Vdc
RL
G
"Shoot through fault" .Ishort is very large
Ishort
Power Electronics and Drives (Version 3-2003):
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19
Single-phase, full-bridge (1)• Full bridge (single phase) is built from two half-
bridge leg.
• The switching in the second leg is “delayed by 180 degrees” from the first leg.
S1
S4
S3
S2
+
-
G
+
2dcV
2dcV
-
2dcV
2dcV
dcV
2dcV−
2dcV−
dcV−
π
π
π
π2
π2
π2
tω
tω
tω
RGV
GRV '
oV
GRo VVVRG '−=
groumd" virtual" is G
LEG R LEG R'
R R'- oV+
dcV
+
-
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20
Three-phase inverter
• Each leg (Red, Yellow, Blue) is delayed by 120 degrees.
• A three-phase inverter with star connected load is shown below
ZYZR ZB
G R Y B
iR iYiB
ia ib
+Vdc
N
S1
S4 S6
S3 S5
S2
+
+
−−−−
−−−−
Vdc/2
Vdc/2
Power Electronics and Drives (Version 3-2003):
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21
Three phase inverter waveforms
13
2,4
23,54
35
4,6
41,56
51
2,6
61,32
Inverter PhaseVoltage
(or pole switchingwaveform)
VRG
2400
IntervalPositive device(s) on
Negative device(s) on
2VDC/3
VDC/3
-VDC/3
-2VDC/3
VDC
-VDC
VDC/2
-VDC/2
Quasi-square wave operation voltage waveforms
1200
VDC/2
VDC/2
-VDC/2
-VDC/2
VYG
VBG
lIne-to -ineVoltage
VRY
Six-stepWaveform
VRN
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22
Pulse Width Modulation (PWM)
Modulating Waveform Carrier waveform
1M1+
1−
0
2dcV
2dcV−
00t 1t 2t 3t 4t 5t
• Triangulation method (Natural sampling)– Amplitudes of the triangular wave (carrier) and
sine wave (modulating) are compared to obtain PWM waveform. Simple analogue comparator can be used.
– Basically an analogue method. Its digital version, known as REGULAR sampling is widely used in industry.
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
23
PWM types
• Natural (sinusoidal) sampling (as shown on previous slide)– Problems with analogue circuitry, e.g. Drift,
sensitivity etc.
• Regular sampling– simplified version of natural sampling that
results in simple digital implementation
• Optimised PWM– PWM waveform are constructed based on
certain performance criteria, e.g. THD.
• Harmonic elimination/minimisation PWM– PWM waveforms are constructed to eliminate
some undesirable harmonics from the output waveform spectra.
– Highly mathematical in nature
• Space-vector modulation (SVM)– A simple technique based on volt-second that is
normally used with three-phase inverter motor-drive
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Dr. Zainal Salam UTM-JB
24
Modulation Index, Ratio
waveformmodulating theofFrequency veformcarrier wa theofFrequency
M
)(MRatio) (Frequency Ratio Modulation
veformcarrier wa theof Amplitude waveformmodulating theof Amplitude
M
:MDepth)n (ModulatioIndex Modulation
R
R
I
I
==
==
=
=
p
p
Modulating Waveform Carrier waveform
1M1+
1−
0
2dcV
2dcV−
00t 1t 2t 3t 4t 5t
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25
( )
(1,2,3...)integer an is and
signal modulating theoffrequency theis where
M:at locatednormally are harmonics The
spectra. in the harmonics of(location)incident thedetermines ratiodulation M
ly.respective voltage,(DC)input and voltage
output theof lfundamenta are , where
M
1, M0 If
component lfundamenta voltageoutput thesdeterrmineIndex Modulation
R
1
I1
I
k
f
fkf
o
VV
VV
m
m
in
in
=
−−−−−−−−−−−−−−−−−−−−−−−−−−−−
=<<
Modulation Index, Ratio
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26
Regular sampling
Regular sampling PWM
Sinusoidal modulatingwaveform, vm(t)
Carrier, vc(t)t1 t2
t'1 t'2
t
t
ππ2
)(tvs
pwmv
Regular sampling waveform,
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27
Asymmetric and symmetric regular sampling
T
samplepoint
tM mωsin11+
1−
4T
43T
45T
4π
2dcV
2dcV
−
0t 1t 2t 3tt
asymmetric sampling
symmetricsampling
t
Generating of PWM waveform regular sampling
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28
Bipolar Switching
Modulating Waveform Carrier waveform
1M1+
1−
0
