Tài liệu Topologies of Power Electronic Converters doc

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Topologies of Power Electronic
Converters
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
18
3. TOPOLOGIES OF POWER ELECTRONIC CONVERTERS
3.1 Introduction
Power electronic converters (PECs) are static devices, without any movable parts, that
convert electric energy of one set of properties into electric energy of another, different set of
properties. The properties that are changed by the action of the converter are one or more of

the following: number of phases, frequency, and voltage rms (or average, in the DC case)
value. Thus any power electronic converter is essentially a transformer in a broad sense. Action
of a power electronic converter is illustrated in Fig. 3.1, in terms of the defined electric energy
properties.
m
i
,f
i
,V
i
m
o
,f
o
,V
o
PEC
Fig. 3.1: Conversion of electric energy by means of a power electronic converter.
Electric circuits that constitute power electronic converters vary to the great extent and
depend on the function that PEC is supposed to perform in terms of electric energy properties.
However, the unique feature of any power electronic converter is that it comprises at least one
(and in reality always more than one) power semiconductor, that is operated in the switched
mode. Operation in the switch mode means that the semiconductor is either fully on in the
circuit (conducting required current with almost zero voltage drop) or it is fully off (blocking
the required voltage with zero current). Power semiconductors are for this reason usually
called switches when PECs are discussed.
In order to arrive to a frequently used classification of PECs, consider the following
couple of examples:
1. A DC motor for its normal operation has to be supplied with a DC voltage. Suppose
that only mains AC voltage is available. In order to connect the DC motor to the mains, an

interface has to be used, that will convert AC into DC. Thus AC→DC conversion is required,
and it is called rectification. PECs that perform rectification are termed rectifiers.
2. Nowadays one frequently meets in homes light sources whose intensity of light can be
varied. Light bulbs are supplied from single-phase AC source of 240 V, 50 Hz and, in order to
have controllable light intensity, a PEC is required. As bulbs require AC supply, then the PEC
has to convert fixed voltage, fixed frequency AC supply into fixed frequency, variable voltage
AC supply (intensity of light will reduce when voltage applied to the bulb is decreased). Hence
the PEC is required to perform AC→AC conversion, in which only rms value of the voltage at
the output can be varied. This type of PEC is usually called AC voltage controller.
3. Milk delivery in early morning hours is done by electric vehicles with on-board source
of electric energy (a battery). The vehicle is powered by a DC motor which has to operate at
variable speed and therefore, as discussed later, requires variable DC voltage for its operation.
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
19
In order to obtain variable DC voltage from the available constant battery DC voltage, a PEC
has to be used, which will perform DC→DC conversion.
4. An induction machine that is to be used as part of a variable speed drive has to be
supplied with voltage of variable rms value and variable frequency. The utility supply is fixed
frequency, fixed voltage (three phase, 50 Hz, 415 V). Therefore a PEC converter is needed
that will converter input AC into variable voltage, variable frequency output AC. This
conversion is usually done in two stages. Input AC is at first rectified, using a rectifier. Next,
another PEC is used to perform DC→AC conversion. This process is called inversion and
PECs that do inversion are known as inverters.
The most frequently utilised classification of PECs is based on properties of electric
energy at the input and at the output of the converter. Electric energy at both input and output
may be either DC or AC, as shown in the four examples above. This leads to the subdivision of
power electronic converters into four categories: AC to DC converters (rectifiers), DC to AC
converters (inverters), DC to DC converters and AC to AC converters. This classification, that
will be used further on, does suffer from one serious disadvantage: certain power electronic

converters may operate as both rectifiers and inverters, making the classification invalid.
However, there is not at present a better way of classifying the converters.
The other possibility, mentioned in the literature, is to classify the converters on the
basis of the way in which the semiconductor devices are switched. In this case there are only
two groups, namely line frequency converters and switching converters. Line frequency
converters are those where utility line voltages are present at one side of the converter (input
or output) and these voltages facilitate the turn-off of the power semiconductor devices.
Similarly, the semiconductors are turned on phase-locked to the line voltage wave-form.
Consequently, the semiconductors are switched on and off at switching frequency equal to the
utility (mains) frequency (50 or 60 Hz). The other group, switching converters, then
encompasses all the converters in which semiconductors are turned on and off at a frequency
that is different from, and usually high when compared to the utility frequency. This means that
switching is independent of the utility frequency, although output of the converter may be DC
or AC at a fundamental frequency that is comparable to the line frequency. Although this
classification of PECs is probably the only fully consistent one, it is usually found impractical
because the vast majority of converters that are nowadays in use would fall into the switching
converter group. This is a consequence of the development of improved semiconductor
devices that allow for higher switching frequencies. As will be shown later, higher internal
switching frequency of a converter is highly desirable, because it leads to an improvement in
the quality of the output and input voltage and current wave-forms. In other words, the output
wave-form will be closer to the ideal one if the switching frequency is higher.
It is worth noting that, depending on the output of the PEC, ideal voltage wave-form is
either a constant voltage (if output is DC) or an ideal sine wave (if the output is AC). This is
illustrated in Fig. 3.2. Unfortunately, PECs are not capable of realising these ideal wave-forms,
as shown shortly. This means that the output voltage is never ideally constant (if output is DC)
nor it is an ideal sine wave (if the output is AC). The consequence of this is that output of any
PEC contains always not only the desired output voltage but higher harmonics of the output
voltage as well. As already pointed out, increase in switching frequency enables reduction in
undesired voltage components (higher harmonics) in the output voltage wave-form.
In what follows basic circuits of the four above mentioned types of PECs, namely AC

to DC, DC to AC, DC to DC and AC to AC converters, will be reviewed and their principles
of operation explained. It should be noted that many applications will require series connection
of more than one converter, as already mentioned in the fourth example. This is typically the
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
20
case in AC motor drives where two (and sometimes even three) converters are connected in
series in order to obtain electric energy of required properties at the output.
v
o
v
o
v
o
=V
o
v
o
= √2V
o
sin ωt
tt
Fig. 3.2: Desired, ideal voltage wave-forms at the output of AC→DC and DC→DC
converters (left) and AC→AC and DC→AC converters (right).
3.2 AC to DC Converters (Rectifiers)
Rectification is undoubtedly the most frequently met application of PECs. The input to
the converter is in this case AC utility voltage of fixed frequency (50 or 60 Hz) and of fixed
rms value. The output voltage is DC and, depending on the application, it may be required to
be constant or variable. Vast majority of rectifiers is based upon utilisation of thyristors (or
diodes if DC voltage is required to be constant) and rectifiers are operated as line frequency

