Valery Vodovozov
POWER ELECTRONIC
CONVERTERS
OF MOTOR DRIVES
Valery Vodovozov, Power Electronic Converters of Motor Drives. 2006.
Valery Vodovozov received PhD Degree in Electrical Engineering from
St. Petersburg Electrotechnical University, Russia, where he works since 1976
as associated professor and senior researcher. His teaching includes electric
drives, computer science, electronics, and programming of electromechanical
and informational systems. The scientific interests and major fields of research
spread on applying object-oriented technologies in industry and education.
In addition, he teaches at St. Petersburg Institute for Continuing Professional Education and
gives a number of courses on disciplines mentioned above in industrial and customs training
centers. He served as a visiting researcher in Scientific Research Laboratory of Ford Motor
Company, USA in 2000. In 1997, 2002, and now from 2005 he is a visiting professor of
Tallinn University of Technology, Estonia.
Professional associations include the Fellow of the Russian Society of Electric Drive
Engineers, Estonian Society of M. H. Jacobi, and the Nordic Network for Electric Drives. He
has been selected as International Man of the Years 1999 and 2000 by the International
Biographical Center of Cambridge, included into Marquis "Who’s Who in the World", "Who’s
Who in Science and Engineering" (1998…2006), and "Famous Russians" (Moscow,
1999…2000)
Valery Vodovozov is the author of the monographs “Power Electronic Converters” (Tallinn:
TUT, 2006), “Теория и системы электропривода” (С.- Петербург: ЛЭТИ, 2004), “Техника
программирования на VBA, Pascal и C++” (С.-Петербург: ЛЭТИ, 2001), “Basics of
Informational Technologies” (Deaborn, MI: SRL, 2000), “Практическое введение в
информационные системы” (С.-Петербург: Поликом, 1995), “Микропроцессорные
системы программного управления” (С.-Петербург: Энергоатомиздат, 1994), “Роботы в
судокорпусных производствах” (С.-Петербург, Судостроение, 1986). More than 200 his
inventions, brochures, papers, and appliances have been published.
Copyright © 2006, Valery Vodovozov
3
Contents
Designations ......................................................................................................................... 4
Symbols ............................................................................................................................ 4
Abbreviations..................................................................................................................... 4
Introduction ........................................................................................................................... 6
1. AC/DC Converters – Rectifiers .........................................................................................13
Topologies........................................................................................................................13
Selection of Rectifier Components....................................................................................15
Gate Circuits.....................................................................................................................20
2. AC/AC Converters – Changers.........................................................................................27
Topologies........................................................................................................................27
Design of Cycloconverter..................................................................................................29
Design of DC Link Converter ............................................................................................31
3. DC/AC Converters – Inverters ..........................................................................................34
Topologies........................................................................................................................34
Block Modulation ..............................................................................................................37
Pulse-Width Modulation....................................................................................................41
Space Vector Modulation..................................................................................................44
4. DC/DC Converters – Choppers ........................................................................................50
Step-Down Choppers .......................................................................................................50
Step-Up Choppers............................................................................................................52
Choppers Calculation .......................................................................................................55
References...........................................................................................................................58
Monographs .....................................................................................................................58
Dictionaries.......................................................................................................................62
List of Journals .................................................................................................................63
Index ....................................................................................................................................64
4
Designations
Symbols
A
С
G
L
R
S
VD
VS
VT
Z
C
amplifier
capacitor
generator
inductor, choke
resistor
switch
diode
thyristor
transistor
load
capacitance
cos ϕ
f
I
k
L
m
P
q
r
R
t
power factor
frequency
current
factor
inductance
number of phases
power
duty cycle
ripple factor
resistance
time
T
U
U,V,W
X
Z
w
period
voltage
phase system
reactance
impedance
number of coils
α
η
ϕ
ω
firing angle
efficiency
phase angle
angular frequency
Abbreviations
A
ADC
ac
BJT
CSI
DAC
dc
DSP
EMC
EMF
F
FET
G
GTO
H
Hz
IGBT
JFET
Ampere
analog-to-digital converter
alternating current
bipolar junction transistor
current source inverter
digital-to-analog converter
direct current
digital signal processor
electromagnetic compatibility
electromotive force
Farad
field-effect transistor
Giga = 109 (prefix)
gate turn-off thyristor
Henry
Hertz
insulated gate bipolar transistor
junction FET
k
M
m
MOS
MCT
n
p
PWM
RAM
ROM
rms
s
SCR
V
VSI
W
kilo = 103 (prefix)
Mega = 106 (prefix)
milli = 10-3 (prefix)
metal-oxide semiconductor
MOS-controlled thyristor
nano = 10-9 (prefix)
pico = 10-12 (prefix)
pulse-width modulation
random access memory
read only memory
root mean square
second
silicon controlled rectifier
Volt
voltage source inverter
Watt
µ
Ω
micro = 10-6 (prefix)
Ohm
6
Introduction
Power electronic converter. An electric drive is an assembly of electronic system,
electrical motor, and mechanical transmission joined together to drive the mechanical load
from electrical energy. A power electronic converter is the part of electronic system, which
converts electrical energy supplying the motor. Successes in the fields of electronics and
materials production determine the situation and trends in the world of drive technology. The
circuit symbols of the four main classes of power electronic converters employed in the
modern electric drives are depicted in Fig. I.1:
•
•
•
•
AC/DC converters called rectifiers that convert input ac voltage Us to dc with
adjustment of output voltage Ud and current Id (Fig. I.1, a);
DC/AC converters called inverters that produce output ac voltage Us of
controllable magnitude and frequency from input dc voltage Ud (Fig. I.1, b);
AC/AC converters called changers that change ac frequency, phase, magnitude,
and shape including an intermediate dc link (Fig. I.1, c);
DC/DC converters called choppers that change the dc voltage and current levels
using the switching mode of semiconductor devices (Fig. I.1, d).
In turn, each converter consists of the primary electronic elements that are: resistors,
capacitors, transformers, inductors (choke coils), etc., and basic classes of semiconductor
devices:
•
•
•
diodes, including Zener, optoelectronic and Schottky diodes, and diacs;
thyristors, particularly silicon-controlled rectifiers (SCR), triacs, gate turn-off
(GTO), and MOS-controlled thyristors (MCT);
transistors, such as bipolar junction (BJT), field-effect (FET), and insulated gate
bipolar (IGBT) transistors.
~
=
Us
Ud
Ud
Us
=
~
a.
b.
~
=
Us in
Us out
Ud in
Ud out
~
=
c.
d.
Fig. I.1
7
An electric motor represents a specific load of power electronic converter as a composition of
three components: ohmic resistance, inductance, and counter-electromotive force (EMF).
Moreover, usually these components change in values during the system operation. Motor
resistance depends on temperature, inductance changes with the rotor position replacement,
and EMF is a function of the rotor speed.
Many applications make high demands to the properties of a drive that arose as a result of
even faster technological processes, increases in machining cycles, and associated
production efficiency. Accordantly, there are some groups of requirements to power
electronic systems: electrical rules, constructional ones, accidental protection needs, and
electromagnetic compatibility.
Electrical requirements. Electrical rules regulate the kind of electrical power and
technical characteristics of the primary supply circuit and the load. In this connection, the
input and output requirements are distinguished.
In the case of ac supply, input voltages, currents, number of phases, and frequencies are
rated as well as the quality of electrical supply as a whole. In the group of quality factors, the
steady state and dynamic stability, possible non-sinusoidal shape of waves, time, and
periodicity of dynamic disturbances are included. In the case of low-power supply source, the
input current harmonic content, power factor, and timing diagrams are limited. In the case of
long cabling distances, the wiring electrical resistance is to be taking in mind. When the high
harmonic currents flow through the cables, the distributed inductances and distributed
capacitances levels are significant also. Thus, the resonant phenomena and signal shape
distortion should be eliminated.
In the case of dc supply, input rated voltages and currents are indicated also as well as the
power supply quality. Among the quality factors, the steady state and dynamic stability,
possible time and periodicity of dynamic disturbances are listed. The ripple level and
frequency are the important features of dc supply. Again, in the case of low-power source,
the supply line dynamic value of electrical resistance is limited. This parameter helps to
evaluate the influence of the input current on the output voltage, commutation spikes and
drops under the load rising and lowering.
A desired converter is the supply source having the necessary outputs range. The output
requirements are similar to the input ones. These properties usually show the result of the
power electronic converter design. The rated output values have to response the standards
and are to be adjusted in accordance with the consumer needs and voltage drops in cable
paths.
Electronics devices are very sensitive to the instantaneous values of currents and voltages.
For this reason, to withstand any short-term overloading and overvoltages, it is required to
increase the number of semiconductor devices or to use more powerful components, which
may operate under the higher currents and voltages. As a result, to meet overload needs
without overtemperature the apparent power of converter should have some derating.
Constructional requirements. Nowadays, when automation is the full swing in all
areas of the engineering sector, the electric drive is dominant, and the mechanics of the
machines have been greatly simplified by using electric drives. Thanks to modern
technology, drive arrangements are much easier to use than they were some years ago.
8
Electronics provides a wide range of application-related options, interfaces to all controls,
and the ability to use computers to commission, optimize, and calibrate equipment.
Converters’ construction depends on the maintenance conditions and functional place of
converter: autonomous, built-in, or a part of other device. Autonomous module is the most
commonly employed type, thus the requirement of standard housing is the typical one.
Mechanical resistances to shocks, vibrations, etc. are another converter characteristics.
Methods of control, repair, and reconstruction processes are very important also.
When humidity and water influence are high as well as aggressive environment, hermetic
sealing is the solution of problem. The same concerns the storage condition of converters.
A designer must take into account the full set of grounding standards. The grounding
methods and elements should have the constant transient rated resistance during the full
time of duty. Other standards and technical rules concern different technological and
production modes. Particularly, they are: electrical connectors and leads, marking, signal
sizes and levels, cabling circuits, metrological devices, ergonomics, etc.
The need in transformers and chokes. In the case of industrial mains, a sine-wave
voltage supplies a power converter. Voltage fluctuations can affect how the drive works.
Within the rated voltage range, the drive functions normally. If the range is exceeded, the
drive will have to shut down to prevent damage occurring. The mains voltage frequency is of
minor significance.
Instead of direct link, two methods of the converter connection to the supply mains are:
through the transformer and through the current-limited chokes (Fig. I.2). When the midpoint
full-wave rectifiers are used, the only decision is the center-tapped transformer. The center
tap serves as an electrical neutral or center of the secondary winding. In other circuits, the
optimum decision must be found.
The use of transformer leads to growing the mass and size of the application as well as
decreasing its efficiency. Inductance of semiconductors’ anode and cathode circuits
becomes the reason of slow commutation and converters’ energy consumption.
U in
U out
Line chokes Input filter
or transformer
Overvoltage
protection
Fig. I.2
Power electronic
converter
Output filter
M
9
On the other hand, the inductive elements in supply lines limit the rate of devices current in
the case of short circuits therefore simplify protection requirements. For this purpose, the line
chokes on the supply side are especially effective. The choke, in conjunction with design
measures in the power section of the converter, completely replaces other customary inrush
current-limiting charging components. It minimizes noise on the supply lines and is part of the
unit security features against transient overvoltages. Moreover, transformers step down
supply voltage level in accordance with the converter capacity, thus provide the most
effective use of electronic components.
Accidental protection. All maintenance accidents are of two kinds: internal and
external. The source of the internal accidents is the component error or parameter instability.
The reason of the external accidents deals with supply power tolerance exceeding. To avoid
the accidental processes rising, different kinds of protection systems are used in electronic
circuits (Fig. I.3). They are distinguished by the method of operation and circuit
implementation.
Rectifiers are the background of electronic circuits in many respects. The source of their
internal accidents is the thyristor and gate driver destroying. Overvoltage and overheating
lead to the great current flow and short-circuiting between phases that destroy other
thyristors, transformers, and other devices connected to the supply line. Maximum short
current value may exceed the double rated current amplitude. That is why the current
derating of semiconductor devices is the first method to save the converter.
The simple protection way is the use of fast fuses. Some fuses include different alarm means
with micro switches. In the case of thyristor parallel connection, the fuses are the main shortcircuit security. But the destroyed fuses require replacing that is their drawback. Moreover,
fuses do not defend the converter against overloading.
Fast circuit breakers built in the converter input circuit provide repetitive converter protection
with possible remote control. They usually switch off the shorts during the units of
milliseconds and switch off the continuous overcurrents as well. As a rule, they provide
sufficient protection against overloading for normal operation with low starting frequencies,
U in
U out
Mains
fuses
Circuit
breaker
Chokes and
filters
Fig. I.3
Switches
blocking
Switch
cabinet
M
10
short run-up times, and starting currents that are not too high. Thus, they are not exclusively
short-circuit protection.
