Chapter 4
DC to AC Conversion
(INVERTER)
•
•
•
•
•
General concept
Singlephase inverter
Harmonics
Modulation
Threephase 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
predetermined sequence so as to generate AC
voltage (or current) output.
• General block diagram
IDC
Iac
Vac
VDC
−
−
• TYPICAL APPLICATIONS:
– Uninterruptible power supply (UPS), Industrial
(induction motor) drives, Traction, HVDC
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Simple squarewave 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 lowpass 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 squarewave 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 “derated”.
• 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 squarewave (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 lowpass
filter design can be very difficult.
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Quasisquare 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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Halfbridge 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 viceversa.
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Shoot through fault and
“Deadtime”
•
In practical, a dead time as shown below is required
to avoid “shootthrough” 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 highquality
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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Singlephase, fullbridge (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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Threephase inverter
•
Each leg (Red, Yellow, Blue) is delayed by 120
degrees.
•
A threephase 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
lIneto ine
Voltage
VRY
Sixstep
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
Quasisquare 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
• Spacevector modulation (SVM)
– A simple technique based on voltsecond that is
normally used with threephase 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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