2dcV
2dcV−
00t 1t 2t 3t 4t 5t
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29
Unipolar switching
Unipolar switching scheme
A BCarrier waveform
(a)
(b)
(c)
(d)
1S
3S
pwmV
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30
Bipolar PWM switching: Pulse-width characterization
k1δk2δ
kα
∆4∆=δ
π π2
carrierwaveform
modulatingwaveform
pulse
kth
π π2
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31
The kth Pulse
pulse PWMkth The
∆
0δ 0δ 0δ 0δ
k1δk2δ
2dcV+
kα2
dcV+
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32
Determination of switching angles for kth PWM pulse (1)
22
11
second,- volt theEquating
ps
ps
AA
AA
=
=
v Vmsin θ( )
Ap2Ap1
2dcV+
2dcV
−
AS2
AS1
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33
PWM Switching angles (2)
( ) ( )
( )( )
( ) ( )
( )( )
[ ]
)sin(2Similarly,
)sin(sin2
cos)2cos(sin
sinusoid, by the supplied second- voltThe
222
half, second for theSimilarly
222
:asgiven is pulse PWM theof cycle halffirst theduring second-Volt The
2
21
2
222
1
111
okmos
okom
kokmms
okdc
kodc
kdc
p
okdc
kodc
kdc
p
VA
V
VdVA
V
VVA
V
VVA
k
ok
δαδ
δαδ
αδαθθ
δδ
δδδ
δδ
δδδ
α
δα
+=
−=
−−==
−=
−�
� �
�−�
� �
�=
−=
−�
� �
�−�
� �
�=
�−
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
34
Switching angles (3)
( )( )
( ) ( )
[ ])sin(1:bygiven is waveformPWM theof
cycle halffirst for the width pulse theThus,
modulation asknown is 2
Ratio, Modulation the,definitionBy
sin(2
)sin(2pulse, PWM of cycle halffirst thefor the Hence,
;strategy, modulation thederive To
)sin(2
)sin(2
, sin
angle smallFor
1
1
1
22
11
2
1
okIok
dc
mI
okodc
mok
okmookdc
sp
sp
okmos
okmos
oo
o
M
)(VV
M
VV
VV
AA
AA
VA
VA
δαδδ
δαδδδ
δαδδδ
δαδδαδ
δδδ
−+=
=
−=−
−=−
=
=
−=
−=
→
Power Electronics and Drives (Version 3-2003):
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35
PWM switching angles (4)
[ ]
[ ]kIok
kk
okIok
kk
M
M
αδδ
δδδ
δδ
δα
δαδδ
δα
sin1 Hence
,Modulation SymmetricFor
different. are andi.e ,Modulation Asymmetricfor validisequation above The
: angle edge trailing theAnd
)sin(1: waveformPWM of cycle half second
theof width pulse method,similar Using
:is pulsekth the
of angle switching edge leading theThus
k 2k 1k
2k 1k
2
2
1
+=
==
+
++=
−
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
36
Example• For the PWM shown below, calculate the switching
angles pulses no. 2.
V5.1V2
π π2
1 2 3 4 5 6 7 8 9
t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12t13
t14t15
t16t17
t18 π2π
1α
carrierwaveform
modulatingwaveform
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
37
Harmonics of bipolar PWM
��
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��
���
�
� �
�−+
��
���
��
���
�
� �
�+
��
���
��
���
�
� �
�−=
�
�
�
�=
�
�
�
�
+
+
+
−
−
−
ok
kk
kk
kk
kk
ok
dnV
dnV
dnV
dnvfb
dc
dc
dc
T
nk
δα
δα
δα
δα
δα
δα
θθπ
θθπ
θθπ
θθπ
2
2
0
2
2
1
1
sin2
2
sin2
2
sin2
2
sin)(1
2
:as computed becan pulse PWM (kth)
each ofcontent harmonic
symmetry, wavehalf is waveform
PWM theAssuming∆
0δ 0δ 0δ 0δ
k1δk2δ
2dcV+
kα2
dcV+
Power Electronics and Drives (Version 3-2003):
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38
Harmonics of Bipolar PWM
{
}
[]
equation. thisofn computatio theshows slideNext
:i.e. period, oneover pulses
for the of sum isthe waveformPWMfor the coefficentFourier ly.Theproductive
simplified becannot equation This
2coscos2
)2(cos)(cos2
Yeilding,
)2(cos)(cos
)(cos)(cos
)(cos)2(cos
: toreduced becan Which
1
11
2
12
1
�=
=
+
−−−=
+−++−−++
−−−−=
p
knkn
nk
ok
kkkkdc
nk
okkk
kkkk
kkokdc
nk
bb
pb
nn
nnnV
b
nn
nn
nnnV
b
δα
αδαπ
δαδαδαδα
δαδαπ
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39
PWM Spectra
p p2 p3 p40.1=IM
8.0=IM
6.0=IM
4.0=IM
2.0=IM
Amplitude
Fundamental
0
2.0
4.0
6.0
8.0
0.1
NORMALISED HARMONIC AMPLITUDES FORSINUSOIDAL PULSE-WITDH MODULATION
ModulationIndex
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40
PWM spectra observations
• The harmonics appear in “clusters” at multiple of the carrier frequencies .