converters. In other words, thyristors are naturally commutated, by means of line voltage
present at the AC side. Thus thyristor ceases conduction and returns to the off state when
either current through the thyristor naturally falls to zero or when the next thyristor is turned
on and it takes over the current from the thyristor which was in on state in the previous
interval. The switching frequency of thyristors equals line frequency, meaning that each
thyristor can be fired and brought into on state only once in a period of the input voltage.
3.2.1 Single-phase, single-semiconductor (half-way) rectifier
Output voltage and current of any rectifier depend greatly on the type of the load at the
DC side. The simplest possible rectifier, that comprises only one thyristor, is illustrated in Fig.
3.3. Characteristic wave-forms in the circuit are shown as well, for two types of loads: purely
resistive load and resistive-inductive load. Operation is illustrated for two values of the
thyristor firing angle α which is measured with respect to the zero crossing of the utility AC
voltage: α = 0 degrees and α = 90 degrees. Note that α = 0 degrees represents at the same
time operation of the same circuit in which a diode is placed instead of the thyristor. For purely
resistive load current is in phase with the AC voltage and therefore thyristor ceases conduction
when current and voltage fall to zero at 180 degrees. However, for resistive-inductive load
current continues to flow for some time after the mains voltage has reversed, so that the output
DC voltage contains negative sections. This indicates that for the same firing angle average DC
voltage is lower when the load is resistive-inductive. Average DC voltage is denoted with
constant value bold straight lines in Fig. 3.3 and with symbol ‘V’ (index o for output is
omitted).
With respect to the basic description of a PEC, equation (1.1), circuit of Fig. 3.3
performs the conversion of the type (assuming 240 V, 50 Hz input): 240 V, 50 Hz, single-
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
21
phase → 0 Hz, V. Average value of the output voltage V can be varied between zero and
maximum value obtainable with zero firing angle.
Consider at first operation with purely resistive load and zero firing angle. Output
voltage and current in the circuit are then determined with (T is period of input voltage)

vx t ivR
v
i
==
==
2 240 2 50
00
sin
π
kt < t < (2k +1)T / 2 k = 0,1,2,.....
otherw
ise
(3.1)
Input voltage wave-form
+v
Th
50 Hz R, ωt
AC v L i
0 180 360 (°)
vv
VV
180 360 (°)
ii
180 360 (°) 180 360 (°)
Zero firing angle
vV v
V
90 180 360 450 (°) 90 450 (°)
ii
90 180 360 450 (°) 90 450 (°)

Firing angle of 90 degrees
Purely resistive load Resistive-inductive load
Fig. 3.3: Single-thyristor rectifier and wave-forms for resistive and resistive-inductive load
for two values of the thyristor firing angle.
Note that input and output current are equal as there is only one current path in the circuit.
Note as well that at all times applied input AC voltage equals sum of the output voltage and
the voltage across thyristor. Thyristor voltage, when thyristor is on, is zero in positive half-
periods of the input voltage, when output voltage equals input voltage. If thyristor is off during
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
22
positive half-cycle, then thyristor voltage equals input voltage, while output voltage is zero. In
negative half-cycles of the input voltage output voltage is zero, while thyristor voltage
becomes equal to the input voltage. Thus thyristor voltage is negative during negative input
voltage half-periods. Operation of the circuit can be described in terms of angle rather than
time, for any value of the firing angle, with the following set of equations (one period of input
voltage is considered;
θ
=
ω
t):
vx ivR v
vivx
Th
Th
== <<
=== <<<<
2 240
00224020
sin /

sin
θαθπ
θπθπ θα
=0
and
(3.2)
Wave-form of the instantaneous output voltage in Fig. 3.3 considerably differs from the ideal
one shown in Fig. 3.2. Although it can be improved, as shown shortly, the ideal one can never
be obtained.
Average value of the output voltage V is given with (T stands for period of the input
voltage - for 50 Hz, T =20ms):
[]
[]
()
V
T
vdt vd V d
VV
V
T
vdt vd V d
VV
T
i
ii
T
i
ii
== = =−=
== = =−=

òòò
òòò
11
2
1
2
2
2
2
2
11
2
1
2
2
2
2
2
2
00
2
0
0
00
2
π
θ
π
θθ
π

θ
π
π
θ
π
θθ
π
θ
π
α
ππ
π
π
α
π
α
π
sin cos
sin cos
for diode case
1+ cos for thyristor case
(3.3)
Variation of average output voltage with thyristor firing angle is illustrated in Fig. 3.4.
V
√2V
i
/π
√2V
i
/2π

0
0 90 180 α(°)
Fig. 3.4: Variation of average voltage with firing angle for single-phase, single thyristor
rectifier with resistive load.
If the load in the circuit of Fig. 3.3 is resistive-inductive, then the current in the circuit
lags voltage and the instant in time when current falls to zero (i.e., when thyristor turns off) is
not known in advance. In order to find this time-instant, it is necessary to at first solve
differential equation of the circuit for current. Once when expression for current is obtained,
instant when current reaches zero value can be calculated. When this instant is known, average
value of the output voltage can be calculated using the procedure given in (3.3): it is only
necessary to change upper border of integration from π to β,whereβ corresponds to time
instant when current falls to zero and β>π.
The rectifier of Fig. 3.3 is very rarely utilised in practice due to pure quality of the
output DC voltage. The rectifier topology that is most frequently met in practice is the bridge
topology, with either single-phase or three-phase input.
3.2.2 Single-phase, bridge (full-wave) rectifier
Both single-phase and three-phase bridge rectifiers, that again utilise thyristors and are
hence once more line commutated rectifiers, are shown in Fig. 3.5. These versions of the
bridge topology are usually called fully controllable bridge rectifiers as all the semiconductors
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
23
are of the thyristor type. Alternatively, in certain applications so-called semi-controllable
bridges are used: in this case upper half of the rectifier is based on thyristors while the lower
half comprises diodes. In semi-controllable rectifiers diodes prevent instantaneous DC voltage
from going negative. In what follows only the fully controllable bridge topology is discussed.
Note that again all the thyristors can be substituted with diodes: in this case output voltage
average value is fixed for given input voltage. Operation of the diode bridge rectifier coincides
with operation of the thyristor bridge rectifier whose firing angle is zero.
All the wave-forms that are to be shown apply to the single-phase fully controllable

bridge rectifier. Operation of the three-phase bridge rectifier, that will be dealt with in the
section on rectifier control of DC motor drives, is in principle the same but the wave-forms are
more complicated due to three-phase input. However, quality of the DC voltage is better in
three-phase rectifier, where so-called six-pulse wave-form is obtained as DC voltage. In the
single-phase rectifier wave-form of the DC voltage is two-pulse.
ii
12
LL
OO
vA v A
DD
34
Single-phase bridge rectifier Three-phase bridge rectifier
Fig. 3.5: Configurations of single-phase and three-phase fully-controllable bridge rectifiers.
Wave-forms in the circuits of Fig. 3.5 greatly depend on the type of the load at the DC
side. Four cases may be distinguished: resistive load, resistive-inductive load, resistive-
inductive load with a DC source, and capacitive filter connected in parallel to the rectifier
output and providing almost constant DC voltage to the subsequent load. The third and the
fourth case will be dealt with later on. The remaining two cases are examined here for the
single-phase bridge fully controllable rectifier.
Figure 3.6 illustrates wave-forms in the circuit for two values of the firing angle, zero
degrees and 90 degrees, for purely resistive load and for resistive-inductive load. In the case of
resistive-inductive load it is assumed that the inductance is sufficiently high to maintain DC
current at almost constant level. Such a situation is met when the rectifier supplies current
source inverter (i.e., load at rectifier output is an inductance connected in series with the
positive DC terminal; DC voltage after the inductance then serves as input into the current
source inverter which performs DC to AC conversion).
The single-phase bridge rectifier is called two-pulse rectifier because the output DC
voltage contains two identical portions of the input sine-wave for one period of the input (Fig.
3.6). During positive half-cycle of the input voltage thyristors 1 and 4 are positively biased,

