Fundamentals of Power Electronics
1. Power Semiconductor Devices (Switches)
There are several power semiconductor devices currently involved in several industrial applications.
This lecture will concentrate mainly on four power devices only, namely; Diodes, SCRs (Thyristors),
MOSEFTs, and IGBTs.
 Diodes: These are two terminal switches, as shown in Fig. 1-a, formed of a pn junction. It is
not controllable and its operating states are determined by the circuit operating point. A
forward positive voltage (vD is positive) will turn it on and a reverse negative current (from
Cathode to Anode, iD is negative) will turn it off. Practically, the diode characteristic consists
of two regions, as shown in Fig. 1-b; a forward bias region (ON state) where both vD and iD
are positive and the current in this region increases exponentially with the increase in the
voltage, and a reversed bias region (OFF state) where both vD and iD are negative and very
small leakage current flow through the diode until the applied reverse voltage reaches the
diode’s breakdown voltage limit VBR. Ideally, the diode is represented by a short circuit when
forward biased and as an open circuit when reversed biased with the ideal characteristic
shown in Fig. 1-c.
Fig. 1 Diode: a) symbol, b) characteristic, and c) ideal characteristic [1]
 Silicon Controlled Rectifiers “SCRs” (Thyristors): These are three terminal switches as
shown in Fig. 2-a, formed of three pn junctions (pnpn). This is a controllable switch that
usually required to be latched to conduct. This latching (triggering) process is carried out by
1
injecting current to the gate terminal (ig) at the required latching instant provided that the
device is forward biased (vAK is positive). Practically, the thyristor characteristic has three
main regions as shown in Fig. 2-b; the Conduction Region where the thyristor is operating in
its ON state, the Forward Blocking Region where the thyristor is forward biased but not yet
triggered or the voltage didn’t reach the forward breakover voltage, and a Reverse Region that
consists of the reverse blocking region and the reverse avalanche region similar to the diode
characteristic. Among the important points along the SCR characteristic:
Fig. 2 Thyristor: a) symbol, b) characteristic, and c) ideal characteristic [1]
Fig. 3 Thyristor gate circuit [1]
2
o Latching Current: This is the minimum required current to turn on the SCR
device and convert it from the Forward Blocking State to the ON State.
o Holding Current: This is the minimum forward current flowing through the
thyristor in the absence of the gate triggering pulse.
o Forward Breakover Voltage: This is the forward voltage required to be applied
across the thyristor to turn it ON without the gate signal application.
o Max Reverse Voltage: This is the maximum reverse voltage to be applied across
the thyristor before the reverse avalanche occurs.
Ideally, SCRs are represented by a short circuit when operating within the conduction region
and as an open circuit when operating within the blocking region. The ideal characteristic is
shown in Fig. 2-c. It is also worth mentioning that once the SCR is triggered and turned ON
the gate signal can be removed without turning it OFF. SCRs are turned OFF when reversing
the terminal voltage and current.
 Metal Oxide Semiconductor Field Effect Transistors “MOSFETs”: These are three
terminal switches as shown in Fig. 4-a. This is considered the fastest power switching device.
It is a controllable switch that requires a gate-source voltage (vGS) higher than a threshold
value (vTh) for the device to conduct. Practically, MOSFET’s characteristic consists of three
regions, as shown in Fig. 4-b; a cut OFF region (OFF state) when vGS < vTh, a linear region
when vDS < vGS – vTh, and an active region when vDS > vGS – vTh. Ideally, MOSFETs are
represented by a short circuit when operating within the ON State and as an open circuit when
operating within the OFF State.
 Insulated Gate Bipolar Transistors “IGBTs” (Thyristors): These are also three terminal
switches as shown in Fig. 5. Their operation modes and characteristics are almost similar to
those for MOSFETs, shown in Fig. 4-b, except for the operating ranges.
 Other Semiconductor Devices: These include; Bipolar Junction Transistors (BJTs), Gate
Turn Off Thyristors (GTO Thyristors), Triode ac switches (Triacs), Static Induction
Transistors (SITs), Static Induction Thyristors (SITHs), and MOS-Controlled Thyristors
3
(MCTs). Comparisons between different types of semiconductor devices from the point of
view of ratings and power and frequency ranges are given in Table 1 and Fig. 6, respectively.
Fig. 4 MOSFET: a) symbol and b) characteristic [1]
Fig. 5 IGBT symbol [1]
Table 1 Power semiconductor devices ratings comparison [1]
4
Fig. 6 Frequency and power ranges for different power semiconductor devices [1]
2. Important Parameters for Periodic Waveforms
For any periodic waveform as shown in Fig. 7, the following parameters can be determined:
 Peak Value: This represent the maximum value of the periodic waveform.
 Peak – to – Peak Value: This represents the difference between the maximum and the
minimum values of the waveform.
Fig. 7 Periodic waveform and its parameters
5
 Average Value: This represent the DC component content of the waveform and can be
calculated from the following expression;
f avr 
1
T
T

f t  dt
f avr 
or
0
1


 f  t  d t
0
where favr is the average value of the periodic function f(t) { f( t)} over a period T {}.
 Root Mean Square (RMS) Value: this represent the effective value of the periodic function
and can be expressed by; (sometimes it is referred to by Effective Value)
f rms
1

