Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
1
Chapter 4
DC to AC Conversion
(INVERTER)
• General concept
• Basic principles/concepts
• Single-phase inverter
– Square wave
– Notching
– PWM
• Harmonics
• Modulation
• Three-phase inverter
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
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.
• TYPICAL APPLICATIONS:
– Un-interruptible power supply (UPS), Industrial
(induction motor) drives, Traction, HVDC
• General block diagram
I
DC
I
ac
+
−
V
DC
V
ac
+
−
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
3
Types of inverter
• Voltage Source Inverter (VSI)
• Current Source Inverter (CSI)
"DC LINK"
I
ac
+
−
V
DC
Load Voltage
+
−
L
I
LOAD
Load Current
I
DC
+
−
V
DC
C
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
4
Voltage source inverter (VSI)
with variable DC link
DC LINK
+
-
V
s
V
o
+
-
C
+
-
V
in
CHOPPER
(Variable DC output)
INVERTER
(Switch are turned ON/OFF
with square-wave patterns)
• DC link voltage is varied by a DC-to DC converter
or controlled rectifier.
• Generate “square wave” output voltage.
• Output voltage amplitude is varied as DC link is
varied.
•
Frequency of output voltage is varied by changing
the frequency of the square wave pulses.
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
5
Variable DC link inverter (2)
• Advantages:
– simple waveform generation
– Reliable
• Disadvantages:
– Extra conversion stage
– Poor harmonics
T
1
T
2
t
V
dc1
V
dc2
Higher input voltage
Higher frequency
Lower input voltage
Lower frequency
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
6
VSI with fixed DC link
INVERTER
+
−
V
in
(fixed)
V
o
+
−
C
Switch turned ON and OFF
with PWM pattern
• DC voltage is held constant.
• Output voltage amplitude and frequency
are varied simultaneously using PWM
technique.
• Good harmonic control, but at the expense
of complex waveform generation
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
7
Operation of simple square-
wave inverter (1)
• To illustrate the concept of AC waveform
generation
V
DC
T1
T4
T3
T2
+ V
O
-
D1
D2
D3
D4
SQUARE-WAVE
INVERTERS
S1
S3
S2
S4
EQUAVALENT
CIRCUIT
I
O
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
8
Operation of simple square-
wave inverter (2)
V
DC
S1
S4
S3
+ v
O
−
V
DC
S1
S4
S3
S2
+ v
O
−
V
DC
v
O
t
1
t
2
t
S1,S2 ON; S3,S4 OFF
for t
1
< t < t
2
t
2
t
3
v
O
-V
DC
t
S3,S4 ON ; S1,S2 OFF
for t
2
< t < t
3
S2
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
9
Waveforms and harmonics of
square-wave inverter
FUNDAMENTAL
3
RD
HARMONIC
5
RD
HARMONIC
π
DC
V
4
V
dc
-V
dc
V
1
3
1
V
5
1
V
INVERTER
OUTPUT
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
10
Filtering
• Output of the inverter is “chopped AC
voltage with zero DC component”.In some
applications such as UPS, “high purity” sine
wave output is required.
• An LC section low-pass filter is normally
fitted at the inverter output to reduce the
high frequency harmonics.
• In some applications such as AC motor
drive, filtering is not required.
v
O 1
+
−
LOAD
L
C
v
O 2
(LOW PASS) FILTER
+
−
v
O 1
v
O 2
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
11
Notes on low-pass filters
• In square wave inverters, maximum output voltage
is achievable. However there in NO control in
harmonics and output voltage magnitude.
• The harmonics are always at three, five, seven etc
times the fundamental frequency.
• Hence the cut-off frequency of the low pass filter is
somewhat fixed. The filter size is dictated by the
VA ratings of the inverter.
• To reduce filter size, the PWM switching scheme
can be utilised.
•
In this technique, the harmonics are “pushed” to
higher frequencies. Thus the cut-off frequency of
the filter is increased. Hence the filter components
(I.e. L and C) sizes are reduced.
• The trade off for this flexibility is complexity in
the switching waveforms.
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
12
“Notching”of square wave
Vdc
Vdc
−
Vdc
Vdc
−
Notched Square Wave
Fundamental Component
• Notching results in controllable output
voltage magnitude (compare Figures
above).
• Limited degree of harmonics control is
possible
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
13
Pulse-width modulation
(PWM)
• A better square wave notching is shown
below - this is known as PWM technique.