they are connected in series, and can be fired to start the conduction at any time between 0 and
180 degrees. Thyristors 2 and 3 are negatively biased and cannot conduct in positive half-cycle.
During negative half-cycle of the input voltage situation is reversed: thyristors 1 and 4 are now
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
24
negatively biased and they cannot conduct; however, thyristors 2 and 3 are positively biased
and therefore they can be fired to start conduction at any time instant between 180 and
Input AC voltage
v 1,4 2,3 1,4 v 1,4 2,3
VV
180 360 (°)
ii
180 360 (°) 180 360 (°)
Input AC current Input AC current
Zero firing angle
vV v
V
90 180 360 450 (°)
ii
90 180 360 450 (°) 90 450 (°)
Input AC current Input AC current
Firing angle of 90 degrees
Purely resistive load Highly inductive load
Fig. 3.6: Wave-forms in single-phase bridge rectifier: purely resistive and highly inductive
load.
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
25
360 degrees. Both half-cycles of the input voltage are now utilised and the output voltage

contains, for zero firing angle, rectified input AC voltage (i.e., absolute value of the input).
From Fig. 3.6 it is evident that neither output DC voltage nor current are pure DC nor
is the current drawn from the utility pure sine wave (except for purely resistive load with zero
firing angle). All these quantities contain considerable amount of undesirable higher harmonics.
For highly inductive load input AC current is a square wave, displaced by the firing angle with
respect to the input AC voltage. Thus the firing angle determines phase displacement between
AC current and voltage at rectifier input terminals and the rectifier always appears to the mains
as consumer of reactive energy (current is lagging voltage by the firing angle). Note that input
AC current in bridge rectifier is no longer of the same wave-form as it was in the case of a
single thyristor rectifier. It is AC, while output current is DC.
It follows from Fig. 3.6 that output current can be either discontinuous (pure resistive
load) or continuous (highly inductive load). Thus average voltage across the load has to be
determined separately for these two cases. Average voltage is:
[]
[]
()
single- phase diode bridge rectifier
single- phase thyristor bridge rectifier, purely resisitve load
1+cos
single- phase thyristor bridge rectifier, highly inductive load
V
T
vdt vd V d
VV
V
T
vdt vd V d
VV
V
T

vdt vd V d
V
T
i
ii
T
i
ii
T
i
i
== = =−=
== = =−=
== = =
òò ò
òò ò
òò ò
+
11
2
2
1
2
2
222
11
2
2
1
2

2
22
11
2
2
1
2
2
2
00
2
0
0
00
2
00
2
π
θ
π
θθ
π
θ
π
π
θ
π
θθ
π
θ

π
α
π
θ
π
θθ
π
ππ
π
π
α
π
α
π
π
α
πα
sin cos
sin cos
sin
[]
−=
+
cos cos
θ
π
α
α
πα
22V

i
(3.4)
One notes from (3.4) that average output voltage as function of the firing angle considerably
differs depending on whether the load is purely resistive or purely inductive. One notes as well
that average output voltage of the bridge rectifier is doubled with respect to the values
obtainable with single thyristor rectifier. Average output voltage is illustrated for these two
cases in Fig. 3.7.
V purely resistive load
2√2V
i
/π
0
90 180 α (°)
highly inductive load
-2√2V
i
/π
Fig. 3.7: Average output voltage of a single-phase fully controllable rectifier for purely resisti-
ve and highly inductive loads.
The average output voltage for highly inductive load is thus directly proportional to the
cosine of the firing angle for continuous DC current (i.e., V = k cos α). The average DC
voltage is positive for firing angles between zero and 90 degrees and negative for firing angles
between 90 and 180 degrees. As current flow is unidirectional then the power supplied to the
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
26
load is positive for firing angles between zero and 90 degrees and the circuit operates in
rectifying mode. However, for firing angles greater than 90 degrees power supplied to the load
attains negative sign. The circuit now operates in inverting mode and the meaning of the
negative sign of power is that the power is transferred actually from DC to AC side. The

operation of the circuit in inverting mode requires that a DC voltage source is present at the
DC side and that its polarity is such that it supports current flow in the direction indicated in
Fig. 3.6. Operation of the circuit in inverting mode is widely utilised in DC motor drives, where
the circuit operates as rectifier during motoring and as a line-commutated inverter during
regenerative braking. Inversion is illustrated in Fig. 3.8, assuming continuous DC current flow.
Extreme case, with firing angle equal to 180 degrees, is shown. Average output voltage has
now maximum, but negative value; output voltage direction arrow in the circuit shows the
direction in which acts absolute value of the average output voltage. As current can flow only
in the direction shown, then value of the DC source voltage E must be greater than the
absolute value of the converter output voltage. Instantaneous voltage at converter DC side is
at all times negative. Power is transferred from DC side to AC side, so that the converter now
operates as a line commutated inverter (DC to AC conversion).
iR
v
L 0 180 360 degs
V
EV
+
Fig. 3.8: Operation of the single-phase bridge converter in inverting mode.
Note that the situation shown for highly inductive load with 90 degrees firing angle in
Fig. 3.6 is the one for which average DC voltage is zero. Thus it denotes transition from
rectifying to inverting mode, providing that there is a DC voltage source of adequate polarity
at DC side.
It should be noted that average voltage across any inductor at DC side equals zero.
This means that the whole of the average voltage, assuming resistive-inductive load, is
developed across the resistor. Hence the average value of the DC current delivered to the DC
load equals
I=V/R (3.5)
regardless of the rectifier type. In other words, it is always necessary to find average value of
the voltage only, using expressions of the type (3.4). Once when average voltage is known,

average current follows from (3.5).
Example:
Aresistive5Ω load is to be supplied from a single-phase AC supply of 240 V, 50 Hz,
through a rectifier. The required power which has to be delivered to the load is 500 W. There
are two rectifiers available: the single thyristor rectifier and the single-phase bridge thyristor
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
27
rectifier. Calculate the firing angles for both thyristor rectifiers. Evaluate the power that would
have been supplied to the load by applying the same rectifier topologies, however with diodes
instead of thyristors. Perform all the calculations by neglecting contribution of the higher
voltage and current harmonics to the output power (this is an unrealistic assumption!).
Solution:
The power required by the load is, neglecting the contribution of the harmonics, given
with P=VI=RI
2
=V
2
/R = 500 W
Hence required average value of the rectifier output voltage is
V=
√
PR =
√
500x5 = 50 V
Average output voltage yields required firing angle for both rectifiers, as follows.
()
()
()
single thyristor rectifier

degrees
bridge rectifier
degrees
V
V
VV
V
V
VV
i
i
i
i
=+
=−= −=−=−
=− =
=+
=−= −=−=−
=−=
−
−
2
2
1
2 1 250 240 1 0 925 1 0 0744
0 0744 94 26
2
1
2 1 50 2240 1 0 462 1 0537
0537 122 5