T
T
f
2
t  dt
f rms 
or
0

f

1
2
 t  d t
0
where frms is the rms value of the periodic function f(t) { f( t)} over a period T {}.
 Peak Reverse Voltage (PRV): This represents the maximum reverse voltage applied to a
semiconductor device during its operation in the off state. Sometimes referred to as the Peak
Inverse Voltage (PIV).
 Conduction Period (Angle): The period of time (angle) during which a semiconductor switch
is conducting (operating in its ON state).
 Extinction Angle: This is the angle ( t) at which the semiconductor switch stops conducting
(switched to the OFF state).
 Firing Angle: This is the angle ( t) at which controlled semiconductor switch starts
conducting (switched to the ON state). Sometimes referred to as the Delay Angle.
3. Power Electronic Converters
In general, power electronics converters can be classified into four main categories namely; Rectifiers,
DC – to – DC Converters, AC – to – AC Converters, and Inverters.
3.1 Rectifiers
These converters are used to convert fixed AC power to fixed or variable DC power. They are
classified into two main categories; Uncontrolled Rectifiers and Controlled Rectifiers.
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3.1.1 Uncontrolled Rectifiers
In this type, the generated DC power is fixed with the converter used and the input AC power.
They usually use diodes as their power switches. The following subsections deal with the basic
operation of some examples of uncontrolled rectifiers.
Single-phase half-wave rectifier loaded with resistive load:
Fig. 8-a presents the basic circuit for a single-phase, half-wave, rectifier loaded with a resistive
load. The circuit is supplied by a single phase transformer whose secondary represents the
rectifier’s circuit AC source (vs) that is represented by a sinusoidal wave given by,
vs = Vm sin ( t)
where vs is the supply voltage, Vm is the peak value of the supply voltage,  is the
angular frequency, and t and is the time.
For this configuration, the diode will conducts (becomes forward biased) whenever the supply
voltage (vs) is positive to force the current in the diode from the anode to the cathode.
Fig. 8 Single-phase half-wave rectifier: a) circuit and b) waveforms [2]
For one total period of operation of this circuit, the corresponding waveforms are shown in Fig.
8-b where two operating states occur as presented in Table 2.
7
Table 2 Operation states
Period
Diode
Representation
Diode State
Output
Voltage (vo)
Load / Supply
Current (io / is )
Diode
Voltage (vD)
vs
vs / R
Zero
Zero
Zero
vs
ON (Forward
SC*
Biased)
OFF (Reverse
OC*
   t < 2
Biased)
*
SC = Short Circuit, and OC = Open Circuit
0t<
The average value of the load voltage Vdc can be calculated as follows,
Vdc 
1
2
V dc 


v s t  dt 
0
Vm


1
Vm sin  t  dt
2

0
 0.318 V m
Since the load is resistive load, therefore the load voltage and current are in phase and they are
related by is = vs / R. Consequently, the average value of the load current Idc is
I dc 
V dc V m
0.318 V m


R
R
R
The rms value of the load voltage Vrms can be calculated as follows,
1
2
V rms 
V rms 

2
 v s t  dt 
1
2
0

 V
m
sin  t 2 dt
0
Vm
 0 .5 V m
2
Therefore the rms value of the load current Irms is
I rms 
V rms V m 0.5 V m


