DC to AC Conversion
(INVERTER)
• General concept
• Basic principles/concepts
• Single-phase inverter
– Square wave
– Notching
– PWM
• 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.
• TYPICAL APPLICATIONS:
– UPS, Industrial drives, Traction, HVDC
• General block diagram
IDC
Iac
+
+
Vac
VDC
−
−
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Types of inverter
• Voltage Source Inverter (VSI)
• Current Source Inverter (CSI)
"DC LINK"
Iac
+
+
C
VDC
Load Voltage
−
−
L
+
VDC
IDC
ILOAD
Load Current
−
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Voltage source inverter (VSI)
with variable DC link
DC LINK
+
Vs
C
-
CHOPPER
(Variable DC output)
+
Vin
-
+
Vo
-
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.
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Variable DC link inverter (2)
• Advantages:
– simple waveform generation
– Reliable
• Disadvantages:
– Extra conversion stage
– Poor harmonics
Vdc2
Higher input voltage
Higher frequency
Vdc1
Lower input voltage
Lower frequency
T1
T2
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5
VSI with fixed DC link
INVERTER
+
Vin
+
Vo
C
(fixed)
−
−
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
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Current Source Inverter (CSI)
L
IDC
IO
+
Vin
RL
-
Io
CHOPPER
CSI
Iac
-Iac
t
Current waveform
• Input (DC) current is “chopped”to obtain
AC output current.
• Need large L.
• Less popular compared to VSI
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Power flow consideration
I dc
+
Io
+
Vdc
Vo
-
-
Vo
i, v
io
t
(4)
(1)
(2)
(3)
• Assume load is drawing lagging Power
Factor.:
• (+) io and (+) vo:
(+) power flow (1)
• (-) io and (-) vo:
(+) power flow (3)
• (+) io and (-) vo:
(-) power flow (2)
• (-) io and (+) vo:
(-) power flow (4)
• Positive power flow indicates power
transfer from input (Vdc.Idc) to load.
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4-quadrant operation
• Negative power flow
indicates that the
power is fed back
from load to source.
io
(2)
RECTIFIER
(1)
INVERTER
vo
(3)
INVERTER
• Hence, inverter must
have “4 quadrant”
capability to cater
for all possible load
types.
(4)
RECTIFIER
4-QUADRANT
OPERATION
T1
• Practically, this can
be achieved by
placing an antiparallel diode across
each switching
device.
LOAD
T2
ANTI-PARELELL
DIODES
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Operation of simple squarewave inverter (1)
• To illustrate the concept of AC waveform
generation
SQUARE-WAVE
INVERTERS
T3
T1
D1
D3
+ VO -
VDC
IO
T4
T2
D2
S1
D4
S3
EQUAVALENT
CIRCUIT
S4
S2
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Operation of simple squarewave inverter (2)
S1,S2 ON; S3,S4 OFF
for t1 < t < t2
vO
S1
VDC
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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Waveforms and harmonics of
square-wave inverter
INVERTER
OUTPUT
Vdc
-Vdc
FUNDAMENTAL
V1
4VDC
π
V1
3
3RD HARMONIC
5RD HARMONIC
V1
5
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Filtering (1)
• 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.
(LOW PASS) FILTER
L
+
vO 1
+
C
vO 2
−
vO 1
LOAD
−
vO 2
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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.
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“Notching”of square wave
Notched Square Wave
Vdc
−Vdc
Fundamental Component
Vdc
−Vdc
• Notching results in controllable output
voltage magnitude (compare Figures
above).
• Limited degree of harmonics control is
possible
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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
pwm waveform
desired
sinusoid
1
SINUSOIDAL PULSE-WITDH MODULATED
APPROXIMATION TO SINE WAVE
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PWM- output voltage and
frequency control
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Output voltage harmonics
• Why need to consider harmonics?
– Waveform quality must match TNB supply.
“Power Quality” issue.
– In some applications, harmonics cause
degradation of equipment. Equipment need to
be “de-rated”.
• Total Harmonic Distortion (THD):
∞
THD =
∑ (Vn, RMS )
∞
2
n=2
V1, RMS
=
∑ (VRMS )2 − (V1,RMS )2
n =2
V1, RMS
where n is the harmonics number.
Current THD can be obtained by replacing
the harmonic voltage with harmonic current :
Vn
In =
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
1 2π
ao = ∫ f (v)dθ
π 0
1 2π
an = ∫ f (v) cos(nθ )dθ
π 0
1 2π
bn = ∫ f (v) sin (nθ )dθ
π 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=ωt
-Vdc
2π