1
1
π
α
απ π
α
π
α
απ π
α
cos
cos / / . .
cos ( . ) .
cos
cos / ( ) . .
cos ( . ) .
Had diode rectifiers been used instead of controllable thyristor rectifiers, the following
power would have been supplied to the load:
single- diode rectifier
V
W
bridge diode rectifier
V
W
VV
PVR
VV
PVR
i
i

== =
== =
== =
== =
2 2240 108 04
108 04 5 2334 44
2 2 2 2240 216 08
216 08 5 93378
22
22
ππ
ππ
.
..
.
..
3.3 DC to AC Converters (Inverters)
Depending on the basic operating principle, which determines inverter output
frequency, inverters can be subdivided into two groups. The first one encompasses line
frequency inverters, where the utility line voltages present at the output side of the converter
facilitate the turn off of switches. The inverter output frequency is in this case fixed and equal
to the mains frequency. As already noted, this is one of the possible operating regimes of fully
controllable rectifier circuits and this type of inversion is of interest only in DC motor drives
with regenerative braking. Line commutated inverters are built utilising thyristors.
The second type of inverter is so-called autonomous inverter. Inverter output is
connected to a single-phase or three-phase system which is independent of mains, so that
output frequency is variable. The inverter output frequency may be both higher and lower than
the mains frequency. Switches within the inverter may be turned on and off at the output
frequency or at a frequency that is significantly higher than the output frequency. In what
follows principle of operation of the most frequently utilised voltage source inverter (VSI) is

explained. The input into the inverter is either a DC voltage source or DC voltage across a
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
28
capacitor connected to the output terminals of a rectifier. A single-phase bridge inverter is
illustrated in Fig. 3.9. Here switch ‘S’ denotes essentially two semiconductor devices
connected in anti-parallel, as shown in Fig. 2.12. The first one is nowadays a fully controllable
semiconductor, whose instants of both turn on and turn off may be controlled (there are still
VSIs based on thyristors that are in use; in that case additional commutation circuit is needed
in order to turn the thyristor off) while the second one is a power diode. This is shown in Fig.
3.9 as well. The existence of a diode in anti-parallel connection with each of the controllable
switches is necessary in order to enable current flow with inductive and capacitive loads.
The inverter is operated in such a way that switches 1 and 2 are on and 3 and 4 are off
during the first half-cycle of the output frequency. In the second half-cycle 1 and 2 are off
while 3 and 4 are on. Thus each switch is on for 180 degrees. This leads to connection of the
reversed DC input voltage to the load in the second half-cycle so that the resulting output
voltage is square-wave AC. Output voltage and current of the single-phase inverter of Fig. 3.9
are shown in Fig. 3.10 for purely resistive and resistive-inductive load. The role of back-to-
back diodes is evident from the inverter output current wave-form for resistive-inductive load.
Change of operating frequency is illustrated in Fig. 3.10 as well. This is achieved simply
by means of changing the duration of the interval during which a pair of switches is on (and the
other pair off). However, it is obvious that amplitude of the output voltage remains the same
and equal to input DC voltage regardless of the output frequency. Thus, if variable output
voltage is required (as the case is when inverter supplies an AC machine), variation of the
output voltage magnitude has to be done by variation of the DC voltage across the capacitor.
In other words, variable voltage, variable frequency operation of the VSI operated in 180
degrees conduction mode asks for application of controllable rectifier as the DC voltage input
source. The rectifier is then used to vary the inverter output voltage magnitude, while the
inverter controls output frequency.
Another possibility of the control of the single-phase inverter is to apply Pulse-Width-

Modulation (PWM) instead of continuous 180 degrees conduction. This will be discussed in
conjunction with inverter fed variable-speed induction motor drives.
Rectifier i
S1 S3
VC vLoad S=
S4 S2
0<t<T/2 T/2<t<T
S1 S3
VR,VR,
vL vL
S2 S4
Fig. 3.9: Single-phase bridge autonomous inverter and its equivalent circuits in two
half-periods of the output frequency.
ENGNG3070 Power Electronics Devices, Circuits and Applications
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29
Input voltage
t
VV
vv
tt
-V -V
ii
tt
T1,T2 T3,T4 T1,T2 T3,T4
D1,D2 D3,D4
Resistive load Resistive-inductive load
vv
tt
Change of output voltage frequency

Fig. 3.10: Output waveforms for a single-phase bridge inverter operating in 180 degrees con-
duction mode.
Total rms value of the output AC voltage of an inverter is determined with
V
T
vdt vd
T
==
òò
11
2
2
0
2
0
2
π
θ
π
(3.6)
where T and V denote period of the output voltage and total output voltage rms value.
Example:
A purely resistive 5 Ω load is to be supplied from a single-phase inverter, which
provides square-wave output voltage shown in Fig. 3.10. Output voltage total rms value is
required to be 240 V and the output frequency is 100 Hz. Determine the necessary input
average DC voltage value.
Solution:
The output voltage total rms value is
Vvd v
V

V
V vd vd VdV dV
VV
i
i
iiii
i
== =
−
== ===
==
ò
òòòò
1
2
240
1
2
2
1
2
11
240
2
0
2
2
0
2
2