2R
R
R
The output power is given by
Pac  I rms  V rms 
8
Vrms 2
R

0.5 Vm 2
R
The PRV of the diode in this configuration is Vm.
Single-phase half-wave rectifier loaded with resistive load and a battery:
Fig. 9-a presents the basic circuit for a single-phase, half-wave, rectifier loaded with a resistive
load and a battery. For this configuration, the diode will conducts (becomes forward biased)
whenever the supply voltage (vs) is positive and greater than the battery voltage “E” to force the
current in the diode from the anode (point 2) to the cathode (point 4).
Vm
PRV = Vm + E
E
Fig. 9 Single -phase half-wave rectifier: a) circuit and b) waveforms [3]
For one total period of operation of this circuit, the corresponding waveforms are shown in Fig.
9-b where two operating states occur as presented in Table 3.
Table 3 Operation states
Period
Diode State
Diode
Representation
OFF (Reverse
OC*
Biased)
ON (Forward
SC*
t1  t < t2
Biased)
OFF (Reverse
OC*
t2  t < t4
Biased)
*
SC = Short Circuit, and OC = Open Circuit
0   t < t1
Output
Voltage (vo)
Load / Supply
Current (io / is )
Diode
Voltage (vD)
E
Zero
vs - E
vs
(vs - E)/ R
Zero
E
Zero
vs - E
In the analysis of this circuit, point 1 is considered the grounded reference for all node voltages
and consequently the following voltages can be defined:
v1 (Voltage at point 1) = zero
9
vs (Supply Voltage) = v2 – v1 = v2
vD (Diode Voltage) = v2 – v4
vR (Resistive Load Voltage) = v4 – v3 = v4 – E
vo (Output Load Voltage) = v4 – v1 = v3 + vR = E + vR
Moreover, the load (supply) {since the load, the battery, and the supply are connected in
series}) current may be defined as
io (Load Current) = is (Supply Current) = vR / R
The angle at which the diode starts conducting   is the same angle at which the supply
voltage is equal to the battery voltage. Therefore, at  t   we have,
v s  t     E  V m sin  
 E
  sin 1 
 Vm




Since the wave form during the first half cycle is symmetric around  t 

2
. Therefore, the
angle at which the diode stops conducting   is be given by,
    
The average value of the load voltage Vdc can be calculated as follows,
V dc
2 
 
2 



1 
1 


v s t  dt 
E d t 
V m sin  t  d t 
E dt 

 2 
2 


 

 



V dc 



1
2 Vm cos      2    E 
2
The average value of the load current Idc is
I dc
 1
2 Vm cos      2    E   E

2 Vm cos    2      E 
V  E 2 


 dc

R
R
2 R
10
The PRV of the diode in this configuration is (Vm + E) which represent the maximum value of
vD = v2 – v4 when there is no current flowing in the load as shown in Fig. 9-b.
Single-phase full-wave rectifier loaded with resistive load:
Fig. 10-a presents the circuit connection for a single-phase, full-wave, rectifier loaded with a
resistive load. It is sometimes referred to as the full-wave bridge rectifier. For this
configuration, two diodes always conducting during the same interval to provide a closed loop
for the current. D1 and D2 conduct whenever the supply voltage (vs) is positive while D3 and D4
conduct whenever the supply voltage (vs) is negative as illustrated by Fig. 10-b.
Fig. 10 Single -phase full-wave rectifier loaded with resistive load [2]
For one total period of operation of this circuit, the corresponding waveforms are shown in Fig.
10-b where two operating states occur as presented in Table 4.
Table 4 Operation states
Period
Conducting
Diodes
Output Voltage
(vo)
Load Current
(io )
Supply Current
(is )
0t<
D1 & D2
vs
vs / R
vs / R
   t < 2
D3 & D4
- vs
- vs / R
vs / R
11
Diode Voltage
(vD)
- vs
for D3 & D4
vs
for D1 & D2
Since the load is a resistive load. Then, the load current will have the same waveform as the
load voltage but with current scale according the load current-voltage characteristic,
io (Load Current) = vo / R
Table 4 reveals that, during the negative half cycle of the supply voltage, the load current is
positive (io = - vs / R) (vs itself is negative) whereas the supply current is negative (is = vs / R).
The average value of the load voltage Vdc can be calculated as follows,
V dc 


V dc 

 v  t  d t    V
1
1
s
0
m
sin  t  d t d t
0
2 Vm

 0.6366 V m
The average value of the load current Idc is
I dc 
V dc 2 V m 0.6366 V m


R
R
R
The rms value of the load voltage Vrms can be calculated as follows,
1
V rms 

V rms 

 v  t 
2
s
1
d t 

0
Vm
2

 V
m
sin  t 2 dt
0
 0.707 V m
Therefore the rms value of the load current Irms is
I rms 
0.707 V m
V rms
V
 m 
R
R
2R
The PRV for any diode in this configuration is (Vm) as shown in Fig. 10-b.
Single-phase full-wave rectifier loaded with highly inductive load:
Fig. 11-a presents the circuit connection for a single-phase, full-wave, rectifier loaded with a
highly inductive load. Highly inductive loads are basically R-L loads where L >>> R.
12
Therefore, the load time constant  
L
is very high and can be considered infinity.
R
Consequently, the load current is assumed constant. For one total period of operation of this
circuit, the corresponding waveforms are shown in Fig. 11-b where two operating states occur
as presented in Table 5.
Fig. 11 Single -phase full-wave rectifier loaded with highly inductive load
Table 5 Operation states
Period
Conducting
Diodes
Output Voltage
(vo)
Load Current
(io )
Supply Current
(is )
0t<
D1 & D2
vs
Ia
Ia
   t < 2
D3 & D4
- vs
Ia
- Ia
Diode Voltage
(vD)
- vs
for D3 & D4
vs
for D1 & D2
Table 5 reveals that, during the negative half cycle of the supply voltage, the load current is
positive (io = Ia) whereas the supply current is negative (is = - Ia).
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The average value of the load voltage Vdc can be calculated as follows,
Vdc 
1
1