1 π
ao =  ∫ Vdc dθ + ∫ − Vdc dθ  = 0
π 0

π
2π

Vdc π
an =
 ∫ cos(nθ )dθ − ∫ cos(nθ )dθ  = 0
π 0

π
2π

Vdc π
bn =
 ∫ sin (nθ )dθ − ∫ sin (nθ )dθ 
π 0

π
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Harmonics of square wave (2)
Solving,
[
]
Vdc
π
2π
− cos(nθ ) 0 + cos(nθ ) π
nπ
V
= dc [(cos 0 − cos nπ ) + (cos 2nπ − cos nπ )]
nπ
V
= dc [(1 − cos nπ ) + (1 − cos nπ )]
nπ
2V
= dc [(1 − cos nπ )]
nπ
bn =
when n is even, cos nπ = 1
bn= 0
when n is odd, cos nπ = −1
4V
bn= dc
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
7
5
9
11
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.
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Quasi-square wave (QSW)
Vdc
α
α
α
π
2π
-Vdc
Note that an = 0.
Due to half - wave symmetry,
[
 1 π −α
 2Vdc
π −α
bn = 2 ∫ Vdc sin (nθ )dθ  =
− cos nθ α
π α
 nπ
2V
= dc [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
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Harmonics control
2Vdc
[cos(nα ) − cos nπ cos nα ]
nπ
2V
= dc cos(nα )[1 − cos nπ ]
nπ
⇒ bn =
If n is even, ⇒ bn = 0,
If n is odd, ⇒ bn =
4Vdc
cos(nα )
nπ
In particular, amplitude of the fundamental is :
4Vdc
cos(α )
b1 =
π
The fundamental , b1 , is controlled by varying á
Harmonics can also be controlled by adjusting α ,
For example if α = 30o , 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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Half-bridge inverter (1)
S1 ON
Vdc S2 OFF
+
Vdc
2
S1
VC1
-
−V +
o
0
G
+
VC2
-
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.
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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.
+
S1
S1
signal
(gate)
Ishort
G
Vdc
RL
−
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
half-bridge 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'
VR 'G
Vdc
2
−
Vdc
2
+
Vdc
2
Vdc
2
Vo
Vdc
−
S4
S2
Vo = VRG − 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
ia
ZR
S2
ib
ZY
ZB
N
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Square-wave inverter
waveforms
VDC/2
VAD
t
-VDC/2
VB0
t
VC0
t
(a) Three phase pole switching waveforms
VDC
600
120 0
VAB
t
-VDC
(b) Line voltage waveform
2VDC/3
VDC/3
VAPH
t
-VDC/3
-2VDC/3
(c) Phase voltage waveform (six-step)
Interval
Positive device(s) on
Negative devise(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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Three-phase inverter
waveform relationship
• VRG, VYG, VBG are known as “pole
switching waveform” or “inverter phase
voltage”.
• VRY, VRB, VYB 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 VRN ,VYN,
VBN. It is also popularly termed as “sixstep” waveform
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MODULATION: 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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Natural/Regular sampling
MODULATION INDEX = M I :
Amplitude of the modulating waveform
MI =
Amplitude of the carrier waveform
M I is related to the fundamental (sine wave)
output voltage magnitude. If M Iis high, then
the sine wave output is high and vice versa.
If 0 < M I < 1, the linear relationship holds :
V1 = M I Vin
where V1 , Vin are fundamental of the output
voltage and input (DC) voltage, respectively.
MODULATION RATIO = M R (= p )
MR =
Frequency of the modulating waveform
Amplitude of the carrier waveform
M R is related to the " harmonic frequency". 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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Asymmetric and symmetric
regular sampling
T
+1
M1 sin ω mt
sample
point
3T
4
T
4
5T
4
π
4
t
−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 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.
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Bipolar PWM switching
∆
δ =
∆
4
modulating
waveform
carrier
waveform
π
2π
π
2π
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kth
pulse
δ 1k
δ 2k
αk
Pulse width relationships
∆
δ =
∆
4
modulating
waveform
carrier
waveform
π
2π
π
2π
kth
pulse
δ 1k
δ 2k
αk
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Characterisation of PWM
pulses for bipolar switching
∆
+ VS
2
δ0
δ0
δ1k
V
− S
2
δ0
δ0
δ 2k
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 average voltage during each half cycle
of the PWM pulse is given as :
 Vdc  δ 1k − ( 2δ o − δ1k ) 
V1k = 