0
2
00
π
θ
θπ
πθ π
π
θ
π
θ
π
θ
π
θ
π
ππππ
Vwhere
0< <
<<2
Hence
V
ENGNG3070 Power Electronics Devices, Circuits and Applications
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30
3.4 DC to DC Converters
The DC to DC converters are widely used in regulated switch-mode DC power
supplies and in DC motor drive applications. The input to these converters is often an
unregulated DC voltage, which is obtained by rectifying AC mains voltage, and therefore it will
fluctuate due to changes in the AC voltage magnitude. Switch mode DC to DC converters are

used to convert the unregulated DC input into a controlled DC output at a desired voltage
level. Alternatively, input into a DC to DC converter may be from a constant DC source such
as a battery. Switch-mode DC to DC converter is then used to provide regulated DC voltage
output from the constant DC voltage input. Such a situation is met in electric vehicles with on-
board electric source (battery) and regulated DC voltage is then required for the supply of a
DC motor drive system. Structure of a system with DC to DC converter is shown
schematically in Fig. 3.11.
AC line voltage DC DC DC
Diode Filter DC to DC Load
rectifier (unregulated) capacitor (unreg.) converter regulated
Control
Fig. 3.11: A DC to DC converter system.
The available DC to DC converters are numerous and encompass step-down (buck)
converter, step-up (boost) converter, step-down/step-up (buck-boost) converter, Cuk
converter, full-bridge converter etc. However, only the first two types (step-down and step-up
converter) are basic converter topologies while all the other types are either combinations of
these two or are derived from one of the basic two topologies. The discussion is in what
follows for this reason restricted to step-down (chopper) converter, while step-up converter is
dealt with later on.
As the name suggests, step-down DC to DC converter is capable of producing output
DC voltage whose average value is smaller or at most equal to the average value of the input
DC voltage. On the contrary, step-up converter produces output DC voltage whose average
value is greater or at least equal to the input voltage average value.
The basic idea of a step-down converter is illustrated in Fig. 3.12, where constant input
DC voltage is assumed. Switch ‘S’ may be any of the fully controllable power semiconductors
(it is shown as a transistor Fig. 3.12); alternatively, it may be a thyristor with additional circuit
components that would provide forced commutation. The load is for this idealised discussion
shown as a pure resistance. When switch is closed input DC voltage appears across the resistor
and therefore output voltage equals input voltage. When switch is open there is no current
flow through the resistor and the output voltage therefore equals zero. Hence average voltage

at the output can be varied by varying the on and off times of the switch, while keeping the
period of operation of the switch constant (i.e., switching frequency is kept constant).
Alternatively, on-time may be kept fixed, while off-time and hence switching frequency are
varied. Creation of output voltage with different average values is shown in Fig. 3.12,
assuming that period of operation (switching frequency) is constant.
ENGNG3070 Power Electronics Devices, Circuits and Applications
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31
The average voltage at the converter output is proportional to the product of the input
DC voltage and the so-called duty ratio. Duty ratio is defined as ratio of switch on-time to the
switch operating period. Hence the duty ratio can be varied from zero up to unity, giving
variation of chopper average output voltage from zero up to at most input voltage. Indeed,
from Fig. 3.12 it follows that the average value of the output DC voltage is
V
T
vdt
T
VdtV
t
T
V
tT
T
DC
t
DC
on
DC
on
on

== ==
==
òò
11
00
δ
δ
duty cycle
(3.7)
The converter is essentially operated in PWM mode since the output voltage is varied
by means of pulse width modulation. Normally period of operation of the switch is very small,
so that switching frequency is nowadays well into kHz region. As the switching frequency is
constant, harmonic content of the output voltage is well defined.
Circuit of Fig. 3.12 is of no practical importance for two reasons. First of all, in every
circuit there is some inductance and it is therefore necessary to provide path for current flow
when the switch is turned off. Secondly, voltage applied to the load has step changes from zero
up to the input voltage value (i.e., high voltage ripple). The first problem is solved by addition
of a flywheel diode. The second problem is overcome by insertion of a low-pass filter between
the chopper and the load. Configuration of a chopper used in practise is shown in Fig. 3.13.
The voltage across the flywheel diode (FD) remains to be of pulsed nature and is identical to
voltage waveforms of Fig. 3.12. The role of the low pass filter is to smooth this voltage so that
voltage across the load becomes more or less constant and equal to the required average value.
Step-down DC to DC converter may operate in continuous conduction mode or
discontinuous conduction mode. In continuous conduction mode current through the filter
inductance is continuous, while in discontinuous mode it falls to zero at certain point in time in
each operating period. In which mode the chopper operates depends on a number of factors
including filter inductance, required average load output current, chopper on-time and average
input and output voltage values.
V
average

=V
V
DC
+v
St
t
on
T2T3T
V
DC
v
Rv
t
V
average
=V
V
average
=V
V
DC
v
t
t
Period of chopper operation = T
Fig. 3.12: Step-down (buck, chopper) converter and its principle of operation.
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
32
Low-pass filter

S
L
V
DC
vCLoadv
0
Fig. 3.13: Step-down DC to DC converter - practical outlay.
Example:
A purely resistive 5 Ω load is to be supplied from a step-down DC to DC converter.
The required average DC voltage value is 50 V. The chopper is of the form shown in Fig. 3.12
and it can be operated either with a) fixed switching frequency of 1 kHz, or with b) fixed on-
time of 0.5 ms. The input DC voltage is 250 V. Calculate on and off time for the fixed
switching frequency control, and switching frequency for the fixed on-time control.
Solution:
Average output voltage of the chopper is
V=V
DC
t
on
/T 50 = 250 t
on
/T t
on
/T =0.2
Hence for fixed switching frequency control, with 1 kHz, T =1msand
t
on
= 0.2 ms, t
off
=0.8ms.

For fixed on-time control t
on
=0.5ms;thusT = 2.5 ms and switching frequency is
f=1/T=400 Hz.
3.5 AC to AC Converters
Input into an AC to AC converter is either single-phase or three-phase mains voltage of
fixed magnitude and frequency. Depending on the characteristics of the output AC voltage, AC
to AC converters may be subdivided into two groups. The first group encompasses so-called
AC to AC voltage controllers whose role is to provide an output AC voltage of variable rms
value at the frequency equal to the input voltage frequency. The second group encompasses
so-called AC frequency changers which provide at the output an AC voltage whose both rms
value and frequency are variable. AC frequency changers can further be subdivided into
cycloconverters and matrix converters. Cycloconverters have very limited output frequency
range, which is in three-phase case usually restricted to at most one third of the input
frequency. Predominant application of cycloconverters is in induction and synchronous motor
drives in very high power range where low speed operation is required. Matrix converters are a
relatively new class of converters which do not have restriction on output frequency range.
They are intended for direct AC to AC conversion without intermediate DC link which is
required in cascaded connection of a rectifier and an inverter. Matrix converters are still at
development stage and there is not at present evidence of their wider application.
AC to AC voltage controllers and cycloconverters are based on thyristors and thyristor
commutation is achieved in a natural manner, by means of mains voltages present at the input
side of the converter. Thus they belong to the class of line commutated converters. In contrast
to this, matrix converters are based on fully controllable switches.
As matrix converters are still at development stage, while application of
cycloconverters is mostly restricted to high power, low speed operated AC motor drives, only
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
33
AC to AC voltage controllers are discussed in more detail here. The basic idea of the

