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22.5. DC TO AC CONVERSION: INVERSION
Dc to ac converters constitute a significant portion of power electronic
converters. These converters, also called inverters, are used in applications
such as electric motor drives, uninterruptible power supplies (UPS), and
utility applications such as grid connection of renewable energy sources.
Inverters for single phase ac and three-phase three-wire ac systems are
described in this section.
22.5.1. Single-Phase AC Synthesis
In an ac system both the voltage and the current should be able to reverse in
polarity. Further, the voltage and current polarities may or may not be the
same at a given time. Thus, a dc to ac converter implementation should be
able to output a voltage independent of current polarity. In the full bridge dc
to dc converter shown in Fig. 22-19a the primary circuit consisting of four
controlled switches, also called H-bridge, has two bi-positional switch
implementations. Each bi-positional switch has bidirectional currentcapability but only positive output voltage ( >, > 0). However, based
on the duty cycles, the difference of the outputs, V = , can
reverse in polarity. Thus the H-bridge is used for synthesizing single phase ac
voltage from a dc voltage.
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AB AN BN
DC TO AC CONVERSION: INVERSION
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Figure 22.19. Single-phase inverter: (a) circuit, (b) quasi-square wave
synthesis.
22.5.1.1. Quasi-Square Wave Inverter.
The simplest form of dc to ac conversion, albeit with poor quality, is synthesis
of quasi-square wave ac instead of a pure sine wave. Diagonally opposite
switches in the H-bridge are turned on simultaneously. The pulse width of
each pair is controlled to adjust the magnitude of the fundamental
component, while the switching frequency is equal to the required output
frequency. The synthesized voltage waveform is shown in Fig. 22-19b. The
peak value of fundamental and harmonic components are
(22-31)
where d is the duty ratio and n the harmonic number. This converter is
widely used for low cost low power UPS applications where the voltage
waveform quality is not important. Incandescent lighting, universal input
motors, and loads with a diode bridge or power factor corrected front end
(discussed in Sec. 22.8) are not affected by the voltage waveform quality. The
load current, i , has harmonics based on the load characteristics.
Sometimes an LC filter is added at the output to reduce the harmonic
content. Low power low cost inverters such as those used to generate ac
from 14 V dc in automobiles usually have quasi square wave voltage output.
22.5.1.2. Single Phase Sinusoidal Voltage Synthesis.
AB
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For applications requiring low voltage and current distortion high-frequency
PWM is utilized to generate a sinusoidally varying average voltage. The
power converter used is the H-bridge shown in Fig. 22-19a. The duty ratio for
each bipositional switch, also called one leg of the inverter, is varied
sinusoidally. The switching signals are generated by comparison of a
sinusoidally varying control voltage with a trianglewave as shown in Fig. 22-
20. Equations relating the control voltages, duty ratios, and the averaged
output voltages are as follows:
Figure 22.20. Single-phase sinusoidal ac synthesis waveforms.
(22-32)
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(22-33)
(22-34)
(22-35)
(22-36)
(22-37)
(22-38
(22-39)
Here and are peak values of control voltage and the triangle wave
respectively, is the modulation index, = 2f is the
angular frequency of the sinusoid to be synthesized, while d (t) and d (t)
are duty ratios of switches S1 and S3, respectively. In Eq. (22-39) k may
be regarded as the gain of the power converter that amplifies the control
signal (t) to the average output voltage . The maximum peak value of
the output voltage, obtained for m= 1, is V . This is significantly lower than
that obtainable with the quasi square wave inverter (4V /). However,
harmonics in the output voltage are significantly reduced and are at much
higher frequencies: k f l f , where k and l are integers such that k +
l is odd. The switching frequency is much higher than the output frequency
f , which has a maximum value of about 50/60 Hz for standard applications
or 400 Hz for aerospace applications.
m m
A B
PWM
c
in
in
s m
4
m
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If the load is inductive, such as a motor, the current harmonics are much
lower than the voltage harmonics. For several applications maximum
harmonic content for the voltage and current output from the inverter is
specified. In these cases an L-C filter similar to the buck converter is used.