More effective are the gate pulses blocking and the transmitting of the rectifier to inverter
mode. These methods require the additional current and voltage sensor circuits or the
temperature-dependent appliances using thermistors or bimetallic switches in the motor
windings. Such arrangements respond at the maximum permissible load temperature that
they measure where it occurs. They defend against the excess current, intensive switching,
moving load stalling, 1-phase start-up, voltage and frequency deviation, insufficient motor
cooling, and motor bearing damage.
Snubbers, chokes, and reactors are the protection equipment also. The overvoltage
protection is implemented by means of capacitors, surge arresters, and varistors. Inside the
power supply, the surge suppressor circuits defend the power section against damage that
may be caused by voltage peaks, which occur when inductive and capacitive loads are
connected to the mains.
Line-fed inverters need additional security methods against the pull-out mode. Pull-out is the
simultaneous conduction of rectifying and inverter groups that leads to short between ac
phases and dc chain. The fast circuit breaker in the dc line protects the converter from this
accident.
The main idea of offline inverters safety is to switch the inverter off the dc bas. The dc circuit
breakers and static contactors help to avoid accidents in these circuits. Instantaneous and
mean current sensors serve as another way of inverter protection. Their signals change
modulation modes by a way that block the transistors and stop their conduction in
overcurrent.
Electromagnetic compatibility. The presence of unwanted voltages or currents in
electronic equipment, which can damage the system or degrade its performance, is called
interference. A frequency spectrum of electromagnetic interference covers a wide range from
dc up to the GHz range. Electromagnetic compatibility (EMC) refers to the ability of
equipment to function satisfactory without producing emissions that degrade the performance
of other equipment and also are not affected by emissions from other equipment.
Harmonics suppression is a matter for the power electronic designer and suitable internal
measures can keep such emission under control. There are three methods of reducing the
harmonic currents:
•
•
•
the first method needs the installation of chokes and capacitors between the
power supply and converter;
the second method is to use a harmonic series LC filter tuned for particular
frequencies close to the equipment;
the third one deals with the implementation of multiphase devices.
A choke can be built internally in the dc link or connected externally at the input terminals of
the converter. The input bulk capacitor is usually placed between the power supply and
converter in the case of dc power converter. Being relatively large in value, it has the
responsibility of storing high and low frequency energy required by the supply during each
power cycle. It is usually made up of at least two capacitors, an electrolytic capacitor for the
11
current harmonic components and a ceramic capacitor for the switching frequency
harmonics. The input capacitor charges at low frequency and sources current over a much
higher frequency range. A further measure, which reduces emission into both supply and
load circuits is to fit a ferrite ring around the output cable power conductors. The ring fits
around the power cores but not the earth.
The most common types of harmonic filters employed in industry are series LC filters with
some damping resistance. Filters may be of relatively simple single-tuned construction, but
are usually the more sophisticated, 2nd or 3rd order filters to provide a wider frequency band.
LC filter between the input line and the converter usually serves a dual purpose. First, this
tank circuit acts as a, which reduces the conducted noise leaving the switching supply back
into the input line giving a reduction of typically 30 dB in overall emission into the supply line.
The low-pass cutoff frequency of this filter should be no higher than 2 or 3 times the supply’s
operating frequency. The capacitors must be safety types with voltage rating suited to the
supply voltage with respect to earth. Values in the range 100 nF to 2.2 µF can be used. The
second purpose of this stage is to add small impedance between the input line and the bulk
input capacitor (if presents). It reduces any transient voltages, spikes or surges. The filter
specified by the load manufacturer should be used, and any limits on cable length or
capacitance and on switching frequency adhered to. The output LC filter section of ac load
uses a series inductor. In the case of dc load, it is a series inductor followed by a shunt
capacitor. Its purpose is to store energy for the load during the times when the power
switches are non-conducting. It basically operates like an electrical equivalent of a
mechanical flywheel.
The use of converters of higher pulse numbers will greatly reduce the lower order harmonics.
The frequencies of high order harmonics increase, therefore the shapes of input and output
current approach to sinusoidal waveforms. Alternately, two converters of lower pulse
numbers can be combined with a phase shift of π / 6 radians to produce a system of higher
pulse numbers. When several similar controlled converters are connected to the same bus
some cancellation of harmonic currents takes place due to phase shifts between the firing
angles of converters running of different speeds.
Braking. A motor serves as a specific converter load having the braking modes of
operation. When the motor is decelerating, kinetic energy is converted into electrical energy
and this is fed back into the dc link. As the capacity of the compensative capacitor is limited,
the voltage in the dc link rises. To enable the motor to decelerate, this additional energy must
be dissipated. It is therefore necessary to store regenerated energy or to convert it into other
forms of energy. There are basically three possibilities for this (Fig. I.4):
•
•
•
energy feedback to the mains (electrical energy becomes accessible to other
consumers);
brake chopper and braking resistor (energy is converted into heat);
exchange of energy in multi-motor applications (electrical energy feeds other
motors connected to the same converter).
The advantage of the mains energy feedback is that energy is fed back into the supply
network and therefore remains available as electrical energy. For this form of braking, the
12
U in
U out
Braking
converter
Brake chopper
and braking resistor
M
Power electronic
converter
Fig. I.4
converter is expanded by additional circuits those mass and capacity often exceed the main
converter.
In contrast to mains energy feedback, energy of braking resistor is not fed back into the
supply. If only small braking energy is produced, it may be less expensive to use a brake
chopper with an external resistor rather than the additional braking converter.
Summary. An electric motor acts like a specific load of power electronic converter
being a composition of three components: ohmic resistance, inductance, and EMF.
Moreover, usually these components change in values during the system operation.
Accordantly, there are a number of special requirements to power electronic system of
electric drives. They include: electrical rules, mechanical restrictions, accidental protection
needs, electromagnetic compatibility standards, and braking arrangements.
13
1. AC/DC Converters – Rectifiers
Topologies
Rectification. AC/DC converters serve as rectifiers. They convert ac to dc in a
number of industrial, domestic, agricultural, and other applications.
The basic rectifier topologies are given in Fig. 1.1. The systems built on diodes are called
uncontrolled rectifiers, and those built on thyristors are known as controlled rectifiers
because their dc output can be changed. The rectification processes are quite varied, and
there are different types of rectifying circuits:
•
•
•
midpoint and bridge rectifiers,
single-phase and three-phase rectifiers,
half-wave and full-wave rectifiers.
They differ by the shape of dc signal, ripples, and efficiency that are, rms, average, and
amplitude values of voltage, current, and power. Their power range is very wide, from
milliwatts to megawatts. Low-power devices operate usually from a single-phase supply
while high-power rectifiers are mainly used in a three-phase configuration.
Single-phase rectifiers. The simple single-phase half-wave rectifier circuits (M1) are
represented in Fig. 1.1, a, b. During the positive alternation of the ac sinusoidal wave the
anode of the diode VD is positive and the cathode is negative, the diode will conduct since it
is forward biased. The positive alternation of the ac will then appear across the load motor M.
During the negative alternation of the ac cycle, the anode becomes negative while the
cathode is positive. The diode is reversed biased by this voltage and no significant current
will flow through the load. Therefore, no voltage will appear across the load. Such type of the
converter is called a half-wave converter because the negative half cycles have been clipped
off. Since the load voltage has only a positive half cycle, the load current is unidirectional and
discontinuous, meaning that it flows in only one direction and has breaks.
Bi-thyristor single-phase full-wave rectifier (midpoint rectifier, or M2) is shown in Fig. 1.1, c. It
produces a rectified rippled output voltage for each alternation of the ac input. The rectifier
utilizes a center-tapped transformer that transfers alternating source voltage to the diode
rectifier circuit. The anodes of each device VS1 and VS2 are connected to the opposite ends
of the transformer’s secondary winding. The cathodes are then joined together to form a
common positive output. The load motor M is connected between the common cathode point
and the center-tap connector of the transformer. The transformer, two diodes or thyristors,
and the load form a complete path for the current.
Four diodes or thyristors are interconnected to form a full-wave single-phase bridge rectifier
(B2) shown in Fig. 1.1, d. By using four devices instead of two, this design eliminates the
need for a center tap. During the performance of a bridge rectifier, two diodes are forward
biased in each alternation of the ac input. When the positive alternation occurs, the devices
VS2 and VS3 are forward biased, while VS1 and VS4 are reverse biased.
14
VD
Us
Ud
VS1
M
U2
a.
U1
VS
Ud
M
U2
Us
Ud
VS2
M
c.
b.
U1
VS1
U2
VS1
U
VS3
VS2
Ud
Us
VS2
M
V
VS3
VS4
M
W
Ud
e.
d.
VS1
U1
VS2
VS3
U2
U
V
M
Ud
W
VS4
VS5
VS6
f.
Fig. 1.1
This biasing conduction is due to the instantaneous voltage that occurs during the positive
alternation. The conduction path is from the ac source, through VS3, the load, then through
VS2, and back to the source. This causes the same alternation to appear across the load.
During the negative alternation, the current flows from the source through VS1, the load, then
through VS4, and back to the supply line. This causes the second alternation to appear
across the load in the same direction as the first alternation. This means that voltage
developed across the load is the same for each alternation. As a result, both alternations of
the input appear as output across the load and pulsating current flows via the dc output.
15
Three-phase rectifiers. Three-phase three-diode rectifier circuit (M3) produces a
purer direct voltage output than the single-phase rectifier circuits do, thus wasting less
power. In Fig. 1.1, e, phases U, V, and W of the three-phase source are connected to the
anodes of thyristors VS1, VS2, and VS3. The load motor M is connected between the
cathodes of the thyristors and the neutral of the wye-connected source. When the phase U is
at its peak positive value, maximum conduction occurs through VS1, since it is forward
biased. No conduction occurs through VS1 during the negative alternation of phase U. Other
thyristors operate in similar manner, conducting during the positive ac input alternation and
not conducting during the associated negative ac alternation.
The 6-pulses counterpart of this rectifier circuit is represented in Fig. 1.1, f. This three-phase
bridge rectifier (B6) uses six diodes. The anodes of the thyristors VS4, VS5, and VS6 are
connected together at one point, while the cathodes of VS1, VS2, and VS3 are jointed at
another point. The load is connected across these two points. This circuit does not require
the neutral line of the three-phase source; therefore, a delta-connected source as well as a
wye-connected source could be used. The voltage ripple is low because the output voltage
consists of six pulses per unit voltage period. The switching order of the thyristors in
Fig. 1.1, f is VS1, VS6, VS2, VS4, VS3, VS5. At least two devices are simultaneously in the
open state here.
Summary. Single-phase half-period rectifier is the simplest one. Nevertheless, it has
poor secondary current shape, very high ripple level, and very low power factor.
The main disadvantage of the two-diode full-wave rectifier is the requirement of center-taped
transformer.
Single-phase bridge rectifier better uses the transformer and semiconductor devices; its
current shape is more sinusoidal. That is why it is the best decision for low-power (up to
1 kW) applications.
Low degree of the transformer use and low power factor are the main disadvantages of the
three-phase three-diode rectifier. Nevertheless, enough high quality of rectified voltage with
small ripples is its main advantage.
The three-phase bridge rectifiers are predominant because of their good technical properties:
low ripple, high power factor, simple construction, and low price. Nowadays, they are used
both in powerful and in small-power suppliers as well as in AC/AC converters with dc link.
Selection of Rectifier Components
Transformer. In Fig. 1.2, a, a transformer-isolated rectification circuit is shown. Here,
the transformer steps the rms supply voltage U1 down to lower level U2, which sometimes is
more suitable for use in rectifiers. The number of turns of the primary winding is w1; the
number of turns on the secondary winding is w2. The voltage induced in the secondary
winding is equal to:
U2 = U1w2 / w1,
therefore the secondary current is given by:
I2 = I1w1 / w2.
16
w1
∼
w2
U1
U2
Ud
=
Fig. 1.2
Here, w2 / w1 is the transformer turns ratio. The secondary output apparent power of a
transformer almost equals the primary input power:
U2I2 = U1I1.
The rated power PS that feeds the load is the arithmetic mean of the secondary and primary
powers:
PS = (π / m) UdId,
I2 = Id√(2 / m)
where m = 2 or 3 – the number of phases.
In rectifier calculations, transformer inductance Ltr and resistance Rtr may be approximately
obtained as particles of the load values Ld and Rd:
Ltr ≈ (0.11…0.16) Ld,
Rtr ≈ (0.10…0.12) Rd.
Commonly, transformers are designed together with converters therefore their data sheets
are not represented in reference sources.
Chokes. When chokes are used instead of transformer, their current and inductance
are calculated as follows:
I = √(2Id / 3),
L = kUs / (2√2dIF / dt)
where Us is the phase supply voltage; dIF / dt is the current slew rate of the rectifier device;
k = 1.2…1.3 – safety factor.
Rectifiers’ data. In the table below, the main data of different non-controlled rectifier
circuits with a resistive load are given.