cycloconverter operation is given at the end of this section.
Topologies of a single-phase and of a three-phase AC to AC voltage controllers are
shown in Fig. 3.14. A back-to-back (anti-parallel) connection of two thyristors of the type
shown in Fig. 2.13 constitutes the voltage controller in each phase of the input AC supply.
Thyristor firing is symmetrical in the two half-periods of the input voltage. The upper thyristor
provides path for current flow during positive half-cycle of the input voltage, while the lower
thyristor enables current flow during negative half-cycle of the input voltage. Thyristor firing is
synchronised with the zero crossing of the mains voltage and thyristor firing angle α is
measured from input voltage zero crossing. The operation of the single-phase AC to AC
voltage controller is further illustrated in Fig. 3.14 for two types of the load: purely resistive
load and resistive-inductive load. Note that the current taken from the mains equals the output
current by virtue of converter operation. Output frequency is equal to the input frequency,
while output rms voltage depends on the firing angle and on the angle at which current falls to
zero. Analysis of the converter is simple for purely resistive load, because current falls to zero
at the end of each 180 degrees interval and output voltage becomes at this instant equal to
zero. However, if the load is resistive-inductive, the instant at which current falls to zero (this
instant determines when voltage at the load will become equal to zero) is in general not known.
It has to be determined for each firing angle and load parameters from solution of the
differential equation that governs the behaviour of the circuit. The situation is even more
complicated in the three-phase case, due to mutual interaction of the phases.
In the case of a single-phase AC voltage controller that supplies a pure resistive load,
rms value of the output voltage can be determined as follows.
()
()
V
T
vdt vd vd vd V d
VV dV dV
VV
T

ii
ii i
i
== = = =
==−=
−
−
é
ë
ê
ù
û
ú
=−+
òò ò òò
òò
11
2
1
2
2
1
2
2
1
2
2
1
2
11

2
12 2
1
2
1
2
2
2
2
1
22
2
4
2
0
2
0
2
2
0
2
2
2
π
ϑ
π
θ
π
θ
π

θθ
π
θθ
π
θθ
π
πα
π
θ
α
π
α
π
ππ
α
π
α
π
α
π
α
π
α
π
sin
sin cos
sin
sin
(3.8)
Note that instantaneous values of output and input current are the same and given with i

i
=i=
v/R. Rms value of the input and output current is for purely resistive load I
i
=I=V/R.
Applications of AC to AC voltage controllers are numerous. Single-phase version is
used in light intensity control, where load is resistive (case dealt with here). Three-phase
versions are used for speed control of induction machines and in power systems for static
reactive power compensation. These two cases will be discussed in more detail in appropriate
sections on applications of power electronic converters.
Example:
A purely resistive 5 Ω load is to be supplied from a single phase thyristor AC voltage
controller. The input AC voltage is 240 V, 50 Hz. Determine the required thyristor firing angle
if the required output voltage rms value is 100 V.
Solution:
Equation (3.8) gives the output voltage rms value as function of the firing angle. When
α is known, calculation of V is simple. However, here firing angle is required. Hence
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
34
T1
i
i
i
T2
v
i
vLoad
Three-phase AC to AC voltage controller Single-phase voltage controller
v

i
ωt
αππ+α 2π
vv
tt
i, i
i
i, i
i
tt
T1 T2 T2 T1 T2
Resistive load Resistive-inductive load
Operation of single-phase AC to AC voltage controller with 90 degrees firing angle
Fig. 3.14: Topologies of three-phase and single-phase AC to AC voltage controller and wave-
forms for single-phase controller operating on resistive and resistive-inductive load.
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
35
VV
V
V
V
V
xx
i
i
i
=−+
=− +
−= −

−= −=−
−=
2
1
22
2
4
2
1
22
2
4
2
42 22
2 2 100 4 2 240 2 5192
2 2 5192
2
2
2
2
22
α
π
α
π
α
π
α
π
ππ αα

αα π π
αα
sin
sin
sin
sin / ( ) .
sin .
The last equation is transcendental and cannot be solved analytically for the unknown
firing angle. Solution can be obtained either graphically or by trial and error method.
By plotting functions 2α and sin2α against 2α and then by finding difference of the
two, it is possible to determine value of the firing angle for which the difference is
5.192. The value is obtained as 2α = 4.283 rad, α = 2.1415 rad = 122.7 degrees.
The basic power electronic circuit of an AC-AC voltage controller can be used to
explain the idea of the cycloconverter type of frequency changer. The circuit is illustrated in
Fig. 3.15 for the single-phase case and purely resistive load is assumed for simplicity. It is
important to realise that this circuit is of no practical value and is never used in practice. A
slightly more complicated version applies to the practise in the case of a single-phase
cycloconverter.
T1
i
i
i
T2
v
i
vR
Fig. 3.15: The most basic configuration of a single phase cycloconverter.
Let the firing angle of thyristors be zero degrees (operation with maximum output
voltage), and suppose that an AC voltage of 16.66 Hz is to be produced from the sinusoidal 50
Hz input voltage. Then, during the first three periods of the input voltage only thyristor T1 will

be fired at zero degrees, while thyristor T2 will be idle. In the following three periods only
thyristor T2 will be fired, while thyristor T1 will be idle. The input and output voltage are
illustrated in Fig. 3.16. Output current, assuming 1 Ohm resistor, is identical to the output
voltage in appearance and is the same as the input current. One notes that significant distortion
of the mains current takes place, since during the first three periods current flows only during
the positive half-cycle, while in the next three periods current flows only in the negative half-
cycle.
Regulation of the output voltage is simply accomplished by changing the firing angle of
the thyristors. The situation for the 90 degrees firing angle is illustrated in Fig. 3.16 as well.
Obviously, the output voltage is of only 50% rms value, compared with the zero firing angle
case. The output voltage is highly distorted (i.e. very far from the desired pure sinusoidal
waveform): fundamental harmonic (i.e. sine wave of 16.66 Hz frequency) is shown in Fig. 3.16
for zero firing angle as a dotted curve.
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
36
It is obvious from Fig. 3.16 that output frequency cannot exceed the input frequency.
As already pointed out, in practical applications of the three-phase cycloconverter for electric
motor drives the output frequency is usually at most one third of the input frequency.
v
i
(50Hzsinewave)
t
v (zero firing angle)
t
v (90 degrees firing angle)
t
Fig. 3.16: Input and output voltage wave-form for the circuit of Fig. 3.15, operated as a
cycloconverter: output voltage is shown for zero and 90 degrees firing angle and
for the output frequency equal to 1/3 of the input frequency.

