Chapter 4
DC to AC Conversion
(INVERTER)
•
•
•
•
•
General concept
Single-phase inverter
Harmonics
Modulation
Three-phase inverter
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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
Vac
VDC
−
−
• TYPICAL APPLICATIONS:
– Un-interruptible power supply (UPS), Industrial
(induction motor) drives, Traction, HVDC
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Simple square-wave inverter (1)
• To illustrate the concept of AC waveform
generation
S1
S3
S4
S2
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AC Waveform Generation
S1,S2 ON; S3,S4 OFF
vO
S1
VDC
for t1 < t < t2
VDC
S3
+ vO −
t1
S4
t
t2
S2
S3,S4 ON ; S1,S2 OFF
for t2 < t < t3
vO
S1
VDC
S3
t2
+ vO −
S4
t3
t
S2
-VDC
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AC Waveforms
INVERTER OUTPUT VOLTAGE
Vdc
π
2π
-Vdc
FUNDAMENTAL COMPONENT
V1
4VDC
π
V1
3
V1
5
3RD HARMONIC
5RD HARMONIC
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Harmonics Filtering
DC SUPPLY
INVERTER
(LOW PASS) FILTER
LOAD
L
+
vO 1
C
+
vO 2
−
BEFORE FILTERING
vO 1
−
AFTER FILTERING
vO 2
• 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.
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Variable Voltage Variable
Frequency Capability
Vdc2
Higher input voltage
Higher frequency
Vdc1
Lower input voltage
Lower frequency
t
• 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
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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.
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Total Harmonics Distortion (THD)
Voltage THD : If Vn is the nth harmonic voltage,
∞
(Vn, RMS )2
THDv = n= 2
V1, RMS
=
V2, RMS 2 + V3, RMS 2 + .... + V2, RMS 2
V1, RMS
If the rms voltage for the vaveform is known,
∞
(VRMS )2 − (V1, RMS )2
THDv = n= 2
V1, RMS
Current THD :
∞
(I n, RMS )2
THDi = n =2
I1, RMS
V
In = n
Zn
Z n is the impedance at harmonic frequency.
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Fourier Series
• Study of harmonics requires understanding of wave
shapes. Fourier Series is a tool to analyse wave
shapes.
Fourier Series
ao =
an =
bn =
1 2π
π
1
π
1
π
0
2π
0
2π
f (v )dθ (" DC" term)
f (v) cos(nθ )dθ
(" cos" term)
f (v) sin (nθ )dθ
("sin" term)
0
Inverse Fourier
∞
1
f (v) = ao + (an cos nθ + bn sin nθ )
2
n =1
where θ = ωt
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Harmonics of square-wave (1)
Vdc
π
2π
θ ω
-Vdc
ao =
an =
bn =
1 π
π
0
2π
Vdc dθ + − Vdc dθ = 0
π
Vdc π
π
0
Vdc π
π
0
2π
cos(nθ )dθ − cos(nθ )dθ = 0
π
2π
sin (nθ )dθ − sin (nθ )dθ
π
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Harmonics of square wave (2)
Solving,
V
π
2π
bn = dc − cos(nθ ) 0 + cos(nθ ) π
nπ
Vdc
[(cos 0 − cos nπ ) + (cos 2nπ − cos nπ )]
=
nπ
Vdc
[(1 − cos nπ ) + (1 − cos nπ )]
=
nπ
2V
= dc [(1 − cos nπ )]
nπ
[
]
When n is even, cos nπ = 1
bn = 0
(i.e. even harmonics do not exist)
When n is odd, cos nπ = −1
4Vdc
bn =
nπ
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Spectra of square wave
Normalised
Fundamental
1st
3rd (0.33)
5th (0.2)
7th (0.14)
9th (0.11)
11th (0.09)
1
3
5
n
7
9
11
• 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.