Circuit
type
kU =
Us / Ud
kI =
Is / Id
kP =
Ps / Pd
M1
M2
B2
M3
B6
2.22
1.11
1.11
0.85
0.42
1.57
0.71
1.00
0.58
0.82
3.10
1.34
1.11
1.35
1.05
cos ϕ =
Pd / Ps
0.29
0.75
0.90
0.73
0.95
kR =
UR / Ud
kF =
IF / Id
r=
Ur / (2Ud)
3.14
3.14
1.57
2.09
1.05
1.00
0.50
0.50
0.33
0.33
1.57
0.78
0.78
0.25
0.06
17
Parameters of different midpoint (M) and bridge (B) rectifying circuits are represented in this
table. The input rms voltage Us and current Is supply the rectifier directly, through chokes, or
via transformer. The average rectified load voltage Ud and current Id are pulsating dc signals.
They have 1, 2, 3, or 6 pulses per supply period T. The ripple factor of the output waveform
is usually determined by:
r = Ur / (2Ud)
where Ur is the rectified peak-to-peak ripple voltage. The peak inverse voltage UR of each
rectifier device depends on the ripple also. The power factor of a rectifier is:
cos ϕ = Pd / Ps
where Pd is the output dc power of a rectifier, Ps is the input apparent power, and ϕ is a
phase displacement angle of current relative to voltage.
Voltages and currents. In the selection process, the restricted parameters of the
rectifier devices are to be taking in mind, such as the peak inverse repetition voltage UR and
average direct current IF. In M1, M2, and B2 rectifiers, diode and thyristor rated inverse
voltage must exceed the value
UR = k √2Us
whereas in M3 and B6 circuits
UR = k √2√3Us
where k = 1.7…1.85 – safety factor for the repetitive and short-term overvoltage protection,
Us is the phase supply voltage. In transformer-isolated circuits, Us = U2.
Device rated current exceeds the value
I F = k kF I d
where k = 2…3 – safety factor for the overcurrent protection, kF is the circuit factor taken from
the rectifiers’ data table cited above.
In process of possible current evaluation, the cooling conditions play an important role.
Devices’ connection. When the required current is high, the parallel rectifiers
connection is used as shown in Fig. 1.3, a. Because the devices resistances are different,
the common forward current IF is distributed unevenly (Fig. 1,3, b). To avoid the devices
destroying by overcurrent, the inductive dividers of different kinds are recommended.
In the case of high voltages, the series rectifiers connection is used as shown in Fig. 1.3, c.
Here, the common reverse voltage UR is distributed unevenly (Fig. 1,3, d). To avoid the
devices destroying by overvoltage, the resistive dividers are commonly needed.
Other parameters. Other parameters of great importance are the reverse current and
transient times. From the reverse recovery time point of view, the special-purpose avalanche
diodes are preferable, which can withstand high short-term reverse overvoltages and
currents. Another devices of choice in new high-speed power applications are the Schottky
diodes that are much faster than the ordinary rectifier diodes. The epitaxial and diffused
diodes are very fast and high-voltage ones as well.
18
IF1
VD1
VD1
IF
VD2
UR1
VD2
Us
Ud
UR2
Us
M
UR
Ud
M
IF2
a.
c.
IA
IA
IF1
UR1
UR2
IF2
UAC
IR
UAC
b.
UF
d.
Fig. 1.3
Transformer checking. After diodes or thyristors choosing, the transformer turns
ratio and capacity should be checked. For rectifiers, the output voltage is described as:
Ud = Ud* + kUAC + Id Rtr + kmf Id Ltr
where Ud* is the required load voltage, k = 1 or more – number of the current-conducted
devices, Id is the rectified load current, m = 2 or 3 – number of phases, f is the supply
frequency, Rtr and Ltr – transformer resistance and inductance, UAC is the device forward
voltage drop. Its preliminary value may be evaluates as follows: UAC = 0.7…2 V for the
ordinary rectifier diodes; 1.1…1.6 V for the diffused diodes; 0.8…1.3 V for the epitaxial
diodes; 0.5…0.9 V for the Schottky diodes.
As a result, the transformer secondary voltage should be:
U2 > k kU U0 / cos αmin
where k = 1.1 – safety factor to replenish possible supply voltage drop, kU is the circuit factor
taken from the rectifiers’ data table cited above, U0 is the rectified voltage when Id = 0 (the
infinite load of a rectifier), αmin = 0.1…0.2 – minimum firing angle of the thyristor. In noncontrolled rectifiers,
U2 > k kU Ud.
Smoothing choke. In the thyristor and diode rectifiers shown in Fig. 1.1, the
smoothing choke is often required when the ripple exceeds 10 %. Example is in Fig. 1.4, a.
The full inductance of the load circuit is obtained as follows:
LΣ ≥ rU0 / (r*Id ω)
19
L
~
Us
Ud
=
a.
U1
b.
c.
U2
R1
F
U
~
V
С
Us
W
R2
Ud
=
d.
e.
Fig. 1.4
where Id is the load current, r is the ripple factor taken from the rectifiers’ data table cited
above, U0 is the rectified voltage when Id = 0, r* = 0.02…0.10 – reference ripple factor,
ω = 2πfm – angular frequency, f = 50 Hz – supply frequency, and m = 2 or 3 – number of
phases. The required smoothing reactance L is the rest of this value:
L = LΣ – Ltr – Ld
where Ltr and Ld are the transformer and load inductances.
Snubbers. To overcome the internal periodic overvoltages, the RC circuit may be
connected across the thyristors and transistors as shown in Fig. 1.4, b, c. The capacitance is
of 1 to 2 µF and resistance of
√(Ltr / C) < R < 2√(Ltr / C)
where Ltr is the inductance of the commutation loop (the phases of transformer or choke).
The resistor’s power is
PR = 450CUs2.
To overcome the external commutations, RC snubbers are connected across the supply
lines as shown in Fig. 1.4, d. These snubbers capacitance is given by the equation:
C = 0.05Is m / (2πfUs (kR / ku)2)
with the circuit factors, taken from the rectifiers’ data table cited above. The resistor’s
parameters are defined as:
20
R ≥ 2√(2Ltr / C),
PR = Is2R / 4000.
Compensative capacitors. The electrolytic compensative capacitor C in Fig 1.4, e,
which defends the dc link from overvoltages, smoothes the ripple in the rectifier’s output (the
input of the dc circuit). It is selected using the formula:
C > 0.15⋅10-6 Ud Id.
Other recommendations are 160…170 µF per 1 kW of the load power.
Sometimes, the ballast resistor R1 is placed in front of the capacitor. It limits the rectifier’s
current in the switch-on instant by the value kkF Id. Another resistor R2 discharges the
capacitor when the circuit switches off. The safety fuse F protects the circuit from fire.
Summary. Rectifier circuits differ in voltage and current factors indicated in the
appropriate data table. During the selection process, the restricted parameters of rectifiers
are to be taking in mind. Parallel and series connection of devices require the additional
dividers to avoid overvoltages and overcurrents.
In the thyristor and diode rectifiers, the smoothing choke is needed when the ripple exceeds
10 %. The RC snubbers may be connected across the thyristors and transistors to overcome
the internal periodic overvoltages. To exclude the external commutations, the snubbers are
connected across the supply lines. Compensative capacitors with additional circuitry defend
the dc links from overvoltages and smooth the ripple of rectifiers.
Gate Circuits
Gate circuits functions. The gate circuits of the thyristor dc and ac converters
perform next operations:
•
•
•
•
•
discrete intervals clocking for the system timing,
carrier signals producing,
control pulses generation and conversion them into the firing signals,
firing currents distribution between the thyristors,
galvanic isolation of control and power circuits.
Gate circuit structures. In Fig. 1.5, a, the gate circuit is represented for the
controlled rectifiers shown above in Fig.1.1. This electronic controller influencing the phase of
the firing pulses performs a thyristor switching on. The switching off is produced by means of
natural commutation caused by cycling of the supply voltages. The phase-shifting gate driver
compares the reference signal u* of the controller with the periodic carrier signal uc of the
carrier generator G, synchronized by the line voltage UN.
To stabilize the output voltage, a voltage feedback is used often. It adjusts the driver action
to keep the output voltage at a desired level. The idea of automatic voltage correction is
shown in Fig. 1.5, b. Here, the reference voltage enters the summer simultaneously with the
voltage sensor signal Ud. The voltage difference drives the amplifier A1, which output signal
enters the gate driver. This circuit stabilizes the rectified voltage Ud influenced by supply and
21
UN
G
uc
(∆)u*
Gate
driver
∆u*
u*
A
A1
Ud
a.
b.
UN
G
uc
(∆)u*
Gate
driver 1
PDU
&
A
S
uc
Gate
driver 2
A
&
c.
Fig. 1.5
load disturbances. For instance, if the output voltage rises, the summer difference rises also
to decrease the voltage by the firing angle increasing.
This gate circuit provides a single-quadrant operation of the rectifier, supplying the load by
the positive current under the positive voltage. For the multiphase rectifiers control, the multichannel pulse distribution unit PDU is required that produces the chain of dual pulses by
mean of the logical multiplication.
In the fully controlled rectifiers, the average dc-side voltage should be adjusted from a
positive maximum to a negative minimum value. These are the two-quadrant and fourquadrant operational dual rectifiers. They are accomplished by connecting the rectifiers in
anti-parallel (back-to-back). The first rectifier conducts when the load current is required to be
positive, and the second one when it is to be negative.
There are two common forms of dual rectifiers. In the first, both rectifiers are controlled
simultaneously to give the same mean output voltage. This is the dual rectifier with
circulating current. However, the instantaneous voltage from both devices cannot be
identical, and reactors are to be included to limit the current circulating between them. The
principal advantage of this system is that when the current is required to change direction,
there needs be no delay between the conduction of one rectifier and other.
22
Fig. 1.5, c represents the gate circuit of the three-phase dual rectifier with PDU for the
circulating current-free mode of operation. Here, only one device at time is allowed to
conduct. For such operation, the non-conducting group is blocked when other group
conducts. As shown, the pair of electronic switches & eliminates the simultaneous switching
of both thyristor groups. Logical gate S acts dependently on the current in the switching-off
group.
The cost and losses associated with the smoothing chokes and reactors may be eliminated
and economies can also be made in the control circuits without reactive components.
However, the penalty is a short time delay, as the current passes through zero, while the
thyristors in one device safely turn off before those in the second opened. This delay
introduces a discontinuous-current mode with a current-free period of typically near 10 ms.
Such circuit is by far the most common industrial four-quadrant dc system and is used in
many demanding applications where rapid control is required.
With a dc supply, there is no natural commutation available, and other methods of device
switching off have to be employed.
Operation diagram. An operation diagram of the gate circuit for the three-phase
bridge rectifier is shown in Fig. 1.6. At those very instants when the supply voltages U, V, W
cross the zero level, the carrier signal uc is generated. Its waveform may be different but the
period is exactly equal to the half of the supply voltage period π. Other entry of the gate
driver is the reference signal u*. The amplitudes of both signals, reference and carrier, are
scaled by the supply amplitude, that is their extremes Um are equal one another. Whenever
their difference becomes positive for the first time in each half period (u* > uc), the gate driver
produces a short pulse IG, which, after amplification (A in Fig. 1.5), passes through an
isolating circuit to the gate of an appropriate thyristor to be fired. For the starting and
discontinuous current modes, the broad pulses or the paired pulses are required as shown in
Fig. 1.6.
It is seen that the comparison of the referred signal u* with the carrier signal uc represents a
sampling method and provides a voltage-to-phase conversion with phase modulation. Often,
the carrier signal is a saw-tooth function or, instead of a saw-tooth, alternations of cosine
wave are employed:
uc (θ) = Um cos θ,
α = arcos u* / Um.
Here, θ is calculated from the carrier starting point, which is in the same time the natural
commutation point of the firing thyristor. The similar effect is sometimes achieved by inserting
an arcsine wave generator in the input channel of the firing circuit.
Control curves. The single-phase rectifiers shown above in Fig. 1.1, b, c, d drive the
resistive load with the discontinuous current. Their rectified load voltage depends on the
firing angle α as shows the control curve of Fig. 1.7, a:
Ud = U0 / 2 (1 + cos α).
23
Us
U
W’
U’
V
V’
W
θ
α
u*
uc
θ
θ
IG1
IG6
θ
θ
IG2
θ
IG4
θ
IG3
IG5
θ
Fig. 1.6
In the half-wave circuit shown in Fig. 1.1, b, the average value of the dc output alternation
that a voltmeter reads in non-controlled rectifier is equal to:
U0 = √2Us / π = 0.45Us.