V dc 
 vs  t  d t 
0
2 Vm

 Vm sin  t  d t
0
 0.6366 V m
Since the load is a highly inductive load. Then, the load current is considered constant (ripple
free current) and equal to the average value of the load current Idc as follows,
I dc  I a 
V dc 2 V m 0.6366 V m


R
R
R
In case the load contains a DC battery “E” (or a back emf) in addition to the highly inductive
load, the load current will be
I dc
2 Vm
V E
 I a  dc

R

E
R

2 Vm   E
R
(provided that E < Vdc)
The rms value of the load voltage Vrms can be calculated as follows,
V rms 

v  t 
 
1
2
s
d t 
0
V rms 
Vm
2

V
 
1
m
sin  t 2 dt
0
 0.707 V m
Since the load current is constant over the studied period, therefore the rms value of the load
current Irms is
I rms  I dc  I a
The PRV for any diode in this configuration is (Vm) as shown in Fig. 11-b.
3.1.2 Controlled Rectifiers
In this type, the generated DC power is controllable and variable. They usually use SCRs as
their power switches. For fast switching operation, MOSFETs and IGBTs are used. The
following subsections deal with the basic operation of some examples of controlled rectifiers.
14
Single-phase half-wave controlled rectifier loaded with resistive load:
Fig. 12-a presents the basic circuit for a single-phase, half-wave, controlled rectifier loaded
with a resistive load. For this configuration, the thyristor will conducts (becomes forward
biased) when triggered using gate pulses provided that the supply voltage (vs) is positive to
force the current in the thyristor from the anode to the cathode.
The instant at which the gate pulse occurs is known as the firing angle and represented by ().
The gate pulses are repeated every 2 (one complete cycle). The firing angle can occur at any
instant ranging from 0 to  as the thyristor has to be forward biased when triggered, otherwise
it won’t conduct. For one total period of operation of this circuit, the corresponding waveforms
are shown in Fig. 12-c where three operating states occur as presented in Table 6.
Fig. 12 Single-phase half-wave controlled rectifier [2]
The average value of the load voltage Vdc can be calculated as follows,
V dc 
1
2



v s t  dt 
1
2

 V
m
15
sin  t  dt
V dc 
Vm
1  cos 
2
The average value of the load current Idc is
I dc 
V dc
V
 m 1  cos 
R
2 R
Table 6 Operation states
Period
Thyristor
Representation
Thyristor State
Output
Voltage (vo)
Load / Supply
Current (io / is )
Thyristor
Voltage (vT1)
Zero
Zero
vs
vs
vs / R
Zero
Zero
Zero
vs
OFF (Forward
OC*
Blocking)
ON (Forward
SC*
t<
Biased)
OFF (Reverse
OC*
   t < 2
Biased)
*
SC = Short Circuit, and OC = Open Circuit
0t<
Therefore, the average output voltage can vary from 0 to
vary from 0 to
Vm

and the average load current will
Vm
when varying  from  to 0, respectively. Moreover, since the load
R
voltage and current for this configuration are always positive, therefore, this converter operates
in the first quadrant only as revealed by Fig. 12-b.
The rms value of the load voltage Vrms can be calculated as follows,
V rms 
1
2
V rms 
Vm
2

 v t 
2
s
dt 
1
2

 V
m
sin 2   
1

   

2

Therefore the rms value of the load current Irms is
I rms 
V rms V m

R
2R
sin 2   
1
   


2

The PRV of the thyristor for this configuration is Vm.
16
sin  t 2 dt
Single-phase full-wave controlled rectifier loaded with highly inductive load:
Fig. 13-a presents the circuit connection for a single-phase, full-wave, controlled rectifier
loaded with a highly inductive load. For one total period of operation of this circuit, the
corresponding waveforms are shown in Fig. 13-c where two operating states occur as presented
in Table 7.
Table 7 Operation states
Period
Conducting
Thyristors
Output
Voltage (vo)
Load Current
(io )
Supply Current
(is )
   t <  +
T1 & T 2
vs
Ia
Ia
 +   t < 2 +
T3 & T 4
- vs
Ia
- Ia
Thyristor
Voltage (vT)
- vs
for T3 & T4
vs
for T1 & T2
Fig. 13 Single -phase full-wave rectifier loaded with highly inductive load [2]
The average value of the load voltage Vdc can be calculated as follows,
V dc 
1