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Quasi-square wave (QSW)
Vdc
α
α
α
π
2π
-Vdc
Note that an = 0. (due to half - wave symmetry)
[
2V
1 π −α
π −α
bn = 2
Vdc sin (nθ )dθ = dc − cos nθ α
π α
nπ
]
2Vdc
[cos(nα ) − cos n(π − α )]
=
nπ
Expanding :
cos n(π − α ) = cos(nπ − nα )
= cos nπ cos nα + sin nπ sin nα = cos nπ cos nα
bn =
2Vdc
[cos(nα ) − cos nπ cos nα ]
nπ
2Vdc
=
cos(nα )[1 − cos nπ ]
nπ
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Harmonics control
If n is even,
bn = 0,
4Vdc
If n is odd, bn =
cos(nα )
nπ
In particular, amplitude of the fundamental is :
b1 =
4Vdc
π
cos(α )
Note :
The fundamental , b1 , is controlled by varying
Harmonics can also be controlled by adjusting α ,
Harmonics Elimination :
For example if α = 30 o , then b3 = 0, or the third
harmonic is eliminated from the waveform. In
general, harmonic n will be eliminated if :
90o
α=
n
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Example
A full - bridge single phase inverter is fed by square wave
signals. The DC link voltage is 100V. The load is R = 10R
and L = 10mH in series. Calculate :
a) the THDv using the " exact" formula.
b) the THDv by using the first three non - zero harmonics
c) the THDi by using the first three non - zero harmonics
Repeat (b) and (c) for quasi - square wave case with α = 30
degrees
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Half-bridge inverter (1)
S1 ON
Vdc S2 OFF
+
VC1
Vdc
G
+
VC2
-
2
S1
− V +
o
0
t
RL
S2
−
Vdc
2
S1 OFF
S2 ON
•
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.
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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.
+ S1
Ishort
G
Vdc
RL
−
S1
signal
(gate)
S2
signal
(gate)
S2
"Shoot through fault" .
Ishort is very large
td
td
"Dead time' = td
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Single-phase, full-bridge (1)
•
Full bridge (single phase) is built from two halfbridge leg.
•
The switching in the second leg is “delayed by 180
degrees” from the first leg.
LEG R
VRG
Vdc
2
LEG R'
π
2π
ωt
π
2π
ωt
π
2π
ωt
+
+
Vdc
2
S1
-
Vdc
G
-
R
S3
+ Vo -
R'
+
Vdc
2
VR 'G
Vdc
2
−
S4
S2
−
Vdc
2
Vdc
2
Vo
Vdc
Vo = V RG − VR 'G
G is " virtual groumd"
− Vdc
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Three-phase inverter
•
Each leg (Red, Yellow, Blue) is delayed by 120
degrees.
•
A three-phase inverter with star connected load is
shown below
+Vdc
+
Vdc/2
G
S1
S3
−
+
Vdc/2
S5
R
Y
iR
iY
S4
B
iB
S6
S2
−
ZR
ia
ib
ZY
ZB
N
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Three phase inverter waveforms
Inverter Phase
Voltage
VDC/2
(or pole switching
waveform)
VRG
-V /2
DC
1200
VDC/2
VYG
-VDC/2
2400
VDC/2
VBG
-VDC/2
lIne-to -ine
Voltage
VRY
Six-step
Waveform
VRN
VDC
-VDC
2VDC/3
VDC/3
-VDC/3
-2VDC/3
Interval
Positive device(s) on
Negative device(s) on
1
3
2,4
2
3,5
4
3
5
4,6
4
1,5
6
5
1
2,6
6
1,3
2
Quasi-square wave operation voltage waveforms
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Pulse Width Modulation (PWM)
Modulating Waveform
+1
M1
Carrier waveform
0
−1
Vdc
2
0
−
•
t 0 t1 t2
t3 t4 t5
Vdc
2
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.
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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 motordrive
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Modulation Index, Ratio
Modulating Waveform
+1
M1
Carrier waveform
0
−1
Vdc
2
0
−
t0 t1 t 2
t 3 t 4 t5
Vdc
2
Modulation Index (Modulation Depth) = M I :
Amplitude of the modulating waveform
MI =
Amplitude of the carrier waveform
Modulation Ratio (Frequency Ratio) = M R (= p )
MR = p =
Frequency of the carrier waveform
Frequency of the modulating waveform
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Modulation Index, Ratio
Modulation Index deterrmines the output
voltage fundamental component
If 0 < M I < 1,
V1 = M I Vin
where V1 , Vin are fundamental of the output
voltage and input (DC) voltage, respectively.