In the single-phase full-wave rectifier shown in Fig. 1.1, c and in the single-phase bridge
rectifier shown in Fig. 1.1, d,
U0 = 2√2Us / π = 0.9Us
When the resistive-inductive load with ωL = ∝ (infinite inductance) is adjusted,
Ud = U0 cos α
with U0 = 0.9Us and continuous current flowing via the load in all single-phase circuits. The
corresponding control curve is shown in Fig. 1.7, a also. In between the two control curves,
the discontinuous current area lies.
24
Ud
Ud
U0
U0
Resistive load
0.5U0
π/ 2
Resistive load
Resistiveinductive
load
Resistiveinductive
load
α
α
π
π/6
π/2
a.
5π / 6
b.
Ud
U0
Ud
U0
Resistive load
α
π/ 2
Resistiveinductive
load
π
α
π / 3 π / 2 2π / 3 π
–U0
d.
c.
Fig. 1.7
In the three-phase full-wave rectifier shown in Fig. 1.1, e when 0 ≤ α ≤ π / 6 the continuous
current flows through the resistive and inductive loads. The average value of the dc output
voltage is equal:
Ud = U0 cos α
where
U0 = Us / kU = 3√6Us / (2π) = 1.17Us.
The corresponding control curve is shown in Fig. 1.7, b. Between π / 6 and 5π / 6, the
continuous current flows through the resistive-inductive load with infinite inductance ωL = ∝
and the control curve stores its previous shape:
Ud = U0 cos α.
In the case of resistive load, the current becomes discontinuous in this area, and the control
curve equation converts into the formula:
Ud = U0 / √3 (1 + cos (π / 6 + α))
where
U0 / √3 = 3√2Us / (2π) = 0.67Us.
The corresponding control curve is shown in Fig. 1.7, b also.
25
In the three-phase bridge rectifier shown in Fig. 1.1, f when 0 ≤ α ≤ π / 3 the continuous
current flows through the resistive and inductive loads. The average value of the dc output
voltage is equal:
Ud = U0 cos α
where
U0 = Us / kU = 3√6Us / π = 2.34Us.
The corresponding control curve is shown in Fig. 1.7, c. Between π / 3 and 2π / 3, the
continuous current flows through the resistive-inductive load with infinite inductance ωL = ∝
and the control curve stores its previous shape:
Ud = U0 cos α.
In the case of resistive load, the current becomes discontinuous in this area, and the control
curve equation converts into the formula:
Ud = U0 (1 + cos (π / 3 + α))
with U0 = 2.34Us The corresponding control curve is shown in Fig. 1.7, c also.
In the case of resistive-inductive load having counter-EMF, the gate driver may arrange the
line-fed inverting process. It is the typical breaking mode of electric drives operation. Here,
the firing angle should rise more than π / 2 as shown in Fig. 1.7, d. In the inverting mode, it is
more convenient to express the firing angle in terms of the angle of advance from the end
limit of the interval available for successful commutation than as a delay α from the
beginning of the interval. This angle of advance is usually denotes as β. Then,
Ud
Ud
α=0
U0
Id
b.
Ud
Discontinuous
current bound
Id
Id
–U0
c.
α = max
a.
Fig. 1.8
26
α + β = π,
cos α = –cos β.
Output curves. The output curve of a converter describes the relation of the load
voltage versus load current, Ud (Id). It depends on the inner resistances of converter circuit:
Ud = U0 – ∆U
where ∆U = kUAC + Id Rtr + kmf Id Ltr – the sum of voltage drops discussed earlier in the
transformer checking formula. In the case of continuous current, the output curves are joined
in the set of parallel straight lines shown in Fig. 1.8, a. They demonstrate that the more the
output current the less the output voltage. Dependently on the power, the active and reactive
components influence differently on ∆U. Usually, in the low-power rectifiers the ohmic
resistances predominate, whereas in high-power converters the inductive components serve
the main role. In common cases of restricted load currents, voltage usually drops no more
than 15…20 % of U0. In overloading modes, this value grows extensively.
When the loading is low, the current approaches discontinues mode, and the output curves
change their shape significantly. The ellipsoidal line on the output characteristics in
Fig. 1.8, a, shows the discontinuous current boundary. The discontinuous current occurs to
the left of this line and the continuous current occurs to the right. Consequently, the
characteristics in the continuous current region are linear, exhibiting only a slight droop. In
contrast, in the discontinuous current region the curves are strongly nonlinear with the loss in
output voltage. The discontinuous current boundary equation seems as follows:
Idb = U0 sin α / (2πf L) (1 + π / m ctg (π / m))
where m is the number of pulses in the rectified voltage, L is the inductance of the rectified
loop, and f is the rectified ripple frequency.
In the no-load point (zero current and idle operation) when 0 < α < π / m,
U(0) = U0 – ∆U,
thereas when α > π / m,
U(0) = U0 cos α – ∆U.
In accordance with the mode of operation, different rectifier circuits provide various output
characteristics. They may be single-quadrant (1Q, Fig. 1.8, b), two-quadrant (2Q, Fig. 1.8, a),
or four-quadrant (4Q, Fig. 1.8, c). In the first case, the load voltages and currents are
unipolar. In the second one, the load voltage may change the sign under the constant current
direction. In the last system, both the load voltage and the load current are bi-directional.
Summary. The core of a thyristor gate circuit is the phase-shifting gate driver that
compares the reference signal of controller with the periodic carrier signal of the generator,
synchronized by the supply voltage. Simple gate circuit provides a single-quadrant operation
of the rectifier whereas for more complex systems are required for the multi-channel pulse
distribution. Operation diagrams and control curves of the gate circuits depend on the
continuous or discontinuous mode of operation.
27
2. AC/AC Converters – Changers
Topologies
Classification. A switching converter that changes an ac supply to the ac supply with
alternative voltage, frequency, phase, or shape is called an AC/AC converter or changer. The
first group of such converter is the direct frequency converters that change frequency and
voltage shape. Another group joins dc link frequency converters where a rectifier is used as a
voltage regulating or constant-voltage front-end system and an inverter generates an ac
voltage of certain frequency and magnitude.
Cycloconverter. Cycloconverters are the naturally commutated direct frequency
converters that are synchronized by supply line. Commonly, they are allowed in high-power
applications up to tens of megawatts for frequency lowering. A thyristor, closing on natural
commutations, i.e. turns off on zero current, is the almost only device that can meet the
switch voltage and current rating needed at this power levels. 3-, 6-, 12-, and 24-pulse
cycloconverters are used.
One of possible circuits of the direct frequency converter is given in Fig. 2.1, a. This
polyphase cycloconverter incorporates three three-pulse controlled converters, which create
the three output voltages. Each converter has six thyristors, three to carry positive load
current and three to carry negative load current. A transformer with three complete sets of
three-phase secondary windings is used for the thyristors supply. There is no difference
whether the load is active or passive because operation in all four quadrants is possible.
DC link converter. Converters with a dc voltage link circuit are the most common on
account of their applicability. They can be used for individual or multiple motor drive
applications in all fields of machine-building and construction, mainly where the emphasis is
on load-independent stability, maintenance-free operation, and high efficiency. Due to the
characteristic of the impressed voltage in the dc link, the converters are stable when no-load
and can be disengaged from the load without damage. Usually, they come standard as 1, 2,
and 4-quadrant operation if the corresponding accessories are employed. A variable threephase output voltage of such converter is commonly rises up to the level of the input voltage
with a proportionally rising output frequency 0.1…10 to 120…400 Hz.
Fig 2.1, b represents the frequency converter having the non-controlled rectifier, the dc link,
and the inverter. Traditionally, the converter’s three-phase bridge rectifier VD1…VD6 is
connected to the supply line through the chokes or transformer that defend the mains from
the converter’s non-linear distortions. The ripple of the rectifier’s output voltage has a low
value because of filtering by the smoothing inductor L. The inductor reduces the pulse spikes
and limits the fault currents. Sometimes, the freewheeling diode VD7 shunts the inductor to
decrease its influence when the switches are off. The large electrolytic compensative
capacitor C protects the dc link from overvoltages. It “stiffs” the link voltage and provides a
path for the rapidly changing currents drawn by the inverter. In some circuits, this capacitor is
shunted by an additional RC circuit, which decreases the high-frequency obstacles. Ones the
converter is switched on, the capacitor charges and limits the circuit startup current.
28
M
a.
L
VD1 VD2 VD3
VD7
VT1
VT2
VT3
VT7
C
M
R
VD4 VD5 VD6
VT4
VT5
VT6
b.
Fig. 2.1
By adjusting the transistor inverter circuit VT1…VT6, the unidirectional link current allows the
use of a two-quadrant operation where the reverse power flow is achieved by the transistors
control. The compensative capacitor permits the dc current to be temporary raised or
lowered during commutation of the load-side inverter under the constant voltage.
The feedback diodes across the transistors provide an alternate path for the inductive current
when the switches are turned off. The diodes return the regenerated power to the dc link,
which will raise the link voltage above its normal value and the measures must be taken to
absorb this regenerated power to prevent a dangerous link voltage buildup. Typically, for the
four-quadrant operation a special brake chopper VT7 switches a braking resistor R across
the dc link capacitor C to absorb this energy. The brake chopper switches automatically on
when the dc link voltage reaches a certain level. Another way of the four-quadrant mode is to
29
arrange the bi-directional input bridge by adding a second inverse parallel transistor bridge
inverter.
Summary. Cycloconverters are used in high-power applications for lowering
frequencies of such low-speed machines as rolling mills, hoists, excavators, and screw
propellers. They do not contain energy storage in the intermediate circuit. Thanks to the
direct conversion of the input to the output frequency, they are very effective. The common
used direct frequency converters are naturally commutated cycloconverters, but their
disadvantages deal with the low frequency output, which cannot be higher than 0.4 of the
supply frequency. The power factor of the cycloconverters is low also.
AC/AC converters with a dc link have the broadest use. The minimum power of such
converters is measured by watts, and maximum may approach megawatts. The best models
can transfer energy in either direction, dependently on the circuit arrangement and switching
sequence. Nevertheless, enough high level of their voltage distortion affects the performance
of other equipment connected to the power supply system.
Design of Cycloconverter
Transformer. The transformer calculation for the circuit shown in Fig. 2.1, a, starts
from the defining of the first harmonic of secondary rms voltage:
U2 = kUs out fs out / f + mUAC / 2 + Rtr Is out
where k = 1.2…1.3 – safety factor; Us out, Is out, and fs out are the load voltage, current, and
frequency, f = 50 Hz – supply frequency, m = 2 or 3 – the number of phases, UAC = 0.7…2 V
– thyristor voltage drop, Rtr is the transformer resistance. While the transformer and thyristors
are not selected, this formula gives the draft result, which should be corrected later. The
required secondary voltage amplitude of transformer is next:
U2m = 2πU2 / (4m√2 sin (π / m) sin (π / (2m)) cos αmin)
where αmin = 0.1…0.2 – minimum firing angle.
The transformer turns ratio is given by:
w2 / w1 = U2 / U1√(3 / 2) .
The transformer rms currents are:
I1 = Is out √2w2 / w1,
I2 = Is out.
The rated powers of the transformer are as follows:
Ps = mUs out Is out,
Ptr = πPs / cos αmin √(1+3√3 / (2π)) / (3√3).
Voltages and currents. The peak inverse repetition voltage UR and the maximum
possible direct rms current IF of the cycloconverter thyristors must exceed the values
UR = k √2Us in,
30
IF = k Is out / √3.
where k = 1.7…1.85 for voltage and 2…3 for current – safety factors for the repetitive and
short-term overvoltage and overcurrent protection, Us is the phase supply voltage, Us in = U2.
When the possible current is evaluated, the cooling conditions play an important role.
When the currents are high, the parallel rectifiers’ connection is used as shown above in
Fig. 1.3, a. In the case of high voltages, the series rectifiers’ connection is used as shown in
Fig. 1.3, c.
Gate circuit of cycloconverter. The cycloconverter shown in Fig. 2.1, a, uses a gate
circuit where the frequency control channel is presented in addition to the voltage control.
Each thyristor group of the cycloconverter has the three-phase mid-point rectifier. Again,
similar to rectifiers, its gate circuit consists of the pulse generator and pulse distribution unit.
The first alternation of the output voltage is produced by the anode thyristor group; the
second by the cathode group. For such operation, the non-conducting group is blocked when
the other group conducts. The blocking gates are constructed with the logical circuit S and
the pair of switches similar to those shown in Fig. 1.5, c.
An operation diagram of the cycloconverter circuit is shown in Fig. 2.2. The supply voltage
Us in is shown by the curves U, V, W. Polarity of the reference voltage u* refers the polarity of
the output voltage Us out, the amplitude of u* displays the desired average output voltage, and
the frequency of u* represents the output frequency of Us out. During the positive half-cycle of
u* the rectified thyristors are fired, and during the negative half-cycle the inverting devices
are fired. The output voltage waveform of Fig. 2.2 displays the case when the output
frequency is the quarter of the input frequency. Because the reference voltage alters with
Us in, Us out
U
V
W
U
V
W
θ
uc
u*
θ
Fig. 2.2
31
time during half-cycle instead of remaining constant, the firing angles change in the half-cycle
as well.