• Both amplitude and frequency can be
controlled independently. Very flexible.
1
1
pwm waveform
desired
sinusoid
SINUSOIDAL PULSE-WITDH MODULATED
APPROXIMATION TO SINE WAVE
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
14
PWM- output voltage and
frequency control
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
15
Output voltage harmonics
• Why need to consider harmonics?
– 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.
• DEFINITION of THD (voltage)
(
)
(
)
(
)
(
)
frequency.
harmonic
at
impedance
the
is
:
current
harmonic
with
voltage
harmonic
the
replacing
by
obtained
be
can
THD
Current
number.
harmonics
the
is
where
,
1
2
2
,
,
1
2
2
,
1
2
,
1
2
2
,
n
n
n
n
RMS
n
RMS
n
RMS
n
RMS
RMS
RMS
n
RMS
n
Z
Z
V
I
I
I
THDi
n
V
V
V
V
V
THDv
=
=
−
=
=
∑
∑
∑
∞
=
∞
=
∞
=
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
16
Fourier Series
• Study of harmonics requires understanding
of wave shapes. Fourier Series is a tool to
analyse wave shapes.
( )
( )
(
)
t
n
b
n
a
a
v
f
d
n
v
f
b
d
n
v
f
a
d
v
f
a
n
n
n
o
n
n
o
ω
θ
θ
θ
θ
θ
π
θ
θ
π
θ
π
π
π
π
=
+
+
=
=
=
=
∑
∫
∫
∫
∞
=
where
sin
cos
2
1
)
(
Fourier
Inverse
sin
)
(
1
cos
)
(
1
)
(
1
Series
Fourier
1
2
0
2
0
2
0
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
17
Harmonics of square-wave (1)
V
dc
-V
dc
θ=ωt
π
2π
( )
( )
( )
( )
−
=
=
−
=
=
−
+
=
∫
∫
∫
∫
∫
∫
π
π
π
π
π
π
π
π
π
θ
θ
θ
θ
π
θ
θ
θ
θ
π
θ
θ
π
2
0
2
0
2
0
sin
sin
0
cos
cos
0
1
d
n
d
n
V
b
d
n
d
n
V
a
d
V
d
V
a
dc
n
dc
n
dc
dc
o
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
18
Harmonics of square wave (2)
( )
( )
[
]
[
]
[
]
[
]
π
π
π
π
π
π
π
π
π
π
π
π
θ
θ
π
π
π
π
n
V
b
n
b
n
n
n
V
n
n
n
V
n
n
n
n
V
n
n
n
V
b
dc
n
n
dc
dc
dc
dc
n
4
1
cos
odd,
is
n
when
0
1
cos
even,
is
n
when
)
cos
1
(
2
)
cos
1
(
)
cos
1
(
)
cos
2
(cos
)
cos
0
(cos
cos
cos
Solving,
2
0
=
=
−
=
=
−
=
−
+
−
=
−
+
−
=
+
−
=
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
19
Spectra of square wave
1
3
5
7
9
11
Normalised
Fundamental
3rd (0.33)
5th (0.2)
7th (0.14)
9th (0.11)
11th (0.09)
1st
n
• Spectra (harmonics) characteristics:
– Harmonic decreases as n increases. It 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 quite difficult.
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
20
Quasi-square wave (QSW)
( )
[
]
( )
(
)
[
]
(
)
(
)
α
π
α
π
α
π
α
π
α
π
α
π
α
π
θ
π
θ
θ
π
α
π
α
α
π
α
n
n
n
n
n
n
n
n
n
n
n
n
V
n
n
V
d
n
V
b
a
dc
dc
dc
n
n
cos
cos
sin
sin
cos
cos
cos
cos
Expanding,
cos
cos
2
cos
2
sin
1
2
symmetry,
wave
-
half
to
Due
.