−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Modulation ratio determines the incident (location)
of harmonics in the spectra.
The harmonics are normally located at :
f = kM R ( f m )
where f m is the frequency of the modulating signal
and k is an integer (1,2,3...)
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Regular sampling
t1 t2
Sinusoidal modulating
waveform, vm(t)
Carrier, vc(t)
2π
π
t
Regular sampling waveform, vs (t )
t'1
t'2
vpwm
t
Regular sampling PWM
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Asymmetric and symmetric
regular sampling
T
+1
M1 sin ω mt
sample
point
3T
4
T
4
5T
4
π
t
4
−1
Vdc
2
asymmetric
sampling
t0
t1
t2
t3
t
symmetric
sampling
V
− dc
2
Generating of PWM waveform regular sampling
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Bipolar Switching
Modulating Waveform
+1
M1
Carrier waveform
0
−1
Vdc
2
0
−
t0 t1 t2
t 3 t 4 t5
Vdc
2
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Unipolar switching
A
Carrier waveform B
(a)
S1
(b)
S3
(c)
V pwm
(d)
Unipolar switching scheme
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Bipolar PWM switching: Pulsewidth characterization
∆
δ=
∆
4
modulating
waveform
carrier
waveform
π
2π
π
2π
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kth
pulse
δ 1k
δ 2k
αk
The kth Pulse
∆
δ0
δ0
+ Vdc
2
δ1k
δ0
δ0
δ 2k
+ Vdc
2
αk
The kth PWM pulse
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Determination of switching angles
for kth PWM pulse (1)
AS2
AS1
v
Vmsin( θ )
+ Vdc
2
Ap1
Ap2
V
− dc
2
Equating the volt - second,
As1 = Ap1
As 2 = Ap 2
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PWM Switching angles (2)
The Volt - second during the first half cycle
of the PWM pulse is given as :
Vdc
Vdc
(δ1k ) −
(2δ o − δ1k )
A p1 =
2
2
= (Vdc )(δ1k − δ o )
Similarly for the second half,
Vdc
(δ 2k ) − Vdc (2δ o − δ 2k )
2
2
= (Vdc )(δ 2k − δ o )
Ap2 =
The volt - second supplied by the sinusoid,
As1 =
αk
Vm sin θdθ = Vm [cos(α k − 2δ o ) − cos α k ]
α k − 2δ o
= 2Vm sin δ o sin(α k − δ o )
Similarly,
As 2 = 2δ oVm sin(α k + δ o )
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Switching angles (3)
For small angle δ o
sin δ o → δ o ,
As1 = 2δ oVm sin(α k − δ o )
As 2 = 2δ oVm sin(α k − δ o )
To derive the modulation strategy,
A p1 = As1;
A p 2 = As 2
Hence, for the the first half cycle of PWM pulse,
(Vdc )(δ1k − δ o ) = 2δ oVm sin(α k − δ o )
(δ1k − δ o ) =
2Vm
(δ o sin(α k − δ o )
Vdc
By definition, the Modulation Ratio,
MI =
Vm
(Vdc 2 )
is known as modulation
Thus, the pulse width for the first half cycle
of the PWM waveform is given by :
δ1k = δ o [1 + M I sin(α k − δ o )]
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PWM switching angles (4)
Thus the leading edge switching angle of
the kth pulse is :
α k − δ1k
Using similar method, pulse width of the
second half cycle of PWM waveform :
δ 2k = δ o [1 + M I sin(α k + δ o )]
And the trailing edge angle :
α k + δ 2k
The above equation is valid for Asymmetric
Modulation, i.eδ1k and δ 2k are different.