The control curve of the cycloconverter supplied by the three-phase voltage Us in is described
by the equation:
Us out = 24Us in / π2 sin π / 3 sin π / 6 cos α.
Design of DC Link Converter
Transistors rating. In AC/AC converters with a dc link shown in Fig. 2.1, b, the
required dc link voltage is defined as:
Ud = mUs out qmax / √2 + mUCE / k
where Us out is the reference load voltage, qmax = 0.95…0.98 – maximum duty cycle (or duty
ratio), k = 1 or 2 – number of current-conducted transistors, UCE = 2…3 V – transistor voltage
drop, m = 2 or 3 – number of phases.
Transistor rated voltage must exceed the value
UR = k Ud
where k = 1.7…1.85 – safety factor for the repetitive and short-term overvoltages protection.
Transistor rated current should be more than
IF = k Is out
where k = 2,,,3 – safety factor for the overcurrent protection, and Is
current.
out
is the rated load
Transistors types. Bipolar junction transistors may be devices of choice in lowfrequency systems. The main parameters of BJT are next: the maximum pulse collector
current, the maximum pulse collector-emitter voltage, the switch-on and switch-off times.
When currents are high, the parallel-connected transistors with additional current-balanced
resistors Rb are used as shown in Fig. 2.3, a. For their calculation, use the maximum
(Usat max) and minimum (Usat min) saturation voltages from the data sheets for the current
misbalance limiting. Usually, the current misbalance ∆I should be less than 10 % of the
maximum collector current IF max:
Rb = (Usat max – Usat min) / ∆I.
MOSFETs are commonly used in low-power high-frequency applications. Their parallel
connection has no problematic because the transistors of the certain type have similar
threshold voltages. As a rule, they are mounted on the single sink with the minimum distance
between the cases. Their gate circuits need the additional resistors (some tens ohms) to
avoid the switch-off ringing (Fig. 2.3, b).
To overcome the overheating coursed by the short circuits, MOSFET often has an additional
signal terminal that is known as current sensor. Thanks to this signal, the protection circuit
may be arranged with a feedback loop, which speeds up the operation and increases its
selectivity.
32
VT1
VT1
VT2
Rb
VT2
Rb
Rb
a.
Rb
b.
Fig. 2.3
The most effective decision is a fully protected power MOSFET switch, which tracks the
crystal temperature and current and switches off the transistor in abnormal situations.
IGBT is the most popular transistor for switching power applications with the average
frequency range 1 to 150 kHz. Usually, IGBT lets no overvoltage but 7…10 times
overcurrent.
In IGBT data sheets the main parameters are listed. The rated collector current IF limits the
maximum possible steady-state current for definite temperatures, usually 25° and 100° C.
Often, a diagram of IF versus the case temperature is available from data sheets that helps in
the transistor choice with the predominant cooling condition. The pulse collector current IF max
is restricted by the data sheet also.
Rated collector-emitter voltage UR and its peak value UR max are the content of the rated data
as well as the maximum dissipated power PF. The maximum (Usat max) and minimum (Usat min)
saturation voltages describe the steady-state losses of the open IGBT. The threshold gate
voltage UG th shows the gate voltage provoking the collector current.
Some rated time intervals define the transistor transients. The turn-on process includes three
time intervals: the turn-on delay t0, the current rise time t1, and the current spreading time t2.
The current rise time is very short. Also, the turn-off transient process duration is the sum of
three time intervals: the turn-off delay t3, the decreasing time t4 of the collector current, and
the stabilizing time t5. The switching off is not so fast as switching on.
The parallel connection of IGBTs passes by the same way as for MOSFETs. The additional
resistors in the gate circuits help to avoid the switch-off ringing. To operate without
overheating, the operation current of the parallel-connected devices is decreased usually by
10…15 % as compared with the rated current.
Braking resistor. Resistance of the braking resistor is given by the electrical braking
power, which flows back into the converter after deduction of the losses in the load. Since the
braking power is usually not continuous but for a limited period only, this aspect can also be
considered in the dimensioning of the braking resistor PR as follows:
33
PR > P
where P is the rms load braking power related to a cycle duration of 120 s.
Summary. Cycloconverters require more complex gate circuits than those of the
thyristor rectifiers because of the frequency control channel in addition to the voltage control.
Due to the high current ripple, low efficiency, and significant noise, these circuits are not
suited for high-performance, high-speed applications.
Approaches of different kinds are used in AC/AC dc link projects with BJT, MOSFET, and
IGBT transistors. BJTs are the devices of choice in low-frequency systems. MOSFETs are
preferable in low-power high-frequency applications. IGBTs are the most popular transistors
for switching power applications with the average frequency range 1 to 150 kHz.
34
3. DC/AC Converters – Inverters
Topologies
Inverting. DC/AC converters are called inverters. They convert dc voltage to ac
voltage of a definite frequency and value. Controlled semiconductor devices, such as SCR,
GTO thyristors, and transistors are used in inverters. The input dc voltage may come from
the rectified output of an ac power supply, in which case it is known as a dc link converter.
Alternately, the input dc may enter from an independent source such as a dc voltage source
or battery.
Inverters are usually designed to provide either single or three-phase output. Larger
industrial applications require three-phase ac. Low-signal half-wave inverters pass electrical
energy during one alternation. These inverters supply the load of 100…200 W power.
Another classification refers to offline and online inverting. If an inverter is the only source of
+ VT1
Ud / 2
VD1
Us
VT2
VT1
+
VT3
Ud
M
Us
VT2
VD2
M
VT4
Ud / 2
–
–
a.
VT1
+
Ud
b.
VT2
VT5
Us
VT3
VT6
Us
VT4
VT7
–
M
c.
Fig. 3.1
VT8
35
the load ac line, it is called an offline inverter or autonomous inverter. On the other hand, if
the inverter is a part of the common power supply line, it is known as an online inverter or a
line-fed inverter.
In accordance with the circuit electromagnetic processes, the voltage source inverters and
the current source inverters are distinguished. A voltage source inverter (VSI, or voltage stiff
inverter) forms the voltage with required properties: magnitude, frequency, and phase. It is
the most commonly used type of inverter. This inverter has the low internal impedance.
Generally, it has a capacitor of high capacity connected across the supply source that keeps
constant input voltage. To provide bi-directional current, the switches of VSI are constructed
on the base of the full controlled devices (transistors, GTO thyristors, or MCT) with
freewheeling diodes.
Alternately, a current source inverter (CSI) is the source of the current with the required
properties: magnitude, frequency, and phase. Usually, it has an inductor connected in series
with the supply source that keeps the current constant. The switches of the inverter
periodically change the output current direction, and the load has very low impedance. Thus,
the output voltage of the CSI is shaped according to the voltage drop on the load caused by
the output current.
Bridge VSI. Fig. 3.1, a, shows the half-bridge configuration of the single-phase VSI.
The role of switches VT1 and VT2 play BJTs, MOSFETs, IGBTs, GTO thyristors, or SCRs
with commutation circuit. The freewheeling diodes VD1 and VD2 are known as the feedback
diodes because they feed the load reactive energy back to the supply line.
Fig. 3.1, b illustrates the single-phase full-bridge VSI. Each its leg includes a pair of
transistors with anti-parallel discharge circuits of reverse current built on the freewheeling
diodes. In the case of motor load, these freewheeling diodes provide an alternate path for the
inductive current, which continues to flow when a switch is turned off. When regeneration
occurs, the diodes return the regenerated power to the dc supply while the switch carries the
reactive voltage.
Fig. 3.1, c shows a converter, which provides the frequency control of the two-phase ac
induction motor. The circuit includes a pair of single-phase full-bridge VSI. The first of them
+
VT1
VT2
VT3
Us
Ud
VT4
VT5
––
Fig. 3.2
VT6
M
36
+
VT1
VT3
LC
Ud
Ud
Us
–
VS1
+
VT2
VT4
VS3
Us
–
VS2
M
VS4
M
a.
b.
Fig. 3.3
drives the control winding of the motor and the second bridge feeds the exciting winding of
the motor.
The most frequently used three-phase bridge VSI is shown in Fig. 3.2. It consists of three
legs, one per each phase. All inverter legs are similar; therefore the output of each leg
depends only on dc supply voltage and the switch status. The output voltage is independent
on the output load current’s magnitude since one switch in a leg is always on at any instant.
Resonant inverter. The resonant inverter displayed in Fig. 3.3, a consists of a
switching circuit VT1…VT4 and LC resonant circuit thus form an alternating voltage for the
load. The maximum frequency of the tank circuit LC is near the communication frequency of
the switches. The frequency of the resonant inverters cannot be changed by the reference
signal of the control system.
CSI. A forced-commutated CSI is shown in Fig. 3.3, b. Here, the single-phase bridge
plays the role of the commutator. For the current source mode, an inductor is included in the
input circuit of the inverter. A capacitor is placed in the output as an energetic buffer between
the pulsing inverter and the load. In addition, the capacitor is the instrument of the forced
commutation of thyristors. When the thyristors VS2 and VS3 conduct the current, the input
voltage charges the capacitor. Ones the thyristors VS1 and VS4 switch on, the previous
thyristors get the reverse voltage of the charged capacitor, which helps them close
immediately. The capacitor begins recharging to other polarity, finishing it before the next
switching instant. More the current, faster the capacitor’s recharging and shorter the time of
the forced commutation.
Summary. In practice, single-phase inverters are used when the power of the load is
100…200 W. This most commonly used inverting approach provides on the output side
functions as a voltage source. The effective method of VSI implementation deals with the use
of the transistor bridges with freewheeling diodes.
In resonant inverters, the controllable switches turn on and off at zero voltage and/or zero
current. Such inverters are used in electro-thermal processes for supply the heating
equipment.
37
CSI can be used for such electrical equipment that needs the control of the current value:
electric arc furnace, induction heating, etc. As compared to the VSI, they are not so popular
because the of the resistive-capacitive load requirement. Instead, in the electric drives the
voltage inverter is used as a current source with appropriate current feedback.
Block Modulation
Transistor gate circuits. The best load would be an inverter that generates pure
sinusoidal waves of symmetrical phases. Unfortunately, any converter shown in Fig. 3.1 is a
source of voltage and current waveforms with more or less distortion. The distortion profile
and level depends on the modulation principle of its gate circuit.
The transistor converter gate circuits perform the operations similar to the thyristor ones:
•
•
•
•
•
prepare the discrete intervals for the system timing,
produce the carrier signals,
generate the control pulses and distribute them between the transistors,
convert the control pulses into the gate signals,
isolate the control and power circuits.
In Fig. 3.4, a, a gate circuit provided these functions is displayed. Here, the carrier generator
G produces the pulse chain uc of the carrier frequency fc. The shape of generated pulses
may be different: triangle, saw-tooth, or rectangle narrow pulses. Unlike the similar generator
for the thyristor circuits, the line voltage should not synchronize these pulses. Another
distinction is that uc may not change its sign during the carrier period Tc = 1 / fc.
G
uc
u*
PDU
Gate
driver
Galvanic
isolators
a.
CT
u*
Reset
=0
&
R
T
uc
R
S
b.
Fig. 3.4
Q
38
The gate driver converts the reference signal u* into the time intervals. For this purpose, a
comparator or a counter may be used. The comparator compares uc with u* by the same way
as in the thyristor gate driver. When a binary counter is used, the carrier generator G
includes a system clock only, without any function generator, which produces the pulse chain
uc, thus the circuit operates in accordance with Fig. 3.4, b. Periodically, the reset signal writes
u* into the counter CT, switches on currier pulses uc, and sets the high level of the output Q
of the flip-flop T. From this instant, the counter starts to subtract pulses from the code u* until
zero. Zeroing pulse of the counter output switches off the carrier chain uc and resets flip-flop
output Q to low until next reset pulse triggers it into the opposite state.
The operation principle of the pulse distribution unit PDU depends on the selected
modulation method.
For the converters shown in Fig. 3.1, some variants of the transistors control exist. For the
converters shown in Fig. 3.2, a lot of different variants of open and close transistors may be
provided, however three transistors are opened together in most commonly used gate
circuits. When the load is wye-connected, each phase is jointed either in parallel to other
phase or in series to other parallel-connected phases.