• Main harmonics located at : f = kp (fm); k=1,2,3....
where fm is the frequency of the modulation (sine) waveform.
• There also exist “side-bands” around the main harmonic frequencies.
• Amplitude of the fundamental is proportional to the modulation index. The relation ship is given as:
V1= MIVin
• The amplitude of the harmonic changes with MI. Its incidence (location on spectra) is not.
• When p>10, or so, the harmonics can be normalised. For lower values of p, the side-bands clusters overlap-normalised results no longer apply.
Power Electronics and Drives (Version 3-2003):
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41
Tabulated Bipolar PWM Harmonics
n MI
0.2 0.4 0.6 0.8 1.0
1 0.2 0.4 0.6 0.8 1.0
MR 1.242 1.15 1.006 0.818 0.601
MR +2 0.016 0.061 0.131 0.220 0.318
MR +4 0.018
2MR +1 0.190 0.326 0.370 0.314 0.181
2MR +3 0.024 0.071 0.139 0.212
2MR +5 0.013 0.033
3MR 0.335 0.123 0.083 0.171 0.113
3MR +2 0.044 0.139 0.203 0.716 0.062
3MR +4 0.012 0.047 0.104 0.157
3MR +6 0.016 0.044
4MR +1 0.163 0.157 0.008 0.105 0.068
4MR +3 0.012 0.070 0.132 0.115 0.009
4MR+5 0.034 0.084 0.119 4MR +7 0.017 0.050
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42
Three-phase harmonics
• For three-phase inverters, there is significant advantage if MR is chosen to be:
– Odd: All even harmonic will be eliminated from the pole-switching waveform.
– triplens (multiple of three (e.g. 3,9,15,21, 27..):All triplens harmonics will be eliminated from the line-to-line output voltage.
• By observing the waveform, it can be seen that with odd MR, the line-to-line voltage shape looks more “sinusoidal”.
• As can be noted from the spectra, the phase voltage amplitude is 0.8 (normalised). This is because the modulation index is 0.8. The line voltage amplitude is square root three of phase voltage due to the three-phase relationship
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43
Effect of odd and “triplens”
2dcV
2dcV
−
2dcV
2dcV
−
2dcV
−
2dcV
−
2dcV
2dcV
dcV
dcV
dcV−
dcV−
π π2
RGV
RGV
RYV
RYV
YGV
YGV
6.0,8 == Mp
6.0,9 == MpILLUSTRATION OF BENEFITS OF USING A FREQUENCY RATIOTHAT IS A MULTIPLE OF THREE IN A THREE PHASE INVERTER
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
44
Spectra: effect of “triplens”
0
2.0
4.0
6.0
8.0
0.1
2.1
4.1
6.1
8.1
Amplitude
voltage)line to(Line 38.0
Fundamental
41 4339
3745
47
2319
21 63
6159
57
6567
69 7779
8183 85
8789
91
19 2343
4741
3761
5965
6783
7985
89
COMPARISON OF INVERTER PHASE VOLTAGE (A) & INVERTER LINE VOLTAGE(B) HARMONIC (P=21, M=0.8)
A
B
Harmonic Order
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
45
Comments on PWM scheme
• It is desirable to have MR as large as possible.
• This will push the harmonic at higher frequencies on the spectrum. Thus filtering requirement is reduced.
• Although the voltage THD improvement is not significant, but the current THD will improve greatly because the load normally has some current filtering effect.
• However, higher MR has side effects:– Higher switching frequency: More losses.– Pulse width may be too small to be constructed.
“Pulse dropping” may be required.
Power Electronics and Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
46
Example
Harmonic number
Amplitude (pole switching waveform)
Amplitude (line-to line voltage)
1 1
19 0.3
21 0.8
23 0.3
37 0.1
39 0.2
41 0.25
43 0.25
45 0.2
47 0.1
57 0.05
59 0.1
61 0.15
63 0.2
65 0.15
67 0.1
69 0.05
The amplitudes of the pole switching waveform harmonics of the redphase of a three-phase inverter is shown in Table below. The inverter uses a symmetric regular sampling PWM scheme. The carrier frequencyis 1050Hz and the modulating frequency is 50Hz. The modulationindex is 0.8. Calculate the harmonic amplitudes of the line-to-voltage(i.e. red to blue phase) and complete the table.