For Symmetric Modulation,
δ1k = δ 2k = δ k
Hence
δ k = δ o [1 + M I sin α k ]
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Example
•
For the PWM shown below, calculate the switching
angles pulses no. 2.
carrier
waveform
2V
1.5V
π
2π
modulating
waveform
1
2
3
4
5
6
7
8
9
π
t1
t2
t3 t4 t5 t6
t13
t15
t17
t7 t8 t9 t10 t11 t12
t14
t16 t18 2π
α1
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Harmonics of bipolar PWM
∆
Assuming the PWM
waveform is half
δ0
δ0
+ Vdc
2
δ1k
wave symmetry,
harmonic
δ0
δ0
δ 2k
content of each
(kth) PWM pulse
can be computed as :
bnk = 2
=
+
+
2
1T
π
α k −δ1k
α k +δ 2 k
π α −δ
k
1k
2
αk
f (v) sin nθdθ
0
π α −2δ
k
o
2
+ Vdc
2
α k + 2δ o
π α +δ
k
2k
V
− dc sin nθdθ
2
Vdc
sin nθdθ
2
V
− dc sin nθdθ
2
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Harmonics of Bipolar PWM
Which can be reduced to :
Vdc
{cos n(α k − 2δ o ) − cos n(α k − δ1k )
bnk = −
nπ
+ cos n(α k + δ 2 k ) − cos n(α k − δ1k )
+ cos n(α k + δ 2 k ) − cos n(α k + 2δ o )}
Yeilding,
2Vdc
[cos n(α k − δ1k ) − cos n(α k − 21k )
bnk =
nπ
+ 2 cos nα k cos n 2δ o ]
This equation cannot be simplified
productively.The Fourier coefficent for the
PWM waveform isthe sum of bnk for the p
pulses over one period, i.e. :
bn =
p
bnk
k =1
Next slide shows the computation of this equation.
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PWM Spectra
M I = 0.2
Amplitude
M I = 0.4
1. 0
0 .8
M I = 0.6
0.6
0.4
M I = 0 .8
Modulation
Index
0.2
0
M I = 1 .0
p
2p
3p
4p
Fundamental
NORMALISED HARMONIC AMPLITUDES FOR
SINUSOIDAL PULSE-WITDH MODULATION
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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.
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Tabulated Bipolar PWM Harmonics
0.2
0.4
0.6
0.8
1.0
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
n
1
MI
MR +4
2MR +1
0.018
0.190
2MR +3
0.326
0.370
0.314
0.181
0.024
0.071
0.139
0.212
0.013
0.033
2MR +5
3MR
0.335
0.123
0.083
0.171
0.113
3MR +2
0.044
0.139
0.203
0.716
0.062
0.012
0.047
0.104
0.157
0.016
0.044
3MR +4
3MR +6
4MR +1
0.163
0.157
0.008
0.105
0.068
4MR +3
0.012
0.070
0.132
0.115
0.009
0.034
0.084
0.017
0.119
0.050
4MR+5
4MR +7
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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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Effect of odd and “triplens”
π
Vdc
2
−
−
2π
V RG
Vdc
2
Vdc
2
VYG
Vdc
2
Vdc
V RY
− Vdc
Vdc
2
−
−
p = 8, M = 0.6
V RG
Vdc
2
Vdc
2
VYG
Vdc
2
Vdc
V RY
− Vdc
p = 9, M = 0.6
ILLUSTRATION OF BENEFITS OF USING A FREQUENCY RATIO
THAT IS A MULTIPLE OF THREE IN A THREE PHASE INVERTER
Power Electronics and
Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
43
Spectra: effect of “triplens”
Amplitude
1.8
0.8 3 (Line to line voltage)
1.6
1.4
1.2
1.0
0.8
0.6
B
0.4
19
0.2
37
23
41
43
47
59
61
65
67
79
83
85
89
0
21
19
Fundamental
A
63
23
37
39
41
43
45
47 57
59
61
83
81
65
79
67
69 77
85
87
89
91 Harmonic Order
COMPARISON OF INVERTER PHASE VOLTAGE (A) & INVERTER LINE VOLTAGE
(B) HARMONIC (P=21, M=0.8)
Power Electronics and
Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
44
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
45
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
1
Amplitude (pole switching
waveform)
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
Amplitude (line-to
line voltage)
Power Electronics and
Drives (Version 3-2003):
Dr. Zainal Salam UTM-JB
46