V dc 
 


v s  t  d t 
2 Vm

1

 
 V
m
cos  
17
sin  t  d t d t
Since the load is a highly inductive load. Then, the load current is considered constant (ripple
free current) and equal to the average value of the load current Idc as follows,
I dc
V E
 I a  dc

R
2 Vm

cos    E
(provided that E < Vdc and Vdc > 0)
R
In case the load doesn’t contain a DC battery “E” (or a back emf) in addition to the highly
inductive load, the load current will be
I dc  I a 
Vdc 2 Vm

cos  
R
R
(provided that Vdc > 0)
Therefore, the average output voltage can vary from
2 Vm

to 
2 Vm

when varying  from 
to 0, respectively. Moreover, since the load voltage for this configuration can be positive or
negative while the load current is always positive because the thyristors prevents a reverse
current flow. Therefore, this converter operates in the first and the fourth quadrants as revealed
by Fig. 13-b.
The rms value of the load voltage Vrms can be calculated as follows,
V rms 
V rms 
1
 
 v  t 

2

s
Vm
2
d t 
1

 
 V

m
sin  t 2 dt
 0.707 V m
Since the load current is constant over the studied period, therefore the rms value of the load
current Irms is
I rms  I dc  I a
The PRV for any thyristor in this configuration is (Vm).
Single-phase semiconverter loaded with highly inductive load:
Fig. 14-a presents the circuit connection for a single-phase semiconverter loaded with a highly
inductive load. This configuration consists of a combination of thyristors and diodes and is
18
used to eliminate any negative voltage occurrence at the load terminals. This is because the
diode Dm is always activated (forward biased) whenever the load voltage tends to be negative.
For one total period of operation of this circuit, the corresponding waveforms are shown in Fig.
14-b where four operating states occur as presented in Table 8.
The average value of the load voltage Vdc can be calculated as follows,
Vdc 


 vs t  dt    Vm sin  t  dt


1
Vdc 
Vm

1
1  cos 
Fig. 14 Single -phase semiconverter loaded with highly inductive load [2]
19
Table 8 Operation states
Period
Conducting
Switches
Output
Voltage
(vo)
Load
Current
(io )
Supply
Current
(is )
Diode Dm
Current
(iDm )
Switch Voltage
0t<
Dm
0
Ia
0
Ia
0.5 vs for T1 & D2
- 0.5 vs for T2 & D1
t<
T1 & D 2
vs
Ia
Ia
0
- vs for T2 & D1 & Dm
   t <  +
Dm
0
Ia
0
Ia
- 0.5 vs for T1 & D2
0.5 vs for T2 & D1
 +   t < 2 
T2 & D 1
- vs
Ia
- Ia
0
vs for T1 & D2 & Dm
The average value of the load current Idc is
Vm
I dc
V E

 dc

R
1  cos   E
(provided that E < Vdc)
R
Therefore, the average output voltage can vary from 0 to
Vm

when varying  from  to 0,
respectively. Moreover, since the load voltage and current for this configuration are always
positive, therefore, this converter operates in the first quadrant only as revealed by Fig. 14-b.
The rms value of the load voltage Vrms can be calculated as follows,
Vrms 

vs t 2 dt 


1
Vrms  Vm

Vm sin  t 2 dt


1
1
sin 2   
   

2


The PRV for any switch in this configuration is (Vm).
3.2 DC – to – DC Converters
These converters are used to convert fixed DC power to controllable, variable DC power. The
following subsections deal with the basic types of these converters. They are sometimes referred to as
DC Choppers. DC Choppers can be classified according to their operation range (load voltage and
current) into five main categories as shown in Fig. 15.
20
Fig. 15 DC Choppers’ classification
3.2.1 First Quadrant DC Chopper
In this type, the load voltage and load currents are always positive. Fig. 16 presents the circuit
of this type. For one total period of operation of this circuit (T), the corresponding equivalent
circuits and waveforms are shown in Fig. 17 where two operating modes occur as presented in
Table 9.
Fig. 16 First quadrant DC Chopper
21
Fig. 17 Modes of operation equivalent circuit and output Waveforms
The equations that governs the operation of this type can be summarized as follows, the
average value of the load voltage Vdc can be calculated as follows,
V dc
1