Depending upon the application a two-stage L-C filter or a two-stage notch
filter (to suppress the dominant switching frequency harmonics) may beused. Further, it has to be ensured that when connected to the load, the filter
is adequately damped by a combination of passive selection and the control
loop. This aspect is particularly important for line connected applications
where the inverter is supplying power to the utility grid.
Equation (22-37) can be rewritten as
(22-40)
This clearly shows that on an average basis the "neutral point" for the output
of one inverter leg is V /2 above "N," i.e., at the mid-point of the input dc
bus. Thus using the same H-bridge a split-phase ac (two ac voltages 180 out
of phase with a common return) can be generated if the center point of the
dc bus is available as the neutral connection for the output. This type of
configuration is commonly used in generating 120/240 from the same
inverter. Furthermore, using three legs instead of two the converter can
generate three phase voltages with a neutral connection, with the flexibility
that the three phases may be loaded independently. Common applications
are inverters for interfacing photovoltaic systems to the utility grid and
exporting power from vehicles.
22.5.2. Three-Phase AC Synthesis
The last observation in the previous section leads us to three-phase inverters
without a neutral connection. The circuit consists of three legs, one for each
output with a common dc link as shown in Fig. 22-21a. Using sine triangle
PWM with control voltages offset by 120 (instead of 180 as in the single-
phase case) we obtain:
in
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(22-41)
(22-42)
k =A, B, C
(22-43)
Figure 22.21. Three-phase ac synthesis: (a) converter, (b) output
voltage vectors, (c) instantaneous waveforms.
The zero sequence component of the output voltages, = ( + +
)/3 = V /2, does not appear in the line-to-line voltages, and since there is
no neutral connection to the inverter, zero sequence currents do not flow.
The maximum peak value of the output line-to-line voltages is .
Using square wave inversion, similar to that for the single-phase case, we canobtain higher magnitude for the fundamental component of the output
voltages at the cost of adding harmonics. However, if, instead of all the
harmonics, only the fundamental and those harmonics of the square wave
z AN BN
CN in
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that contribute zero sequence component (triplen harmonics) are retained,
the output voltage amplitude increases without adding harmonics to the line-
to-line voltages and the line currents. Usually, addition of the third harmonic
component is sufficient. As described in Refs. 37 and 38 the most
effective method is to add the following zero sequence component to the
control voltages for each phase:
(22-44)
In terms of output voltage generation, this is equivalent to space vector
modulation (SVM).
22.5.3. Space Vector Modulation
This method has become extremely popular for three-phase inverters in the
low to medium power range. A very brief description will be presented here
and details can be found in Refs. 31, 35, and 37.
For three-phase systems with no zero sequence component, i.e., = ( +
+ )/3 = 0, the three-phase quantities are linearly dependent and canbe transformed to a two-phase orthogonal system commonly called the
system. Quantities in the system can be represented by complex numbers
and as two-dimensional vectors in a plane, called space vectors. The
transformation from the abc to quantities is given by
(22-45)
With negative sequence components absent, and components of steady
state sinusoidal abcquantities are also sinusoids with constant amplitude
and a 90 phase difference between them. Under transient conditions they
are arbitrary time varying quantities. Thus, for balanced sinusoidal
conditions, the space vector rotates in counter clockwise direction with
angular frequency equal to frequency of the abcvoltages, and describes a
circle of radius(3/2) being the peak of the phase voltage.
The instantaneous output voltages of the three-phase inverter shown in Fig.
22-21acan assume ei ht different combinations based on which of the six
35, 36
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MOSFETs are on. The space vectors for these eight combinations are shown
in Fig. 22-21b. For example, vector V denoted by (100) corresponds to
switch states = V , = 0, and = 0. The vectors V (000) and
V (111) have zero magnitude and are called zero vectors.