Modulation waveforms. Fig.3.5 illustrates the block control modulation method
known also as square-wave modulation or rectangle modulation. The diagrams of the singlephase inverter (Fig. 3.1, c) operation depict the method of simultaneous voltage and
frequency adjustment. In the beginning of the first modulation period, the transistors VT1 and
VT4 switch on whereas the transistors VT2 and VT3 are in off state. During t1, the source dc
voltage supplies the control winding. At the end of this interval, VT4 switches off and VT3
switches on and the control winding remains unsupplied during the time interval t2. In the
next interval t1 VT2 switches on and VT1 switches off, thus the polarity of the control winding
supply changes. The duty cycle
T
t1
t2
U1
θ
U2
θ
U3
θ
U4
θ
Ucon
θ
Uexc
θ
Fig. 3.5
39
q = t1 / T
is proportional to the alternating gate pulses frequency although the on period remains
constant. The corresponding diagram of the excitation winding supply is similar, with π / 2
shift.
For the three-phase VSI shown Fig. 3.2, the phase, phase-to-neutral, line-to-neutral, and
line-to-line voltages of the block control modulation method have the waveforms plotted in
Fig. 3.6. Switching on of the three half-bridges are phase-shifted by π / 3. Each transistor is
open during half a period and closed during another half a period. Duration of the full
switching period (2π) depends on the required output frequency that is referred by the reset
signals and reference code u*.
The sequence of switching is in the order VT1–VT3–VT5, VT1–VT5–VT6, VT1–VT2–VT6,
VT2–VT4–VT6, VT2–VT3–VT4, VT3–VT4–VT5, and back VT1–VT3–VT5. For switching off
the load, two zero states may be used: VT1–VT2–VT3 or VT4–VT5–VT6. Accordantly, a
specific phase is alternately switched from positive pole to negative pole and that it is
alternately in series with the remaining two phases connected in parallel. When VT1 is
switched on, load phase U is connected to the positive terminal of dc supply, making
UU = Ud / 2. When VT4 is switched on, phase U is connected to the negative terminal of dc
Ud /2
UU
π
2π θ
θ
UV
θ
UW
Ud /6
UN
θ
2Ud /3 UUS
θ
UVS
θ
θ
UWS
Ud
UUV
θ
UVW
θ
UUW
θ
Fig. 3.6
40
supply, making UU = -Ud / 2. Waveforms of V and W are exactly the same as those of U,
except that they are shifted by π / 3.
For balanced three-phase operation, the voltage of the load neutral can be written as:
UN = (UU + UV + UW) / 3.
The neutral potential is either positive or negative as two upper or lower transistors are on in
the inverter leg. From here, the load phase voltages may be obtained as follows:
UUS = UU – UN, UVS = UV – UN, UWS = UW – UN.
Therefore, each phase gets the voltage drop equal to ±Ud / 3 or ±2Ud / 3 with the polarity of
the voltage drop across the phase being determined by whether it is connected to the
positive or negative pole. They have a characteristic six-stepped wave shape. The presence
of six steps in the line-to-neutral voltage waveform is the reason for this type of modulation
being called a six-step modulation. The rms value of the load phase voltage is equal to
√2Ud / 3.
The instantaneous load phase currents may be described as follows:
I0<θ<π/3 = Ud / (3Rd) (1 – (1 + k) (2 – k) / (1 + k3) exp (–t / Td)),
Iπ/3<θ<2π/3 = Ud / (3Rd) (2 – (1 + k)2 / (1 + k3) exp (–t / Td)),
I2π/3<θ<π = Ud / (3Rd) (1 – (1 + k) (1 – 2k) / (1 + k3) exp (–t / Td))
where k = exp (1 / (6fin / Td), Td = Ld / Rd, Rd and Ld are the load resistance and inductance, fin
– supply frequency.
The line voltages are related to the phase voltages as follows:
UUV = UU – UV, UVW = UV – UW, UWU = UW – UU,
These voltages are quasi-square waves with π / 3 pulse width. The line-to-line voltage
contains an rms fundamental component of √6UN / π = √(2/3)Ud. The number of pulses is
constant over a predetermined frequency range although this number of pulses one may
change discretely at several prescribed frequencies.
Analysis. Fourier analysis of these waveforms indicates a square-wave type of
geometric progression of the harmonics. That is, the line-to-line and line-to-neutral
waveforms contain 1/5th of the fifth harmonic, 1/7th of the seventh harmonic, and so forth.
Harmonics of order three and multiples of three are absent from both the line-to-line and lineto-neutral voltages and consequently absent from the currents.
Here, the logical structure of the converter is constant because the number of open and
closed transistors equals three and does not change during the full operation period. This is
the main advantage of the described six-step modulation mode. Other switching modes are
possible also. For example, the open state interval may continue π / 3 or 5π / 6 instead of π as
well as switching order VT1–VT6, VT6–VT2, VT2–VT4, VT4–VT3, VT3–VT5, VT5–VT1 or
VT1–VT6, VT1–VT2–VT6, VT2–VT6, VT2–VT4–VT6, VT2–VT4, VT2–VT3–VT4, VT3–VT4,
VT3–VT4–VT5, VT3–VT5, VT3–VT5–VT1, VT5–VT1, VT5–VT1–VT6. The last one leads to
the variable structure of the converter.
41
Summary. The advantages of the square-wave modulation are: high efficiency (near
98 %), potentially good reliability, and high-speed capability. A simple form of block
modulation results in minimum switching duty of the semiconductor switches. Thus, a
constant switching frequency scheme guarantees the fast response together with the limited
steady-state tracking error.
Voltage control is impossible in the described three-phase block modulated driver circuits;
this is their drawback. As a result, square-wave modulation is commonly used in low-power
applications where the voltage range is fixed and dynamic performance is not important.
Examples are frequency changers and inverters with dc controlled input. The need in a
phase-controlled rectifier to control the voltage of the inverter is an inherent weakness of this
circuit. A line-commutated rectifier supplying the dc link is particularly notorious often
because it is not only produces the line currents with low orders of harmonics, but draws also
substantial reactive currents of line frequency. Its large compensative capacitor enlarges the
response time of the system.
Another disadvantage of block modulated ac converters deals with its suffering from lowvoltage pulsations due to non-sinusoidal voltage shape, which leads to the load current
pulsations and instability with extra energy losses especially when the frequency is low. In
such inverters, harmonic voltage amplitudes are inversely proportional to the harmonic order.
Thus, the six-step mode is worst with respect to voltage harmonic content with 20 % of the 5th
harmonic, 14 % of the 7th and so forth. Hence, there are no pronounced high-order
harmonics. These are filtered out by the load inductances.
Pulse-Width Modulation
PWM technique. The pulse-width modulation, or PWM method is now gradually
taking over the inverter market in control applications. This technique combines both voltage
and frequency control. The PWM circuit output is the chain of constant amplitude pulses, in
which the pulse duration is modulated to obtain the necessary specific waveform. In the
DC/AC converters shown above, the dc link voltage is uncontrolled and derived from a
simple diode bridge. In the case of PWM modulation, the controlled output voltage is easily
obtained by switching the transistors on and off many times within a cycle to generate a
variable-voltage output which is normally low in harmonic content.
The pulse-width modulators may be of a variety of designs. A large number of PWM
techniques exists each having different performance notably in respect to the stability and
audible noise of the load.
Sinusoidal PWM. One frequently used PWM algorithm is illustrated in Fig. 3.7 for the
circuit of Fig. 3.2. The objective of sinusoidal modulation is to synthesize voltages that
produce currents as near to a sinusoidal as economically possible. The sinusoidal
modulating signals u* refers the required output waveform. To obtain balanced three-phase
output voltages in a three-phase PWM inverter, three reference sinusoidal modulated
voltages that are 2π / 3 radians out of phase are needed, one per each phase. The highfrequency triangle carrier signal uc is required also. Its frequency is typically 2 kHz to 20 kHz.
The natural intersections of u* and uc determine both the offset and duration of the gate
control signals. In PWM, the waveform of pulse pattern is dependent on the ratio of the peak
42
ton
toff
uc u*
θ
T*
UU
θ
UV
θ
UW
θ
UN
θ
UUS
θ
UVS
θ
UWS
θ
UUV
θ
Fig. 3.7
u* to the peak uc. The carrier ratio (frequency ratio) kf = fc / f* determines the number of
pulses in each half-cycle of the inverter output voltage. The modulation index (modulation
ratio) ku = u*max / uc max determines the width of the pulses and hence the rms value of the
inverter output voltage. The ideal maximum modulation index is equal to unity. Various PWM
schemes allow ku < 1 that represents an important performance criterion as the inverter
maximum power depends on the maximum voltage at load terminals.
Changing of the pulse width of each half-cycle alters the output phase voltages UU, UV, UW of
the inverter (with respect to mid dc link point). They are switched between positive and
negative buses at the intersections of the carrier wave and the modulating waves. Here,
unlike the block modulation scheme, the conduction angle ton of various transistors may be
less than π / 3.
The sequence of switching has no order as in the block modulation case, and zero states are
used regularly that correspond to zero load voltage. When VT1 switches on, load phase U is
connected to the positive terminal of dc supply, making UU = ku Ud / 2. When VT4 switches
on, phase U is connected to the negative terminal of dc supply, making UU = –ku Ud / 2.
Waveforms of V and W are the same as those of U, except that they are shifted.
For the balanced three-phase operation, the voltage of the load neutral can be written as:
43
UN = (UU + UV + UW) / 3.
The load neutral voltage has three times the referred frequency and thus contains the triple
harmonics, which does not appear in the load phase voltages that may be obtained as
follows:
UUS = UU – UN, UVS = UV – UN, UWS = UW – UN.
Therefore, each phase gets the voltage equal to ±ku Ud / 3, ±2ku Ud / 3, or zero. Again, they
have a characteristic six-stepped wave shape.
The corresponding line voltages of the load are next:
UUV = UU – UV, UVW = UV – UW, UWU = UW – UU.
Note, that the positive pulse patterns of the voltages are not quite the same as the negative
ones when f ≠ 6kfc with any integer k, although the two areas are quite similar to give zero
average values. Fourier analysis of the inverter voltage waveforms reveals that they have
less harmonic content than a single pulse per half-cycle inverter block-modulated voltage.
Nevertheless, they have sinusoidal fundamental components but still noticeable losses as
well as objectionable noise emitted by the converter and the load.
Gate drivers. So far as the reference frequency must be of very high frequency, the
digital on-chip modulation sub-processors are used for this purpose. Their goal is to generate
the triangle carrier function uc, compare it with the three reference signals u*, and find the
logical results of this operation (Fig. 3.8, a). The driver opens the required transistor when
u* > uc, and closes it in opposite case.
To produce this operation faster, one calculation per carrier period Tc, the real triangle
function is replaces by symmetrical interpolation procedure illustrated by Fig. 3.8, b:
ton = Tc / 4 (1+ u* / uc),
toff = ton + Tc / 2.
The asymmetrical interpolation procedure shown in Fig. 3.8, c is used as well. Thanks to the
double frequency measurement, the dynamic modulation precision rises and the load current
distortion decreases here, although the higher processor capacity is required.
Summary. When the sinusoidal PWM technique is used, the low-order voltage
harmonics are greatly attenuated although other significant harmonics are represented close
to the carrier frequency. Hence, this is a good solution where an electronic system is to be
used across a wide voltage and frequency range.
Since voltage and frequency are both controlled with the PWM, quick response to changes in
demand voltage and frequency can be achieved.
PWM inverter efficiency typically approaches 98 % but this figure is heavily affected by the
choice of switching frequency – with low frequency, the losses are low, while for higher
switching frequency, the losses are higher.
To counterbalance these advantages, the switching frequency is variable and very intensive
in such circuits; the number of switching per period is as high as 2 / kf. As a result, the
converter losses are higher than for block mode of operation. When f ≠ 6kfc, the phase
44
G
uc
u*
PDU
Gate
driver
Galvanic
isolators
a.
Tc
θ
UU
θ
b.
Tc
θ
UU
θ
c.
Fig. 3.8
voltages are asymmetrical, therefore the bipolar modulation leads to the high current
pulsation and the high reactive energy level.
An attempt to synthesize the best possible sine wave by selecting a higher carrier frequency
may well create more losses in the inverter than in the load.
Pulse-width modulators are now available in a variety of integrated circuits, which greatly
simplifies the design of PWM converters. There is also the possibility of software-based
modulation using fast signal processors that offer unlimited flexibility by combining PWM with
other methods.
Space Vector Modulation
Objective. The unsatisfactory noise situation has given rise to the development of a
multitude of advanced modulation schemes where off-line computed binary switching
sequences are kept in a microelectronic memory to be called up in real time for small
increments of voltage or current. Their objective is to reduce the current harmonics of power
45
losses, the current pulsations or noise under steady-state conditions as well as to avoid the
continued fluctuations of the voltage amplitude, which would disturb the switching
sequences.
There are a number of circuits where the fine and rapid control is obtained through the
frequency input of the modulator but the voltage would be changed more slowly and
temporarily in somewhat coarser steps. Clearly, the more often the voltage is reversed per
period, the more conditions can be satisfied, given a precision fundamental voltage
component.
The restrictions of the converter with regard to the minimum time between two subsequent
switching operations must of course be observed to allow the commutation to be completed.