0
that
Note
=
+
=
−
=
−
−
−
=
−
=
=
=
−
−
∫
π
π
2
α
α
α
V
dc
-V
dc
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
21
Harmonics control
( )
[
]
( )
[
]
( )
( )
n
n
b
b
V
b
n
n
V
b
b
n
n
n
V
n
n
n
n
V
b
o
dc
dc
n
n
dc
dc
n
o
3
1
1
90
:
if
eliminated
be
will
harmonic
general,
In
waveform.
the
from
eliminated
is
harmonic
third
or the
,
0
then
,
30
if
example
For
,
adjusting
by
controlled
be
also
can
Harmonics
α
by varying
controlled
is
,
,
l
fundamenta
The
cos
4
:
is
l
fundamenta
the
of
amplitude
,
particular
In
cos
4
odd,
is
n
If
,
0
even,
is
n
If
cos
1
cos
2
cos
cos
cos
2
=
=
=
=
=
⇒
=
⇒
−
=
−
=
⇒
α
α
α
α
π
α
π
π
α
π
α
π
α
π
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
22
Example
degrees
30
with
case
wave
square
-
quasi
for
(c)
and
(b)
Repeat
harmonics
zero
-
non
e
first thre
the
using
by
THDi
the
c)
harmonics
zero
-
non
e
first thre
the
using
by
THDv
the
b)
formula.
exact"
"
the
using
THDv
the
a)
:
Calculate
series.
in
10mH
L
and
10R
R
is
load
The
100V.
is
ge
link volta
DC
The
signals.
wave
square
by
fed
is
inverter
phase
single
bridge
-
full
A
=
=
=
α
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
23
Half-bridge inverter (1)
V
o
R
L
+
−
V
C1
V
C2
+
-
+
-
S
1
S
2
V
dc
2
Vdc
2
Vdc
−
S1 ON
S2 OFF
S1 OFF
S2 ON
t
0
G
• 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.
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
24
Half-bridge inverter (2)
• 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.
• In practical, a dead time as shown below is
required to avoid “shoot-through” faults.
t
d
t
d
"Dead time' = t
d
S
1
signal
(gate)
S
2
signal
(gate)
S1
S2
+
−
V
dc
R
L
G
"Shoot through fault" .
I
short
is very large
I
short
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
25
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
+
2
dc
V
2
dc
V
-
2
dc
V
2
dc
V
dc
V
2
dc
V
−
2
dc
V
−
dc
V
−
π
π
π
π
2
π
2
π
2
t
ω
t
ω
t
ω
RG
V
G
R
V
'
o
V
G
R
o
V
V
V
RG
'
−
=
groumd"
virtual
"
is
G
LEG R
LEG R'
R
R'
-
o
V
+
dc
V
+
-
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
26
Three-phase inverter
• Each leg (Red, Yellow, Blue) is delayed by
120 degrees.
• A three-phase inverter with star connected
load is shown below
Z
Y
Z
R
Z
B
G
R
Y
B
i
R
i
Y
i
B
i
a
i
b
+V
dc
N
S1
S4
S6
S3
S5
S2
+
+
−
−
V
dc
/2
V
dc
/2
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
27
Square-wave inverter
waveforms
1
3
2,4
2
3,5
4
3
5
4,6
4
1,5
6
5
1
2,6
6
1,3
2
V
AD
V
B0
V
C0
V
AB
V
APH
(a) Three phase pole switching waveforms
(b) Line voltage waveform
(c) Phase voltage waveform (six-step)
60
0
120
0
Interval
Positive device(s) on
Negative devise(s) on
2V
DC
/3
V
DC
/3
-V
DC
/3
-2V
DC
/3
V
DC
-V
DC
V
DC
/2
-V
DC
/2
t
t
t
t
t
Quasi-square wave operation voltage waveforms
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
28
Three-phase inverter
waveform relationship
• V
RG
, V
YG
, V
BG
are known as “pole
switching waveform” or “inverter phase
voltage”.
• V
RY
, V
RB
, V
YB
are known as “line to line
voltage” or simply “line voltage”.
• For a three-phase star-connected load, the
load phase voltage with respect to the “N”
(star-point) potential is known as V
RN
,V
YN
,
V
BN
. It is also popularly termed as “six-
step” waveform
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
29
MODULATION: Pulse Width
Modulation (PWM)
Modulating Waveform
Carrier waveform
1
M
1
+
1
−
0
2
dc
V
2
dc
V
−
0
0
t
1
t
2
t
3
t
4
t
5
t
• 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 2): Dr.