3.6 Suggested Further Reading
Books:
[1] N.Mohan, T.M.Undeland, W.P.Robbins; Power Electronics: Converters, Applications and Design,
John Wiley and Sons, 1995.
[2] M.H.Rashid, Power Electronics: Circuits, Devices and Applications, Prentice-Hall International,
1994.
[3] C.W.Lander, Power Electronics, McGraw Hill, 1993.
[4] B.M.Bird, K.G.King; An Introduction to Power Electronics, John Wiley and Sons, 1983.
[5] R.P.Severns, E.Bloom; Modern DC-to-DC Switchmode Power Converter Circuits, Van Nostrand
Reinhold Company, 1985.
[6] K.Thorborg; Power Electronics, Prentice-Hall International, 1988.
[7] L.Gyugyi, B.R.Pelly; Static Power Frequency Changers, John Wiley and Sons, 1975.
[8] G.Moltgen; Line commutated thyristor converters, Siemens - Pitman, 1972.
[9] J.G.Kassakian, M.F.Schlecht, G.C.Vergese; Principles of power electronics, Addison-Wesley
Publishing Company, 1991.
[10] T.H.Barton; Rectifiers, cycloconverters and AC controllers, Clarendon Press, 1994.
[11] E.Ohno; Introduction to power electronics, Clarendon Press, 1988.
[12] S.K.Datta; Power electronics and control, Reston Pub. Co., a Pretice-Hall Company, 1985.
[13] K.K.Sum; Switch mode power conversion, Marcel Dekker Inc., 1984.
[14] M.Kazimierczuk; Resonant power converters, Wiley, 1995.
[15] P.C.Sen; Power electronics, McGraw-Hill, 1992.
[16] J.Vithayathil; Power Electronics: Principles and Applications, McGraw-Hill, 1995.
[17] B.K.Bose, ed.; Modern Power Electronics: Evolution, Technology, and Applications, IEEE Press,
1991.
[18] P.A.Thollot, ed.; Power Electronics Technology and Applications, IEEE Press, 1993.
[19] S.S.Ang; Power-Switching Converters, Marcel Dekker Inc., 1995.
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
37
4. FOURIER ANALYSIS AND TIME-DOMAIN ANALYSIS

4.1 Introduction
As already pointed out in the previous chapter, desired ideal waveforms at the
converter output are the pure DC and pure sinusoidal AC voltages, for converters whose
output is DC and AC, respectively. Unfortunately, as the discussion of the basic types of
power electronic converters in the previous chapter has clearly shown, such an ideal wave-
form can never be obtained. Regardless of whether the output is DC or AC, the voltage wave-
form will always depart from the ideal one to a smaller or greater extent. The direct
consequence of such a situation is that analysis of circuits supplied from power electronic
converters is more tedious than the analysis of circuits supplied from ideal DC or AC sources.
Such a situation leads to utilisation of Fourier analysis of periodic wave-forms as a
standard way of dealing with the harmonic effects brought in by non-ideal nature of the power
electronic converters. As is well-known, Fourier series represents a non-sinusoidal periodic
wave-form with a sum of the constant DC term (average value of the function) and a series of
sinusoidal functions of different frequencies.
The second problem encountered in many circuits supplied from thyristor based power
electronic converters is that the output current may become discontinuous. The circuit will
normally contain inductors and/or capacitors. In such a case the instant of cessation of the
thyristor conduction is not known. The only way to determine the instant when thyristor turns
off and therefore calculate, say, the average output voltage of the rectifier, is to solve the
differential equation of the circuit. Solution of the circuit differential equations is necessary as
well when the exact waveform of the, say, output current is needed for the certain known
output voltage wave-form.
The method of Fourier analysis and methods of solving the time-domain circuit
differential equations are therefore reviewed in what follows.
4.2 Fourier Analysis of Periodic Waveforms
Let v(t) be a periodic waveform with period of repetition T. Frequency of repetition is
then f = 1/T and this is at the same time frequency of the fundamental (first) AC component in
the waveform. Corresponding angular frequency is
ω
= 2

π
t. Such a periodic waveform can be
expressed as an infinite series of sinusoidal components, whose frequencies are k
ω
, k = 0,1,2...
The wave-form may or may not contain a DC term (k = 0), which represents the average value
of the function. With regard to power electronic converter output, the wave-form will contain
the DC component if the converter delivers DC voltage at the output and this will be the
desired average (‘ideal DC’) voltage. The rest will be the undesired harmonic components,
determined with the converter switching frequency. In other words, the first (fundamental)
harmonic is the unwanted component, as are all the higher frequency components as well.
In contrast to this, if the converter output is AC, then DC component (average value)
will not normally exist. The fundamental frequency is the desired output frequency of the
converter and the fundamental component corresponds to the desired ‘ideal AC’ wave-form.
Higher frequency components are the unwanted components.
A periodic voltage can be represented, using Fourier series, in the following way:
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
38
()
()
()
()
vt V A k t B k t
V
T
vtdt v d
A
T
vt k tdt v k d

B
T
vt k tdt v k d
vt V V k t
VAB AB
VVV
o
kk
k
o
T
k
T
k
T
ok k
k
kkk k kk
o
=+ +
==
==
==
=+ +
=+ =
=+
=
∞
=
∞

−
å
òò
òò
òò
å
cos sin
() ( )
()cos ( )cos
()sin ( )sin
() sin
tan
ωω
π
θθ
ω
π
θθθ
ω
π
θθθ
ωφ
φ
π
π
π
1
00
2
00

2
00
2
1
22 1
2
11
2
21
21
2
2
Alternatively,
Total rms value of the waveform is
1
2
2
2
3
222
+++ +++VV V V
kn
.......... ....
(4.1)
In (4.1) V
o
denotes average value of the waveform. If the waveform is pure AC then average
value is zero. Index 1 denotes the first AC harmonic, which is called fundamental component
of the waveform. The number of higher harmonics that will exist in the waveform can be
anything, from one to infinity. Whether or not both even and odd harmonics will exist, depends

on the type of the waveform. The type of waveform that will be frequently of interest later on
is pure AC, symmetrical in two half-periods, which is additionally either even function or odd
function. In these two cases Fourier series is given with
pure AC waveform, symmetrical in two half - periods, even function
even function
pure AC waveform, symmetrical in two half - periods, odd function
odd function
vt v t
vt V k t
vt v t
vt V k t
k
k
k
k
() ( )
() cos( )
() ( )
() sin( )
=−
=+
=− −
=+
+
=
∞
+
=
∞
å

å
221
221
21
0
21
0
ω
ω
(4.2)
It should be noted that many waveforms can be regarded as either even or odd, depending on
where the reference zero time instant is placed.
Example:
Consider the two square-waves, both of the same amplitude V
DC
and the same period T,
shown in Figure. Determine Fourier series for both cases.
v(t) v(t)
0 90 270 degs 0 180 360 degs
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
39
Solution:
Waveforms are the same, except that the left-hand side one is even, while the right-
hand side one is odd. This difference is due to different selection of the zero time
instant. Note that the same waveform has been shown in Fig. 3.10, as output voltage of
the single-phase voltage source inverter. Consider at first the square-wave given in the
left part of the Figure. It is a pure AC waveform (i.e., V
o
= 0), symmetrical in the two

half-periods and it is an even function of time. Hence its Fourier series contains only
harmonics of the order 2k + 1 (eq. (4.2)) and only coefficients A
2k+1
of the Fourier
series in (4.1) need to be determined. Thus
()
[]
()
[]
()
[]
()
()
[]
vt V k t V A
A v kd v kd V kd
AV
k
k
V
k
k
AV A V AV
k
k
kk
k DC
kDC DC
DC DC DC
() cos( ) /