Also, the losses in the converter caused by each commutation transient should be
considered, which means that there is an upper limit for the switching frequency.
When realizing a suitable modulator it is of course desirable to keep the required memory
volume enough small. This can be achieved by storing only the data for the single phase.
Transposing and inverting of the stored pattern gives the remaining information. The matter
of fact, considerable angular resolution is needed to satisfy the various conditions with
adequate accuracy.
Switching model. A very effective method that is particularly suited for the fast
switching converters is called space vector control or vectorial PWM because it represents
an attempt to reproduce a voltage vector demanded by a controller in a given time interval.
Like the block modulation algorithm and sinusoidal PWM, this method implements the sixstep block control system. For this purpose a switching model of Fig. 3.9, a, simulates a
converter circuit shown in Fig. 3.2. In this model, an inverter allows for six switch triplets that
produce nonzero voltage space vectors.
Each load terminal assumes a potential defined by the control. To avoid the legs shortcircuiting by the link voltage, one transistor in each leg is blocked while the other is
conductive, except for the short protective intervals, when both transistors are blocked and
the load current flows through one of the shunting diodes. The protective interval, which lasts
V axis
+
U3
1
0
Ud
S1
U2
B
U
1
C
S2
A u*
θ*
V
0
1
0
S3
D
U4
F
U5
–
U6
W axis
b.
a.
Fig. 3.9
U1
E
W
U axis
46
only few microseconds, can be assigned to a finite switching time in the proper converter
model.
Each half-bridge is modeled by a reversing switch indicated by a binary variable Si = {1, 0},
depending on whether the switch is in the upper or lower position. The switching state (S1,
S2, S3) of the complete converter is then described by a three-bit binary word having eight
different values, where the load terminals U, V, W are connected to the upper or lower dc
bus Ud. The corresponding base space vectors U0 – U7 describe each switching state of the
converter. These vector set includes six voltage vectors U1 – U6: 100, 110, 010, 011, 001,
101 and two zero voltage space vectors U0, U7: 111 and 000.
U
UUS
U0, U7
U1 (VT1,
U2 (VT1,
U3 (VT4,
U4 (VT4,
U5 (VT4,
U6 (VT1,
(VT1,VT2,VT3;
VT5,VT6) VT2,VT6) VT2,VT6) VT2,VT3) VT5,VT3) VT5,VT3)
VT4,VT5,VT6)
100
110
010
011
001
101
111, 000
0
2Ud / 3
Ud / 3
–Ud / 3
–2Ud / 3
–Ud / 3
Ud / 3
UVS
–Ud / 3
Ud / 3
2Ud / 3
Ud / 3
–Ud / 3
–2Ud / 3
0
UWS
–Ud / 3
–2Ud / 3
–Ud / 3
Ud / 3
2Ud / 3
Ud / 3
0
Switching table and graph. To describe the space vectors in terms of phase
voltages, six distinct voltage vectors and two zero vectors result as seen in the table. The
definite binary word corresponds to each vector. Voltage of each phase is equal to ±2Ud / 3,
±Ud / 3, or zero dependently on which transistors written in brackets are switched on.
The graph of this table (known also as Concordia graph) shown in Fig. 3.9, b includes six
space vectors again, 60° apart. Voltage vectors U1, U3, U5 are oriented along the axes of
phases U, V, W. Supply voltage Ud refers the amplitude of the space vectors.
Control method. The reference inverter output voltage vector is determined by its
value u* and phase θ*. Since u* is normally not coinciding with one of the available space
vectors, it is to be composed by a switching sequence comprising the neighbor space
vectors U1…U6 while filling up the rest of the time interval with zero vectors U0 or U7 during
the voltage alternation. Clearly, when going from one corner of the hexagon to the next, only
one leg of the converter needs to change its state:
•
•
•
A and D sectors – second leg (VT2, VT5),
B and E sectors – first leg (VT1, VT4),
C and F sectors – third leg (VT3, VT6).
If the voltage is not adjusted, other transistors keep their previous state. As a result, the
vector’s end travels on the hexagon or stops at zero. The vector path’s deviation from the
circle corresponds to the voltage and current distortions. Timing the eight voltage space
vectors U0…U7 is, in fact, the art of space vector modulation.
Let the fixed modulation interval is equal to
Tc = 2πf* / fc = 2π / kf.
47
Then, the sub-intervals between the two adjacent vectors are to be computed from the
following equivalence:
u* = fc (ti Ui + ti+1 Ui+1),
ti + ti+1 + t0 = Tc
where Ui is one of the space vectors; Ui+1 is the space vector valid in the next interval Tc;
ti, and ti+1 are the sub-intervals for the two adjacent vectors that are to be computed in real
time; t0 is the zero vector interval. Solving for ti and ti+1 results in:
ti = √3 / 2 u* / Ud Tc sin (π / 3 – θ*),
ti+1 = √3 / 2 u* / Ud Tc sin θ*,
t0 = Tc – ti – ti+1.
Minimum value of ti is zero and maximum ti+1 = Tc.
Of course, these equations describe an idealized situation, where the intervals and the
inherent delays of the switching devices are neglected. For the actual design of modulators,
these effects must be taken into account, particularly the difference between turn-on and
turn-off times, which can course considerable distortion of the converter characteristics at low
output voltage and frequency.
In fact, this technique produces an average of three voltage space vectors Ui, Ui+1, and U0
(U7) over a sub-cycle Tc. Particularly, the maximum value of u* / Ud without delays (when
t0 = 0) may be calculated from these equations as follows:
√3 / 2 u* / Ud Tc sin (π / 3 – θ*) + √3 / 2 u* / Ud Tc sin θ* = Tc.
In each sector, it describes the straight line, which is the side of the hexagon that connects
the ends of space vectors. Taking the sector altitude as the maximum space vector modulus
u* / Ud = 1 / √3, we will get the inscribed circle as the optimum switching path. This means
that in any case except of kπ / 3 (k = 1…6), the zero space vectors should take part in
switching.
For this reason as follow from Fig. 3.10, to control the average modulus of the space vector
(output voltage pausing) it is needed some extra switching in addition to the transistors of the
switching leg. Particularly,
•
•
•
in A and D sectors, VT1 and VT4 preserve their states, VT2 and VT5 switch
periodically to move the space vector along the circle, as well as VT6 is replaced
momentary by VT3 to produce the zero voltage vector;
in B and E sectors, VT3 and VT6 preserve their states, VT1 and VT4 switch
periodically to move the space vector along the circle, as well as VT2 is replaced
momentary by VT5 to produce the zero voltage vector;
in C and F sectors, VT2 and VT5 preserve their states, VT3 and VT6 switch
periodically to move the space vector along the circle, as well as VT4 is replaced
momentary by VT1 to produce the zero voltage vector.
Gate circuits. As a rule, the DSP microcontrollers of “Intel”, “Texas Instruments”, or
“Analog Devices” are used for different space vector modulation algorithms implementation.
48
0 π/3 2π/3 π 4π/3 5π/3 2π π/3 2π/3 π 4π/3 5π/3 2π π/3
A
B C
Tc
D
E
F
A
B
C
D
E
F
A
θ
B
T*
UU
θ
UV
θ
UW
θ
UN
θ
UUS
θ
UVS
θ
UWS
θ
UUV
θ
Fig. 3.10
They have the required processor capacity up to 30 millions instructions per second, include
built-in interface for inverters and sensors connection, as well as universal signal generators.
Particularly, 16-bit TMS320LF2407 of “Texas Instruments”, “Intel” MCS-196/296, and
ADMC300/330 of “Analog Devices” are suitable for building the gate systems of inverters and
rectifiers. Another solutions deal with less expensive “Atmel” crystal AT90PWM3 of 16 mega
instructions per second capacity with 8-bit RISC core, 8-kB system flash memory, 512-byte
static RAM, 512-byte ROM, 8-bit and 16-bit timers, PWM bridge-oriented channels, 10-bit
ADC and 10-bit DAC, built-in comparators, and pulse generators.
The functional gate driver controller algorithm is drawn in Fig. 3.11. Its input block calculates
the angle and sector of reference space vector. Sine table stores the sine values of A sector
angles. Next block solves the equations for ti, ti+1, and t0, and distribute pulses between the
switches.
Summary. This class of modulators produces high performance although can be
implemented only on microprocessors because it requires online computation of the
reference voltage space vector.
Compared to the block modulation and sinusoidal PWM, vectorial PWM allows a higher
phase voltage and thus a higher output power of a converter with minimum switching
frequency. The output voltage amplitude in sinusoidal PWM is Ud / 2. With the space vector
PWM, the amplitude is equal to the inner-circle radius of the hexagon, that is Ud / √3 or
49
G
uc
u*
Angle
and sector
calculation
θ
A–F
Sine
table
Ui, Ui+1
ti, ti+1, t0
calculation
Galvanic
isolators
Fig. 3.11
15.5% higher. However, this is achieved through abandoning the sinusoidal output that
results in higher losses caused by higher harmonic components.
Furthermore, with a diode rectifier as the input circuit a high power factor, approaching unity,
is offered to the incoming ac supply over the entire voltage and load range.
Thus, this technique is well suited for the high-performance high-speed applications.
50
4. DC/DC Converters – Choppers
Step-Down Choppers
Choppers. The switching DC/DC converters are called choppers. As a rule, they
provide changing and stabilizing of the output voltage levels as well as galvanic
disconnection of the input and output electronic circuits.
The basic chopper topologies are step-down (forward) and step-up (flyback). Both may
operate in single-quadrant, two-quadrant, and four-quadrant modes with or without the output
feedback. Forward converters with a feedback are known as buck regulators. Flyback
converters with a feedback are called boost regulators. Any chopper consists of the switching
circuitry and the filter section. In between the switching and filter sections, there may be a
transformer for stepping up or down the voltage.
Single-quadrant operation. In the step-down forward chopper, the power switch VT
is placed directly between the input voltage source Ud in and the filter section (Fig. 4.1, a).
The switch serves to replenish energy lost to the load during its off time. The shunt (flywheel)
diode VD, series inductor L, and shunt capacitor C arrange an energy storage reservoir
whose purpose is to save enough energy to maintain the load current over the entire off-time
of the switch.
VD1
VT
L
L
VT1
+
+
Ud in
VD
VT2
C
Ud
Ud in
M
Ud out
–
Ud out
–
b.
VT1
C
VT2
L
Ud in
M
VT3
Ud out
–
c.
Fig. 4.1
C
UdM
a.
+
VD2
VT4
51
Usually, the switch commutates the output voltage using PWM; therefore the voltage applied
to the load has the form of a square wave of varying periodicity. The graphs of the switch
output voltage Ud and filtered load current Id out and voltage Ud out for the forward converter are
shown in Fig. 4.2, a. The operation can be broken up into two phases. The first is when the
switch is on (ton). During this period, the current passes from the input source through the
inductor to the load. The diode is reverse-biased in this period. Once the switch turns off (toff),
the inductor still expects current to flow through it. The diode now begins to conduct and the
load current freewheels through the diode, thus maintaining a closed current loop through the
load. Then, the switch is turned on again and the cycle repeats. Chopping period of this
circuit is:
Tc = ton + toff
and chopping frequency
fc = 1 / Tc.
The duty cycle is:
q = ton / Tc.
The average and rms output voltage values are:
UVD
Ud out
t
Id out
t
Ud out
t
ton
q
toff
a.
b.
Symmetrical control
Asymmetrical control
VT1
t
VT1
t
VT2
t
VT2
t
t
VT3
t
VT4
t
VT4
ULC
t
ULC
VT3
t
t
c.
Fig. 4.2
52
Ud out = qUd in,
Urms out = √qUd in.
From the control curve (Fig. 4.2, b) it is obvious that the output voltage grows linearly with the
duty ratio of the chopper.
This circuit supplies the load by unidirectional current and voltage.
Two-quadrant operation. A circuit that is capable of two-quadrant operation is
depicted in Fig. 4.1, b. During the first phase, the converter operates as the basic chopper
with VT1 and then VD2 carrying the current. The current flows through the load while the
transistor VT1 is in on state. Since the transistor VT1 is in off state, reactive energy of the
load dissipates through the discharge transistor VT2. During the second phase, VT1 is
inoperative and VT2 controls the current, which builds up negatively, limited by the load
inductance. When VT2 turns off, the only path for the current is via VD1 back into the supply;
hence the circuit is regenerative.
Four-quadrant operation. Fig. 4.1, c shows a basic four-quadrant forward converter.
There are the two methods for this circuit control: a symmetrical control and an asymmetrical
one. In the left timing diagrams of Fig. 4.2, c the symmetrical control mode is shown, where
all switches change their state simultaneously. During the first phase, the transistors VT1 and
VT4 are switched on in the on period and the diodes VD2 and VD4 conduct in the off period.