Zainal Salam, 2002
30
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
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
31
Natural/Regular sampling
( )
(1,2,3...)
integer
an
is
and
signal
modulating
the
of
frequency
the
is
where
M
:
at
located
normally
are
harmonics
The
.
frequency"
harmonic
"
the
to
related
is
M
waveform
modulating
the
of
Frequency
veform
carrier wa
the
of
Frequency
M
)
(
M
RATIO
MODULATION
ly.
respective
voltage,
(DC)
input
and
voltage
output
the
of
l
fundamenta
are
,
where
M
:
holds
ip
relationsh
linear
the
1,
M
0
If
versa.
vice
and
high
is
output
wave
sine
the
then
high,
is
M
If
magnitude.
tage
output vol
wave)
(sine
l
fundamenta
the
to
related
is
M
veform
carrier wa
the
of
Amplitude
waveform
modulating
the
of
Amplitude
M
:
M
INDEX
MODULATION
R
R
R
R
1
I
1
I
I
I
I
I
k
f
f
k
f
p
p
V
V
V
V
m
m
in
in
=
=
=
=
=
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
=
<
<
=
=
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
32
Asymmetric and symmetric
regular sampling
T
sample
point
t
M
m
ω
sin
1
1
+
1
−
4
T
4
3T
4
5T
4
π
2
dc
V
2
dc
V
−
0
t
1
t
2
t
3
t
t
asymmetric
sampling
symmetric
sampling
t
Generating of PWM waveform regular sampling
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
33
Bipolar and unipolar PWM
switching scheme
• In many books, the term “bipolar” and
“unipolar” PWM switching are often
mentioned.
• The difference is in the way the sinusoidal
(modulating) waveform is compared with
the triangular.
• In general, unipolar switching scheme
produces better harmonics. But it is more
difficult to implement.
• In this class only bipolar PWM is
considered.
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
34
Bipolar PWM switching
k
1
δ
k
2
δ
k
α
∆
4
∆
=
δ
π
π
2
carrier
waveform
modulating
waveform
pulse
kth
π
π
2
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
35
Pulse width relationships
k
1
δ
k
2
δ
k
α
∆
4
∆
=
δ
π
π
2
carrier
waveform
modulating
waveform
pulse
kth
π
π
2
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
36
Characterisation of PWM
pulses for bipolar switching
pulse
PWM
kth
The
∆
0
δ
0
δ
0
δ
0
δ
k
1
δ
k
2
δ
2
S
V
+
2
S
V
−
k
α
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
37
Determination of switching
angles for kth PWM pulse (1)
v
Vmsin θ
( )
A
p2
A
p1
2
dc
V
+
2
dc
V
−
A
S2
A
S1
2
2
1
1
second,
-
volt
the
Equating
p
s
p
s
A
A
A
A
=
=
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
38
PWM Switching angles (2)
[
]
)
sin(
sin
2
cos
)
2
cos(
sin
sinusoid,
by the
supplied
second
-
volt
The
where
;
2
Similarly,
where
2
2
2
)
2
(
2
:
as
given
is
pulse
PWM
the
of
cycle
half
each
during
voltage
average
The
2
1
2
2
2
2
1
1
1
1
1
1
1
o
k
o
m
k
o
k
m
m
s
o
o
k
k
dc
k
k
o
o
k
k
s
k
o
o
k
dc
o
k
o
k
dc
k
V
V
d
V
A
V
V
V
V
V
V
k
o
k
δ
α
δ
α
δ
α
θ
θ
δ
δ
δ
β
β
δ
δ
δ
β
β
δ
δ
δ
δ
δ
δ
δ
α
δ
α
−
=
−
−
=
=
−
=
=
−
=
=
−
=
−
−
=
∫
−
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
39
Switching angles (3)
)
sin(
)
2
(
)
sin(
2
2
2
edge
leading
for the
Hence,
;
strategy,
modulation
the
derive
To
2
2
;
2
2
,
waveforms
PWM
the
of
seconds
-
volt
The
)
sin(
2
Similarly,
)
sin(
2
,
small
for
sin
Since,
1
1
2
2
1
1
21
2
1
1
2
1
o
k
dc
m
k
o
k
m
o
o
dc
k
s
p
s
p
o
dc
k
p
o
dc
k
p
o
k
m
o
s
o
k
m
o
s
o
o
o
V
V
V
V
A
A
A
A
V
A
V
A
V
A
V
A
δ
α
β
δ
α
δ
δ
β
δ
β
δ
β
δ
α
δ
δ
α
δ
δ
δ
δ
−
=
⇒
−
=
=
=
=
=
+
=
−
=
→
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
40
PWM switching angles (4)
[
]
[
]
)
sin(
1
and
)
sin(
1
width,
-
pulse
for the
solve
to
ng
Substituti
)
sin(
:
derived
be
can
edge
trailing
the
method,
similar
Using
)
sin(
Thus,
1.