T
kT
V
s
d t  k Vs
0
where Vs is the DC supply voltage, T is the total period of operation, and k is the duty
cycle given by k 
t on
, where ton is the period at which the chopper is ON.
T
Table 9 Operation modes
Period
Mode
Chopper
State
1
ON (SC*)
0  t < kT
OFF (OC*)
2
kT  t < T
*
SC = Short Circuit, and OC = Open Circuit
22
Diode State
Output
Voltage
(vo)
OFF (OC*)
ON (SC*)
Vs
0
Currents
(i )
i1
i2
(is )
i1
0
(iDm )
0
i2
For (L / R) >> T, the load current is continuous and can be expressed by,


V E

t R / L
1  e t R / L
 s
 i1 t   I1 e
R

i t   

E
 (t  kT ) R / L
 1  e (t  kT ) R / L
i2 t   I 2 e
R


where I 1 
Vs
R
0t kT

k T t T
 ek z  1 E
Vs  e k z  1  E
TR




I

,
, and z 
.
2

z

 ez 1  R

R  e 1  R
L


For discontinuous mode of operation (I1 = 0), the load current can be expressed by,


V E

i1 t   s
1  e t R / L

R

i t   

E
 (t  kT ) R / L
 1  e (t  kT ) R / L
i2 t   I 2 e
R


0t kT

k T t T
For the load current to be continuous, I1 should be greater than or equal to zero. Therefore,

V  e kz  1  E 
 0
 I 1  s  z
R  e  1  R 

 e kz  1  E

  z

 e  1  V s


0

E e kz  1
 z
Vs
e 1
3.2.2 Second Quadrant DC Chopper
In this type, the load voltage is always positive while the load current is always negative. Fig.
18-a presents the circuit connection of this type. For one total period of operation of this circuit
(T), the corresponding equivalent circuits and waveforms are shown in Figs. 18-b and 18-c
where two operating modes occur as presented in Table 10.
23
Fig. 18 Second quadrant DC Chopper
The equations that governs the operation of this type can be summarized as follows, the
average value of the load voltage Vdc can be calculated as follows,
V dc 
1
T
T
V
s
d t  1  k  V s
kT
where Vs is the DC supply voltage, T is the total period of operation, and k is the duty
cycle given by k 
ton
, where ton the period at which the chopper is ON.
T
Table 10 Operation modes
Period
Chopper State
Diode State
Load Voltage (vo)
ON (SC*)
OFF (OC*)
0  t < kT
OFF (OC*)
ON (SC*)
kT  t < T
*
SC = Short Circuit, and OC = Open Circuit
0
Vs
Currents
(iL )
i1
i2
(is )
0
i2
For continuous load current operation, the load current is can be expressed by,


E

I1 e  t R / L  1  e  t R / L

R

iL t   

E  Vs
( t  kT ) R / L
1  e (t  kT ) R / L

i2 t   I 2 e
R


where I 1 
E V s  1  e  1 k  z

R R  1  e z

V
, I 2  E  s

R R

24
 e k z  e  z

 1  e z

0t kT

k T t T

 , and z  T R .