()cos ()cos cos
sin
sin
() (
//
=+ =
=+= +=+
=
+
+
é
ë
ê
ù
û
ú
=
+
+
==− =
+
=
∞
++
+
+
å
òòò
221 2
1

21 4
1
21
4
21
4
21
21
4
212
21
4434
21
0
21 21
21
0
2
0
2
0
2
21
0
2
13 5
ω
π
θθθ
π

θθθ
π
θθ
π
θ
π
π
ππ
πππ
π
Thus
()
()
547
4
3
3
5
5
7
7
9
9
21
21
7
ππ
πω
ωωωω
ω

)()
() cos
cos cos cos cos
.........
cos
...
AV
vt V t
tttt
kt
k
DC
DC
=−
=−+−+++
+
+
+
æ
è
ç
ö
ø
÷
Rms values of individual harmonic components can be given as
V
k
V
kDC21
1

2
41
21
+
=
+
π
Consider now square-wave given in the right part of the Figure. It is again a pure AC
waveform (i.e., V
o
= 0), symmetrical in the two half-periods and it is an odd function of
time. Hence its Fourier series contains only harmonics of the order 2k + 1 (eq. (4.2))
and only coefficients B
2k+1
of the Fourier series in (4.1) need to be determined. Thus
()
[]
()
[]
()
[]
()
vt V k t V B
BvkdvkdVkd
AV
k
k
V
k
BV BV BV

k
k
kk
k DC
kDC DC
DC DC DC
() sin( ) /
( ) sin ( ) sin sin
cos
() ()
//
=+ =
=+= +=+
=
−+
+
é
ë
ê
ù
û
ú
=
+
== =
+
=
∞
++
+

+
å
òòò
221 2
1
21 4
1
21
4
21
421
21
41
21
44345
21
0
21 21
21
0
2
0
2
0
2
21
0
2
13 5
ω

π
θθθ
π
θθθ
π
θθ
π
θ
π
ππ π
πππ
π
Thus
()
()
BV
vt V t
tttt kt
k
DC
DC
7
47
4
3
3
5
5
7
7

9
9
21
21
=
=++++++
+
+
+
æ
è
ç
ö
ø
÷
()
() sin
sin sin sin sin
.........
sin
...
π
πω
ωωωω ω
Rms values of individual harmonic components remain to be given with
V
k
V
kDC21
1

2
41
21
+
=
+
π
Square-wave in this example possesses so called quarter-wave symmetry. This has
enabled change of upper border of integration in calculation of coefficients of Fourier
series from 360 degrees to 90 degrees only, and subsequent multiplication of the results
of integration with four.
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
40
Note that the voltage wave-form, analysed in this example, has already been
encountered as the output voltage of the single-phase DC-AC converter (inverter) in
the previous chapter.
Example:
Consider the voltage shown in Figure and determine its Fourier series if the duration of
the non-zero voltage in both half-cycles is determined with angle
β
. Next, consider the
special case when
β
= 120 degrees.
v(t)
V
DC
-270 -90 0 90 degrees
β

β
Solution:
The waveform is once more pure AC, symmetrical in two half-periods and even. Hence
only odd harmonics exist again. Waveform possesses again quarter-wave symmetry.
From (4.1) and (4.2)
()
[]
()
[]
()
[]
()
()
()() ()
vt V k t V A
A v kd v kd V kd
AV
k
k
V
k
k
AV AV
k
k
kk
k DC
kDC DC
DC DC
() cos( ) /

( )cos ( )cos cos
sin
sin
/
/
=+ =
=+= +=+
=
+
+
é
ë
ê
ù
û
ú
=
+
é
ë
ê
ù
û
ú
+
==
+
=
∞
++

+
+
å
òòò
221 2
1
21 4
1
21
4
21
4
21
21
4
21
2
21
4433
21
0
21 21
21
0
2
0
2
0
2
21

0
2
13
ω
π
θθθ
π
θθθ
π
θθ
π
θ
π
β
πβ π
ππ
β
β
Thus
sin 2 sin
() ()()
()
() ()
βπβ
πω
βωβ
ωβ
2sin2AV
vt V t
t

kt
k
k
DC
DC
5
45 5
4
2
3
3
3
2
21
21
21
2
=
=+++
+
+
+
+
æ
è
ç
ö
ø
÷
() cos sin

cos
sin .........
cos
sin ...
In the special case when
β
= 120 degrees,
β
/2 = 60 degrees, and the Fourier series
becomes
A
V
AA
V
A
V
AA
V
vt
V
t
tt t
DC DC
DC DC
DC
135
7911
43
2
0

4
5
3
2
4
7
3
2
0
4
11
3
2
23 5
5
7
7
11
11
===−
===−
=−+−++
æ
è
ç
ö
ø
÷
ππ
ππ

π
ω
ωω ω
() cos
cos cos cos
.... ...
This type of voltage waveform will be met later on, for the case of three-phase voltage
source inverter. The important thing to note is that this specific voltage expression does
not contain any harmonics divisible by 3.
ENGNG3070 Power Electronics Devices, Circuits and Applications
 E Levi, Liverpool John Moores University, 2002
41
Example:
Consider the voltage waveform shown in Figure and determine its Fourier series.
Assuming that the required rms value of the fundamental harmonic is 240 V, find the
necessary DC supply voltage.
v(t)
(2/3)V
DC
(1/3)V
DC
180 210 240 300 330 360
0 30 60 120 150 degrees
Solution:
The waveform shown in Figure can be regarded as being composed of two already
analysed waveforms of the previous two examples. It consists of a full square-wave and
of a quasi square-wave of 60 degrees duration. The Fourier series can therefore be
found using the so-called decomposition principle, which is essentially an application of
the superposition principle. Fourier series is found separately for the two parts of the
waveform and the resulting Fourier series for the complete series is obtained by

summing the two series.
From the first example, taking vertical axis position in the middle of the positive pulse,
Fourier series is
()
[]
()
[]
()
[]
()
()
[]
vt V k t V A
AvkdvkdVkd
AV
k
k
V
k
k
AV A V
k
k
kk
k DC
kDC DC
DC DC
() cos( ) /
( ) cos ( ) cos cos
sin

sin
(
//
=+ =
=+= += +
=
+
+
é
ë
ê
ù
û
ú
=
+
+
==−
+
=
∞
++
+
+
å
òòò
221 2
1
21 4
1

21
1
3
4
21
1
3
421
21
1
3
4
212
21
1
3
4
1
3
43
21
0
21 21
21
0
2
0
2
0
2

21
0
2
13
ω
π
θθθ
π
θθθ
π
θθ
π
θ
π
π
π
ππ π
π
Thus
()
()
ππ π
πω
ωωωω ω
)() ()
() cos
cos cos cos cos
.........
cos
...

AV A V
vt V t
tttt kt
k
DC DC
DC
57
1
3
45
1
3
47
1
3
4
3
3
5
5
7
7
9
9
21
21
==−
= −+−+++
+
+

+
æ
è
ç
ö
ø
÷