to
0
from
It varies
depth.
or
index
modulation
as
known
is
2
ratio,
voltage
The
2
1
1
1
2
1
o
k
I
o
k
o
k
I
o
k
o
o
k
k
o
k
I
k
o
k
I
k
dc
m
I
M
M
M
M
)
(V
V
M
δ
α
δ
δ
δ
α
δ
δ
δ
δ
δ
β
δ
α
β
δ
α
β
+
+
=
−
+
=
⇒
−
=
−
=
−
=
=
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
41
PWM Pulse width
[
]
k
I
o
k
k
k
k
k
M
α
δ
δ
δ
δ
δ
δ
δ
δ
α
δ
α
sin
1
,
Modulation
Symmetric
For
different.
are
and
i.e
,
Modulation
Asymmetric
for
valid
is
equation
above
The
:
edge
Trailing
:
edge
Leading
:
is
pulse
kth
the
of
angles
switching
the
Thus
k
2k
1k
2k
1k
1
1
+
=
⇒
=
=
+
−
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
42
Example
• For the PWM shown below, calculate the switching
angles for all the pulses.
V
5
.
1
V
2
π
π
2
1
2
3
4
5
6
7
8
9
t1
t2
t3 t4 t5 t6 t7 t8 t9 t10 t11 t12
t13
t14
t15
t16
t17
t18 π
2
π
1
α
carrier
waveform
modulating
waveform
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
43
Harmonics of bipolar PWM
{
}
)
2
(
cos
)
(
cos
)
(
cos
)
(
cos
)
(
cos
)
2
(
cos
:
to
reduced
be
can
Which
sin
2
2
sin
2
2
sin
2
2
sin
)
(
1
2
:
as
computed
be
can
pulse
PWM
(kth)
each
of
content
harmonic
symmetry,
wave
-
half
is
waveform
PWM
the
Assuming
2
1
2
1
2
2
0
2
2
1
1
o
k
k
k
k
k
k
k
k
k
o
k
dc
nk
dc
dc
dc
T
nk
n
n
n
n
n
n
n
V
b
d
n
V
d
n
V
d
n
V
d
n
v
f
b
o
k
k
k
k
k
k
k
k
k
o
k
δ
α
δ
α
δ
α
δ
α
δ
α
δ
α
π
θ
θ
π
θ
θ
π
θ
θ
π
θ
θ
π
δ
α
δ
α
δ
α
δ
α
δ
α
δ
α
+
−
+
+
−
−
+
+
−
−
−
−
=
−
+
+
−
=
=
∫
∫
∫
∫
+
+
+
−
−
−
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
44
Harmonics of PWM
[
]
equation.
this
of
n
computatio
the
shows
page
next
on the
slide
The
:
i.e.
period,
one
over
pulses
for the
of
sum
isthe
waveform
PWM
for the
coefficent
Fourier
ly.The
productive
simplified
be
cannot
equation
This
2
cos
cos
2
)
2
(
cos
)
(
cos
2
Yeilding,
1
1
1
∑
=
=
+
−
−
−
=
p
k
nk
n
nk
o
k
k
k
k
k
dc
nk
b
b
p
b
n
n
n
n
n
V
b
δ
α
α
δ
α
π
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
45
PWM Spectra
p
p
2
p
3
p
4
0
.
1
=
M
8
.
0
=
M
6
.
0
=
M
4
.
0
=
M
2
.
0
=
M
Amplitude
Fundamental
0
2
.
0
4
.
0
6
.
0
8
.
0
0
.
1
NORMALISED HARMONIC AMPLITUDES FOR
SINUSOIDAL PULSE-WITDH MODULATION
Depth of
Modulation
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
46
PWM spectra observations
• The amplitude of the fundamental decreases or
increases linearly in proportion to the depth of
modulation (modulation index). The relation ship is
given as: V
1
= M
I
V
in
• The harmonics appear in “clusters” with main
components at frequencies of :
f = kp (f
m
);
k=1,2,3....
where f
m
is the frequency of the modulation (sine)
waveform. This also equal to the multiple of the
carrier frequencies. There also exist “side-bands”
around the main harmonic frequencies.
• The amplitude of the harmonic changes with M
I
. Its
incidence (location on spectra) is not.
• When p>10, or so, the harmonics can be normalised
as shown in the Figure. For lower values of p, the
side-bands clusters overlap, and the normalised
results no longer apply.