Synthesis utilizing the idea of space vectors is done by dividing one switching
time period into several time intervals, for each of which a particular voltage
vector is output by the inverter. These time intervals and the vectors applied
are chosen so that the average over one switching time period is equal to the
desired output voltage vector. For the reference voltage vector , shown in
Fig. 22-21b, the nonzero vectors adjacent to it (V and V ), and the zero
vectors (V and V ) are utilized as shown in Fig. 22-21c. Relative values of
time intervals t and t determine the direction, while ratio oft to the
switching time period determines the magnitude of the output vector
synthesized. The formulae for time intervals are as follows:
(22-46)
(22-47)
t /2 = T /2-(t + t )
(22-48)
where is the angle of the vector measured from the axis. The
maximum obtainable average vector lies along the hexagon connecting the
six nonzero vectors. As stated earlier, balanced three-phase sinusoidal
quantities describe a circle in the plane. Thus, to synthesize distortion
free and balanced three-phase sinusoidal voltages, the circle must be
contained within the hexagon, i.e., with a maximum radius of . This
gives the maximum peak value of line-to-line voltage obtained with SVM as
. This is significantly higher than that obtained using sine triangle
PWM: .
4
AN in BN CN 0
7
1 3
0 7
1 3 0
0 sw 1 3
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Further, the sequence and choice of vectors applied can be optimized to
minimize number of switchings and ripple in the resulting currents. There
are several variations of SVM, each suited to a different application. SVM can
be easily implemented digitally using microcontrollers or DSPs, and is
advantageous in control of three-phase ac machines using vector control
and direct torque control (DTC). Experimental waveforms for an SVMinverter are shown in Fig. 22-22.
Figure 22.22. Experimentally measured PWM signal and line current
for one phase of a three-phase SVM inverter: (a) 60 Hz synthesis; (b)
20 Hz synthesis.
22.5.4. Multilevel Converters
The converter topologies described so far are based on a two-level converter
leg (bi-positional switch), where the output voltage of each leg ( ) can be
either zero or V . The converters are therefore called two-level converters.
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40-44
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In two-level converters, all the switches have to block the full dc bus voltage
(V ). For high-power applications IGBTs and GTOs are used as the
semiconductor switches. These have higher voltage and current ratings and
lower on-state voltage drop compared to power MOSFETs, but cannot switch
as fast. In some applications like some motor drives and utility applications,
even the voltage ratings of available IGBTs and GTOs is not sufficiently high.
Simple series connection, to achieve a higher blocking voltage, has problems
of steady-state and dynamic voltage sharing. Moreover, due to the low
switching frequency of high-power switches, the output voltage and current
quality deteriorates. These issues are addressed by multilevel converters. In
a multilevel converter, the output of each phase leg can attain more than
two levels leading to improved quality of the output voltage and current. The
circuit comprising each leg and its proper operation ensure that voltage
blocked by the switches reduces as the number of levels is increased. In
addition, multilevel converters are modular to some extent, thereby making it
easy to scale voltage ratings by increasing the number of "cells."
22.5.4.1. Multilevel PWM.
For two-level PWM, comparison of the control voltage with a triangle wave
generates the switching signal for the top switch, while the bottom switch iscontrolled in complement to the top switch. Each of these two states
corresponds to the two levels of the output voltage. For multilevel
converters, there are more than two effective switch states, each of which
corresponds to an output voltage level. For example, in a three-level
converter there are three effective states q(t) = 0, 1, 2, corresponding to
output voltage levels (t) = 0, V /2, V . The control voltage (t) is
compared with two triangle waves to obtain two switching signals q (t) andq (t), and the effective switching signal can be obtained as q(t) = q (t) +
q (t) as shown in Fig. 22-23. The output voltage is then given by = q(t)
(V /2). Switching signals for the individual switches are derived using q(t)
and the circuit topology. For the waveforms in Fig. 22-23, f = 60 Hz and V
= 2 kV. Since the v waveform is closer to desired sinusoid in the three level
case, the output voltage has lower THD even if the switching frequency is
low. For three-phase converters, space-vector-based PWM can be used for
generating the switching signals, the advantage in the multilevel case
compared to the two-level case being the significantly higher number of
output voltage vectors.
in
45,46
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1
2 1
2 AN
in
s in
AN
47
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Figure 22.23. Multilevel triangle comparison.