L

(ich )
i1
0
3.2.3 First and Second Quadrants DC Chopper
In this type, the load voltage is always positive while the load current can be either positive or
negative. It is also know as the two quadrant chopper. Fig. 19 presents the circuit connection of
this type. This is considered a combined converter consisting of both the first-quadrant and the
second quadrant DC choppers. It can operate in the first quadrant by controlling S1 and D4 in
the same manner explained in Section 3.2.1. Moreover, it can operate in the second quadrant by
controlling S4 and D1 in the same manner explained in Section 3.2.2. S1 and S4 must not be
switched ON at the same time; otherwise, the source will be short circuited. Table 11 shows the
different operation mode of this converter.
Fig. 19 First and second quadrants DC Chopper
Table 11 Operation modes
Mode
Conducting switch
Load Current (iL)
Load Voltage (vL)
1
2
3
4
S1
S4
D1
D4
+ ve
- ve
- ve
+ ve
Vs
0
Vs
0
Note that the diode D1 is activated once S4 is switched off while the diode D4 is activated once
S1 is switched off. This operation is carried out because of the presence of the inductance in the
load that requires a continuous flow of current at the instant of turning OFF the chopper
switches.
3.2.4 Third and Fourth Quadrants DC Chopper
In this type, the load voltage is always negative while the load current can be either positive or
negative. Fig. 20 presents the circuit connection of this type. It can operate in the third quadrant
by controlling S3 and D2 while it can operate in the fourth quadrant by controlling S2 and D4. S2
25
and S2 must not be switched ON at the same time; otherwise, the source will be short circuited.
Table 12 shows the different operation mode of this converter.
Fig. 20 Third and Fourth quadrants DC Chopper
Table 12 Operation modes
Mode
Conducting switch
Load Current (iL)
Load Voltage (vL)
1
2
3
4
S2
S3
D2
D3
+ ve
- ve
- ve
+ ve
0
- Vs
0
- Vs
The diode D2 is activated once S3 is switched off while the diode D3 is activated once S2 is
switched off.
3.2.5 Four-Quadrants DC Chopper
In this type, both the load voltage and the load current can be either positive or negative. Fig.
21 presents the circuit connection of this type. It can operate in the third quadrant by
controlling S3 and D2 while it can operate in the fourth quadrant by controlling S2 and D4. S2
and S2 must not be switched ON at the same time; otherwise, the source will be short circuited.
Table 13 shows the different operation mode of this converter.
Fig. 21 Four-quadrants DC Chopper
26
Table 13 Operation modes
Mode
Conducting switch
Load Current (iL)
Load Voltage (vL)
1
2
3
4
S2
S3
D2
D3
- ve
+ ve
- ve
+ ve
0
- Vs
0
- Vs
The diode D2 is activated once S3 is switched off while the diode D3 is activated once S2 is
switched off.
3.3 Other Power Converters
In addition to the previously discussed converters, there are also the AC - to - AC Converters, that
convert fixed AC power to controllable, variable AC power, and the DC - to – AC Converters
(Inverters), that convert fixed AC power to controllable, variable DC power. These converters are not
covered within the scope of this course.
4. Fast Switching Modulation Techniques
There are several techniques to control (modulate) fast power semiconductor switches. This section
will introduce the basic principle and the main types of Pulse Width Modulation (PWM) techniques.
This technique is basically based on comparing a reference signal vr with a carrier signal vcr to generate
the control (switch’s gate) signal as shown in Fig. 22. This generated control signal is send to the
control terminal of the power electronic switches to activate them whenever vr < vcr provided that the
switch is forward biased. By varying the carrier signal magnitude the ON period (ton) and the duty
cycle of switching changes. Moreover, the switching frequency (switching period) is varied by varying
the reference signal frequency.
Fig. 22 Basics of PWM
27
4.1 PWM for DC Output
These techniques are used for the converters generating DC outputs. They are classified into two main
categories:

Uniform (Equal width) PWM: The generated pulses have equal width as shown in Fig.
23. They are generated by comparing a triangular wave with a DC signal.
Fig. 23 Uniform PWM for DC outputs [4]

Sinusoidal PWM: The generated pulses have different widths as shown in Fig. 24.
Fig. 24 Sinusoidal PWM for DC outputs [1]
28
They are generated by comparing a triangular wave with a variable DC signal to generate
a sinusoidal variable duty cycle tracking the following function,
d t   D dc  D max sin  o t 
where d t  is the required duty cycle signal to be generated, D dc is the normal
duty cycle with no sinusoidal modulation, D max is the maximum modulation
constant, and  o is the modulation frequency.
The generated DC output voltage in this case can be represented by
v o  d t  V dc  D dc  D max sin  o t  V dc  D dc V dc  D max V dc sin  o t 
where Vdc is the DC source voltage.
4.2 PWM for AC Output
They techniques are used for the converters generating AC outputs. They are classified into three main
categories:

Uniform (Equal width) PWM: The generated pulses have equal width as shown in Fig.
25. They are generated by comparing two inverse triangular waves with two opposite DC
signal levels.
Fig. 25 Uniform PWM for AC outputs [4]
29

Bipolar Sinusoidal PWM: The generated pulses have different widths as shown in Fig.
26. They are generated by comparing a triangular wave with a sinusoidal wave.
Whenever, the reference signal is greater than the triangular signal a pulse is generated by
turning ON switch S1. On the other hand, whenever, the reference signal is lower than the
triangular signal a pulse is generated by turning ON switch S2 as shown in Fig. 27.
Fig. 26 Biopolar Sinusoidal PWM for AC outputs [1]
Fig. 27 Simplified circuit for generating biopolar Sinusoidal PWM for AC outputs [1]

Unipolar Sinusoidal PWM: The generated pulses have different widths as shown in Fig.
28. They are used when a positive and negative sinusoidal control signal are available.
These signals are compared with sawtooth signals to generate to output Vo1 and Vo2 . The
difference generates the total output control signal Vo = Vo1 - Vo2. For example, positive
pulses can be used to trigger S1 while negative pulses can be used to trigger S2, shown in
Fig. 27, to generate the output voltage waveform shown in Fig. 28.
30
Fig. 28 Unipolar Sinusoidal PWM for AC outputs [1]
5. Numerical Examples
Example 1: For the half-wave uncontrolled rectifier circuit shown in Fig. 29. The supply is a 110
V, 60 Hz. The resistive load is 25 . Calculate:
1. The average value of the output voltage and current,
2. The rms value of the output voltage and current,
31
3. The average value of the power delivered to the load,
4. The average value of the power delivered to the load if the source has a
resistance of 60 .
Given:
V = 110 V, f = 60 Hz, R = 25 , and Rs = 60 .
Solution:
For half-wave uncontrolled rectifier circuit shown in Fig. 29.
Fig. 29 Single-phase half-wave rectifier [2]
1. The average value of the load voltage Vdc can be calculated as follows,
Vdc
1