With the transistors VT2 and VT3 conducting, the current is reversed and hence the full fourquadrant operation is obtained. The disadvantage of the method is that amplitude of the
output ripple voltage is twice that of the simple converter, and the current ripple is therefore
worse due to a high ripple factor. This problem can be overcome by a technique known as an
asymmetrical control. If the load is resistive-inductive, the asymmetrical control is preferable.
In this mode depicted by the right timing diagrams of Fig. 4.2, c, the switches VT3 and VT4
change their state while the switch VT1 is open and the switch VT2 is closed. In such a way,
when the switches change their state the current transfer is smoothed.
Step-Up Choppers
Flyback choppers. The step-down converters discussed above produce output
voltages less than the input voltage. However, a change in the chopper configuration
provides higher load voltages. Step-up flyback converter (Fig. 4.3, a) produces the output
voltages higher than the input voltage. Here, the inductor L is situated directly between the
input source Uin and the switch VT. The anode of the rectifier VD is placed on the node
where the switch and inductor are connected, and the capacitor C is connected across the
load.
As explains Fig. 4.4, a, the flyback operation consists of two periods. When the power switch
is on, the current is being drawn through the inductor, which causes energy to be stored
within its coil material. The switch then turns off. Since the current through the inductor
cannot change instantaneously and is forced to flow through the diode and the load, the
inductor’s voltage reverses (flies back). This causes the diode to turn on, thus dumping
inductor’s energy into the capacitor. The inductor current decreases. This process passes
until inductor energy is emptied. Since the inductor voltage flies back above the input
voltage, the capacitor voltage becomes higher than the input voltage. When the capacitor
53
VD1
L
VD
+
VT
Ud in
+
M
Ud in
C
Ud
M
–
Ud out
VT
Ud out
a.
–
VD1
VT1
b.
D
VT
+
U2
Ud out
C
Ud in
M
L
Ud in
U2
M
Ud out
–
d.
VD2
VT2
c.
Fig. 4.3
voltage reaches the desired level, the switch turns on ones more. The capacitor cannot
discharge via the switch, as diode is reverse biased. In this way, a stable voltage typically
twice the Ud in or more can be obtained.
The duty ratio formula describes the input and output voltages relation as follow:
Ud out = Ud in / (1 – q).
Thus, for a variation of q in the range 0 < q < 1, the output voltage will forced to vary in the
range of Ud in < Ud out < ∝. In practice, the parasitic resistance Rp of the circuit restricts the
upper border of the control curve:
Rp = R1 + R2 + R3 + R4
where R1 is the inner resistance of the supply source, R2 is the ohmic resistance of inductor,
R3 is the resistance of the switched-on transistor, and R4 is the diode resistance. Because of
this,
qmax = 1 – √(Rp / Rd) < 0.8 to 0.9
where Rd is the load resistance. The control curve is shown in Fig. 4.4, a.
54
Ud in
Ud in
t
t
Ud in – Ud out
–Ud out
Ud out
Ud out
q
0.5
q
qmax
0.5
a.
qmax
b.
Fig. 4.4
The transformer-isolated flyback converter shown in Fig. 4.3, b passes energy when the
switch is off. During the first phase, the switch is on, the primary winding stores energy, and
the primary current is growing up. Qnce the switch turns off, the polarity of the windings
changes due to the self-induction phenomenon. The diode opens, the secondary current
charges the capacitor, and the primary current falls. There is no high overvoltage in this
case.
Push-pull principle of dc converting helps to build more effective bi-directional circuits. The
two-phase push-pull converter is shown in Fig. 4.3, c. The circuit consists of the transformer
with a center tap and the two-phase rectifier. During the first period, the switch VT1 is on and
the switch VT2 is off. The current flows through the diode VD1 and charges the capacitor.
During the second period, VT1 is off and VT2 is on. The current flows through the diode VD2
and charges the capacitor. In such a way, energy supplies the load both periods.
A chopper configuration, which provides load voltages lower and higher than the supply
voltage (buck-boost converter) is shown in Fig. 4.3, d. Like in the step-down chopper, its
power switch VT is placed directly between the input voltage source Ud in and the filter
section. Diode VD, series inductor L, and shunt capacitor C arrange an energy storage
reservoir. When the switch is on, the inductor is connected to the supply voltage and the
inductor current increases. While the switch is off the inductor current flows through the load
and diode. The inductor voltage changes the polarity and the inductor current decreases.
Fig. 4.4, b shows the output voltage and the control curve of the converter, where
Ud out = Ud in q / (1 – q).
Here, for a variation of q in the range 0 < q < 1, the output voltage will vary in the range
0 < Ud out < ∝. Again, q < qmax.
Summary. The chopping circuits normally operate at the frequencies of 2…20 kHz.
The main features of the step-down choppers are: unlimited current and voltage speed up
55
and speed down during transients that lead to high dynamic power losses. Industrial
applications of these circuits are normally restricted to loads below 5 kW. Traction
applications, however, are designed at ratings of hundreds kilowatt.
DC/DC converters, which produce the voltage higher than supply voltage must accumulate
energy in the input reactive element (inductor) and pass it into the output reactive element
(capacitor) independently, in different time intervals. The control of these processes is
provided by mean of duty cycle changing with or without feedback.
The most universal DC/DC converters step up and step down the load voltage, support
single-, two- and four-quadrant operation, and do not require additional filters and powerful
reactive elements. The powerful and fast switching devices are the necessary components of
such circuits.
Choppers Calculation
Supply voltages. In DC/DC converters shown in Fig. 4.1, the required supply voltage
for asymmetrical mode of operation is defined as follows:
Ud in = Ud out / qmax + k UCE
where Ud out is the reference load voltage, qmax = 0.95…0.98 – maximum duty cycle of PWM,
k = 1 or 2 – number of current-conducted transistors, UCE = 2…3 V – transistor voltage drop.
In the case of symmetrical control,
Ud in = Ud out / (2qmax – 1) + 2UCE.
When the load current is continuous, it is obvious to restrict the input voltage variation by a
value:
∆u = ∆Ud out / Ud out.
where Ud out is the load rated voltage and ∆Ud
rated input voltage is to be in the range:
out
is its possible variation. As a result, the
Ud min = Ud in / (1 – ∆u),
Ud max = Ud in / (1 + ∆u).
Transistor parameters. Transistor rated voltage must exceed the value
UR = k Ud in
where k = 1.7…1.85 – safety factor for the repetitive and short-term overvoltages protection.
Transistor rated current should be more than
IF = k Id
where k = 2…3 – safety factor for the overcurrent protection, and Id is the rated load current.
Filter. The LC filter is placed between the switch and the load. The resonance
frequency of the filter is:
f = 1 / (2π√(LC)),
56
thus to avoid resonance, the chopping frequency should be fc > (2…3)f. To provide the
continuous current mode, the inductance value should be like this:
L > Rd / (2fc ) (1 – qmin)
where Rd is the load resistance and qmin = ton min / Tc. Another formula is:
L > Ud out / (2Id out fc ) (1 – qmin)
where Ud out and Id out are the load rated voltage and current. Then, the peak load current is:
Id out max = (1 – q) Ud / fc L.
Output curves. In the single-quadrant step-down chopper, the output voltage versus
average load current varies linearly with the duty cycle, as show the continuous traces in
Fig. 4.5, a. Dotted traces describe discontinuous current mode. If the load resistance is R
and the rated load current is Id, then the mean value of the voltage should be RId, therefore
Ud out
Ud out
q=1
q=1
q=0.1 Id out
q=0.1
Id out
b.
a.
Ud out
q=1
Id out
q = 0.5
q=0
c.
Ud out
q
q=1
Ud out = 4
q = 0.7
Ud out = 1
q = 0.1
Ud out= 0.1
Discontinuous current boundary
Id out
d.
Id out
e.
Fig. 4.5
57
q = RId / Ud in.
The two-quadrant forward chopper is able to reverse the average current flow of the load but
unable to reverse the load terminal voltage as show the load curve in Fig. 4.5, b. In the fourquadrant forward chopper with symmetrical control, the full four-quadrant operation is
obtained as show curves in Fig. 4.5, c. When asymmetrical control is used, the same curves
correspond to -1 < q < 1.
The output characteristics of the buck converter and boost converter depend on the duty
cycle. When the load current Id out decreases, the converter passes from the continuous
operation to the discontinuous operation and the voltage Ud out changes as shown in
Fig. 4.5, d, e.
The instantaneous load currents depend on the circuit and control mode. In the four-quadrant
forward chopper with symmetrical control,
Ion = Ud / Rd (1 – 2(1 – k2) / (1 – k1k2) exp (–t / TZ)) – E / Rd,
Ioff = Ud / Rd (1 – 2(1 – k1) / (1 – k1k2) exp (–t / TZ)) – E / Rd
and with asymmetrical control
Ion = Ud / Rd (1 – (1 – k2) / (1 – k1k2) exp (–t / TZ)) – E / Rd,
Ioff = Ud / Rd (1 – k1) / (1 – k1k2) exp (–t / TZ)) – E / Rd
where k1 = exp (–qTc / TZ), k2 = exp ((q – 1)Tc / Td), Tc = 1 / fc, fc – carrier frequency,
Td = Ld / Rd, Rd and Ld are the load resistance and inductance, E = qUd – RdId – load EMF.
Examples have been shown in Fig. 4.2, a.
Summary. Supply voltage, transistors parameters, and filter features of choppers
depend on duty cycle and continuous or discontinuous current mode of operation.
The main features of choppers are: unlimited current and voltage speed up and speed down
during transients that lead to the high dynamic power losses; the absence of inverse voltage
on the switch; the output curves dependence on the load parameters (inductance,
resistance), and non-linearity of the load curves.
58
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List of Journals
AEU – International Journal of Electronics and Communications
Chip News
Computers and Electrical Engineering
Electric Power System Research
Electromechanical and Power Systems
EPCOS Components
EPE Journal
IEEE Industry Applications Magazine
IEEE Power Engineering Review
IEEE Transactions on Components, Hybrids and Manufacturing Technology
IEEE Transactions on Education
IEEE Transactions on Energy Conversion
IEEE Transactions on Industrial Electronics
IEEE Transactions on Industry Applications
IEEE Transactions on Fuzzy Systems
IEEE Transactions on Mechatronics
IEEE Transactions on Power Electronics
International Journal of Electrical Power & Energy Systems
Power Systems World
Solid-State Electronics
Известия вузов. Электроника
Компоненты и технологии
Силовая электроника
Практическая силовая электроника
Электронные компоненты
64
Index
accident, 9
breaker, 9
capacitor, compensative, 20
carrier generator, 20
carrier ratio, 42
carrier signal, 20
changer, 27
choke, smoothing, 18
chopper, 50
chopper, buck-boost, 54
chopper, fly-back, 50
chopper, forward, 50
chopper, step-down, 50
chopper, step-up, 50
chopping frequency, 51
chopping period, 51
circulating current mode, 21
circulating current-free mode, 22
contactor, 10
control curve, 22
converter, AC/AC, 27
converter, AC/DC, 13
converter, dc link, 27
converter, DC/AC, 34
converter, DC/DC, 50
converter, direct frequency, 27
converter, push-pull, 54
cycloconverter, 27
discontinuous current boundary, 26
discontinuous-current mode, 22
duty cycle, 31
duty ratio, 31
electric drive, 6
electromagnetic compatibility, 10
electromagnetic interference, 10
frequency ratio, 42
fuse, 9
gate circuit, 20, 37
gate driver, 20
interference, 10
interpolation, asymmetrical, 43
interpolation, symmetrical, 43
inverter, 34
inverter, autonomous, 35
inverter, CSI, 35
inverter, CSI forced-commutated, 36
inverter, line-fed, 35
inverter, offline, 35
inverter, online, 35
inverter, resonant, 36
inverter, voltage stiff, 35
inverter, VSI, 35
inverter, VSI single-phase full-bridge, 35
inverter, VSI three-phase bridge, 36
modulating signal, 41
modulation index, 42
modulation method, 38
modulation ratio, 42
modulation, block, 39
modulation, phase, 22
modulation, PWM, 41
modulation, rectangle, 38, 39
modulation, sinusoidal, 41
modulation, six-step, 40
modulation, space vector, 45
modulation, square-wave, 38, 39
modulation, vectorial, 45
power electronic converter, 6
pulse distribution unit, 21
rectifier, 13
rectifier, controlled, 13
rectifier, half-wave, 13
rectifier, midpoint, 13
rectifier, single-phase bridge, 13
rectifier, single-phase full-wave, 13
rectifier, single-phase half-wave, 13
rectifier, three-phase bridge, 15
rectifier, three-phase three-diode, 15
rectifier, uncontrolled, 13
regulator, boost, 50
regulator, buck, 50
resistor, ballast, 20
sensor, 10
snubber, 19
space vector, 46
voltage feedback, 20
zero space vector, 46