22.5.4.2. Multilevel Converter Topologies.
The chief multilevel converter topologies are diode clamped, flying capacitor,
and cascaded full bridge.
22.5.4.2.1. DIODE CLAMPED CONVERTER.
Figure 22-24ashows one phase leg of a three-level diode-clamped
converter. The input dc bus is split by means of capacitors. Pairs of
switches are turned on to obtain three different voltage levels for the output
voltage = 0, V /2, V as shown in Fig. 22-24c. It is evident that this
circuit acts like a tri-positional switch connecting the output to one of three
positions of the input dc bus. The minimum voltage at point "b1," and the
maximum voltage at point "b2," is clamped to V /2 by the blocking diodes
D and D , respectively. Thus, all the switches have to blockV /2 during
their off state. This topology can be extended to more number of levels.
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AN in in
in
b1 b2 in
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However, it is eventually limited by the voltage rating of blocking diodes,
which have to block increasing voltages as the number of levels is increased.
One-phase leg of a five-level version is shown in Fig. 22-24b.
Figure 22.24. Diode clamped converters: (a) one phase of a three-
level converter, (b) one phase of fivelevel converter, (c) switching
states in a three-level converter.
22.5.4.2.2. FLYING CAPACITORCONVERTER.
Figure 22-25 shows the topology of a three-level flying capacitor converter.
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The basic idea here is that the capacitor C is charged to half the input dc
voltage by appropriate control of the switches. The capacitor can then be
inserted in series with the output voltage, either adding or subtracting V /2,
and thereby giving three output voltage levels.
Figure 22.25. Three-level flying capacitor converter.
22.5.4.2.3. CASCADED FULL BRIDGE CONVERTERS.
In this scheme, single-phase H-bridges shown in Fig. 22-19aare connected
in series at the output to form one single-phase circuit as shown in Fig. 22-
26a. Three separate circuits are required for a three-phase implementation.
A delta connection of cascaded converters is shown in Fig. 22-26b. Since all
the H-bridges are same, the circuit is modular and can be scaled by adding
more H-bridges. However, dc sources at the input of all H-bridges have to be
isolated from each other. It is also possible to combine different types of H-
bridgesIGBT-based fast switching type and GTO-based slower switching
typeor have different dc bus voltage magnitudes in different bridges to
optimize losses or increase effective number of levels. One example of the
cascaded approach is the multilevel drive offering from Robicon, now a part
of Siemens. In some solar inverters the dc input (PV panel) is common and
the isolation is carried out by transformers at the output of the H-bridges;
in
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the transformer secondaries are then connected in series to obtain the
stepped waveform construction of the AC voltage.
Figure 22.26. Cascaded converters: (a) one phase; (b) three-phase
connection in delta.
22.5.4.2.4. OTHERMULTILEVEL CONVERTERS.
The recently proposed modular multilevel converter uses series connected
cells that together generate the required voltage for each phase. The dc
voltages to each cell have to be isolated similar to the case of cascaded
converters. The major advantage of this approach is scalability and
redundancy. Other types of multilevel converters proposed recently are the
interconnected multilevel converter and the Hexagram converter.
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Citation
H. Wayne Beaty; Donald G. Fink: Standard Handbook for Electrical Engineers,
Sixteenth Edition. DC TO AC CONVERSION: INVERSION, Chapter (McGraw-Hill
Professional, 2013), AccessEngineering
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