2
Vdc 
Vm



0


1
v s t  dt 
Vm sin  t  dt
2

0
2 x 110

 49.517 V
The average value of the load current Idc is
I dc 
V dc V m


R
R
2 x 110
 1.981
25 
A
2. The rms value of the load voltage Vrms can be calculated as follows,
32
1
2
V rms 
V rms 

2
 v s t  dt 
0
Vm

2
1
2

 V
m
sin  t 2 dt
0
2 x 110
 77.782 V
2
Therefore the rms value of the load current Irms is
I rms 
V rms 77.782

 3.111 A
R
25
3. The average power delivered to the load is
Pavr
1

2
 Pavr 


0
1
v o  t   io  t d t 
2
77.7822
25


v s  t 2 d t  Vrms 2
R
0
R
 242.002 W
4. If the source has a resistance of 60 , the load voltage will be related to the total
voltage using the voltage divider equation as follows,
v o  t   v s  t 
R
R  R s 
Therefore the value of the average power delivered to the load is
Pavr 
1
2


v o  t   i o  t d t 
0
 Pavr  77.782 2 
25
25  602
1
2

 v  t 
2
s
0
R
R  R s 
2
d t  V rms 2 
R
R  R s  2
 20.934 W
Example 2: For the single-phase, full-wave uncontrolled rectifier circuit shown in Fig. 30. The
supply is a 110 V, 60 Hz. The resistive load is 25 . Calculate:
1. The average value of the output voltage and current,
2. The rms value of the output voltage and current,
3. The average value of the power delivered to the load,
Given:
V = 110 V, f = 60 Hz, and R = 25 .
33
Solution:
For full-wave uncontrolled rectifier circuit shown in Fig. 30.
Fig. 30 Single-phase, full-wave rectifier [2]
1. The average value of the load voltage Vdc can be calculated as follows,
V dc 


1

V dc 
v s  t  d t 
0
2 Vm


1

2 2 x 110


V
m
sin  t  d t d t
0
 99.035 V
The average value of the load current Idc is
I dc 
V dc 99.035

 3.961 A
R
25
2. The rms value of the load voltage Vrms can be calculated as follows,
V rms 
V rms 
1


2
 v s  t  d t 
0
Vm
2

2 x 110
1


 V
m
sin  t 2 dt
0
 110 V
2
Therefore the rms value of the load current Irms is
34
I rms 
V rms 110

 4 .4
R
25
A
3. The average power delivered to the load is
Pavr
Vrms 2

1  vs  t 2

 vo  t   io  t d t    R d t  R
0
0
1
 Pavr 

110 2
25
 484 W
Example 3: For the first quadrant DC chopper shown in Fig. 31. The supply is a DC source of
220 V. The load parameters are as follows: E = 0 V, L = 7.5 mH, and R = 5 . The
chopper is switched at 1 kHz with a duty cycle of 50 %. Calculate:
1. The average value of the load voltage,
2. The maximum and the minimum values for the instantaneous load current,
3. The peak-to-peak load current ripple.
4. The approximated average value of the load current,
Given:
Solution:
Vs = 220 V, E = 0 V, f = 1 kHz, L = 7.5 mH, R = 5 , and k = 0.5.
For the first quadrant DC chopper shown in Fig. 31.
Fig. 31 First quadrant DC chopper [2]
1. The average value of the load voltage Vdc can be calculated as follows,
Vdc 
1
T
kT
V
s
d t  k V s  0.5 x 220  110 V
0
35
2. Since the back emf (battery voltage) is zero. Then, the load current will satisfy the
continuity equation
E e kz  1
 z
. Therefore, the load current is continuous and the
Vs
e 1
crrosponding waveforms are shown in Fig. 32.
Fig. 32 First quadrant DC chopper waveforms [2]
Therefore the maximum and the minimum values for the instantaneous load
current can be determined using the following equations
I1 
Vs
R
 ek z  1 E


 ez 1   R ,


I2 
Vs
R
 e k z  1  E


 e z  1  R ,


z
and
TR
L
using the circuit parameters and substitute in the above equation yield to,
z
TR
5
R


 0.667 ,
L
f L 1000 x 7.5 x 10 3
I1 
220  e 0.5 x 0.667  1 
 1.396  1 
 0  44 x 
  18.38
0
.
667


5  e
1 
 1.948  1 
36
A,
and