2δ o
 2 

 Vdc  δ1k − δ o 
 Vs 
=
 = β1k  

 2  δ o 
2
where β1k
 δ 1k − δ o 

= 
 δo 
Similarly,
 δ 2k − δ o 
 Vdc 
V2 k = β 2 k 

 ; where β 2 k = 
 2 
 δo 
The volt - second supplied by the sinusoid,
As1 =
αk
∫ Vm sin θdθ = Vm [cos(α k − 2δ o ) − cosα k ]
α k − 2δ o
= 2Vm sin δ o sin(α k − δ o )
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Switching angles (3)
Since,
sin δ o → δ o for small δ o ,
As1 = 2δ oVm sin(α k − δ o )
Similarly,
As 2 = 2δ oVm sin(α k + δ o )
The volt - seconds of the PWM waveforms,
Vdc 
Vdc 


Ap1 = β1k 
Ap 2 = β 21k 
2δ o ;
2δ o
 2 
 2 
To derive the modulation strategy,
Ap1 = As1; Ap 2 = As 2
Hence, for the leading edge
V 
β1k  dc 2δ o = 2δ oVm sin(α k − δ o )
 2 
Vm
⇒ β1k =
sin(α k − δ o )
(Vdc 2)
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PWM switching angles (4)
The voltage ratio,
Vm
MI =
is known as modulation
(Vdc 2 )
index or depth. It varies from 0 to 1.
Thus,
β1k = M I sin(α k − δ o )
Using similar method, the trailing edge
can be derived :
β 2k = M I sin(α k − δ o )
Substituting to solve for the pulse - width,
δ1k − δ o
β1k =
δo
⇒ δ1k = δ o [1 + M I sin(α k − δ o )]
and
δ 2k = δ o [1 + M I sin(α k + δ o )]
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PWM Pulse width
Thus the switching angles of the kth pulse is :
Leading edge : α k − δ1k
Trailing edge : α k + δ1k
The above equation is valid for Asymmetric
Modulation, i.eδ1k and δ 2k are different.
For Symmetric Modulation, δ1k = δ 2k = δ k
⇒ δ k = δ o [1 + M I sin α k ]
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Example
•
For the PWM shown below, calculate the switching
angles for all the pulses.
carrier
waveform
2V
1.5V
π
2π
modulating
waveform
1
t1
t2
2
3
t3 t4 t5 t6
4
5
6
7
8
9
π
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 - wave
symmetry, harmonic content of each (kth)
PWM pulse can be computed as :
1T


bnk = 2 ∫ f (v) sin nθdθ 
π

 0

α −δ

2  k 1k  Vdc 
=  ∫ −
 sin nθdθ 
π α −2δ  2 

 k o
α +δ

2  k 2 k  Vdc 
+  ∫ 
 sin nθdθ 
π α −δ  2 

 k 1k
α + 2δ

2  k o  Vdc 
+  ∫ −
 sin nθdθ 
π α +δ  2 


k
2k
Which can be reduced to :
Vdc
bnk = −
{cos n(α k − 2δ o ) − cos n(α k − δ1k )
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 )}
Power Electronics and
Drives: Dr. Zainal Salam,
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Harmonics of PWM
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
The slide on the next page shows the
computation of this equation.
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Drives: Dr. Zainal Salam,
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PWM Spectra
M = 0.2
Amplitude
M = 0.4
1.0
M = 0.6
0.8
0.6
M = 0.8
0.4
Depth of
Modulation
0.2
M = 1.0
0
p
2p
3p
4p
Fundamental
NORMALISED HARMONIC AMPLITUDES FOR
SINUSOIDAL PULSE-WITDH MODULATION
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Drives: Dr. Zainal Salam,
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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: V1= MIVin
•
The harmonics appear in “clusters” with main
components at frequencies of :
f = kp (fm);
k=1,2,3....
where fm 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 MI. 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: Dr. Zainal Salam,
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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: Dr. Zainal Salam,
FKE, UTM Skudai, JB
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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: Dr. Zainal Salam,
FKE, UTM Skudai, JB
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Waveform: effect of “triplens”
π
Vdc
2
−
−
2π
VRG
Vdc
2
Vdc
2
VYG
Vdc
2
Vdc
VRY
− Vdc
p = 8, M = 0.6
Vdc
2
−
−
V RG
Vdc
2
Vdc
2
VYG
Vdc
2
Vdc
VRY
− 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
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Drives: Dr. Zainal Salam,
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Harmonics: 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
37
23
41
43
47
59
61
65
67
83
79
85
89
0.2
0
21
19
A
63
23
37
39
41
43
45
47 57
59
61
65
81
79
67
69 77
83
85
87
89
91 Harmonic Order
Fundamental
COMPARISON OF INVERTER PHASE VOLTAGE (A) & INVERTER LINE VOLTAGE
(B) HARMONIC (P=21, M=0.8)
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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(fm), where
fm 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.
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Drives: Dr. Zainal Salam,
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