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
47
Bipolar PWM Harmonics
h
M
I
0.2
0.4
0.6
0.8
1.0
1
0.2
0.4
0.6
0.8
1.0
M
R
1.242
1.15
1.006
0.818
0.601
M
R
+2
0.016
0.061
0.131
0.220
0.318
M
R
+4
0.018
2M
R
+1
0.190
0.326
0.370
0.314
0.181
2M
R
+3
0.024
0.071
0.139
0.212
2M
R
+5
0.013
0.033
3M
R
0.335
0.123
0.083
0.171
0.113
3M
R
+2
0.044
0.139
0.203
0.716
0.062
3M
R
+4
0.012
0.047
0.104
0.157
3M
R
+6
0.016
0.044
4M
R
+1
0.163
0.157
0.008
0.105
0.068
4M
R
+3
0.012
0.070
0.132
0.115
0.009
4M
R
+5
0.034
0.084
0.119
4M
R
+7
0.017
0.050
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
48
Bipolar PWM harmonics
calculation example
( )
harmonics.
dominant
the
of
some
and
voltage
frequency
-
l
fundamenta
the
of
values
the
Calculate
47Hz.
is
lfrequency
fundamenta
The
39.
M
0.8,
M
100V,
V
inverter,
PWM
phase
single
bridge
-
full
In the
:
Example
M
of
function
a
as
2
ˆ
:
from
computed
are
harmonics
The
2
PWM,
bipolar
phase
-
single
bridge
full
for
:
Note
R
I
DC
I
'
,
=
=
=
=
−
=
=
DC
n
RG
RG
G
R
RG
RR
o
V
V
v
v
v
v
v
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
49
Three-phase harmonics:
“Effect of odd triplens”
• For three-phase inverters, there is
significant advantage if p is chosen to be:
– odd and multiple of three (triplens) (e.g.
3,9,15,21, 27..)
– the waveform and harmonics and shown on the
next two slides. Notice the difference?
• By observing the waveform, it can be seen
that with odd p, the line voltage shape
looks more “sinusoidal”.
• The even harmonics are all absent in the
phase voltage (pole switching waveform).
This is due to the p chosen to be odd.
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
50
Spectra observations
• Note the absence of harmonics no. 21, 63
in the inverter line voltage. This is due to p
which is multiple of three.
• In overall, the spectra of the line voltage is
more “clean”. This implies that the THD is
less and the line voltage is more sinusoidal.
• It is important to recall that it is the line
voltage that is of the most interest.
• Also can be noted from the spectra that 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.
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
51
Waveform: effect of “triplens”
2
dc
V
2
dc
V
−
2
dc
V
2
dc
V
−
2
dc
V
−
2
dc
V
−
2
dc
V
2
dc
V
dc
V
dc
V
dc
V
−
dc
V
−
π
π
2
RG
V
RG
V
RY
V
RY
V
YG
V
YG
V
6
.
0
,
8
=
= M
p
6
.
0
,
9
=
= M
p
ILLUSTRATION OF BENEFITS OF USING A FREQUENCY RATIO
THAT IS A MULTIPLE OF THREE IN A THREE PHASE INVERTER
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
52
Harmonics: 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
3
8
.
0
Fundamental
41
43
39
37
45
47
23
19
21
63
61
59
57
65
67
69 77
79
81
83
85
87
89
91
19
23
43
47
41
37
61
59
65
67
83
79
85
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 2): Dr.
Zainal Salam, 2002
53
Comments on PWM scheme
• It is desirable to push p to as large as
possible.
• The main impetus for that when p is high,
then the harmonics will be at higher
frequencies because frequencies of
harmonics are related to: f = kp(f
m
), where
f
m
is the frequency of the modulating
signal.
• Although the voltage THD improvement is
not significant, but the current THD will
improve greatly because the load normally
has some current filtering effect.
• In any case, if a low pass filter is to be
fitted at the inverter output to improve the
voltage THD, higher harmonic frequencies
is desirable because it makes smaller filter
component.
Power Electronics and
Drives (Version 2): Dr.
Zainal Salam, 2002
54
Example
The amplitudes of the pole switching waveform harmonics of the red
phase of a three-phase inverter is shown in Table below. The inverter
uses a symmetric regular sampling PWM scheme. The carrier frequency
is 1050Hz and the modulating frequency is 50Hz. The modulation
index is 0.8. Calculate the harmonic amplitudes of the line-to-voltage
(i.e. red to blue phase) and complete the table.
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