power electronics mod

Fundamentals of Power Electronics MTE 320 Spring 2006 E.F. EL-Saadany

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Fundamentals of Power Electronics

1. Power Semiconductors Devices (Switches) There are several power semiconductors 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

diodes 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 junction (pnpn). This is a controllable switch that

usually required to be latched to conduct. This latching (triggering) process is carried out by


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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 didnt 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]


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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, MOSFETs 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


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(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]


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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


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Average Value: This represent the DC component content of the waveform and can be calculated from the following expression;

( )= Tavr dttfTf0

1 or ( )=

0

1 tdtff avr

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)

( )= Trms dttfTf0

21 or ( )= 0

21 tdtff rms

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 Electronics 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

rectifiers 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.


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Table 2 Operation states

Period Diode State Diode Representation Output

Voltage (vo) Load / Supply Current (io / is )

Diode Voltage (vD)

0 t < ON (Forward Biased) SC* vs vs / R Zero

t < 2 OFF (Reverse Biased) OC* Zero Zero vs * SC = Short Circuit, and OC = Open Circuit

The average value of the load voltage Vdc can be calculated as follows,

( ) ( ) == 00

sin21

21 tdtVtdtvV msdc

mmdc VV

V 318.0==

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

R

VR

VR

VI mmdcdc

318.0===

The output DC power is given by ( ) ( )

RV

RV

VIP mdcdcdcdc22 318.0===

The rms value of the load voltage Vrms can be calculated as follows,

( ){ } ( ){ } == 0

2

0

2 sin21

21 tdtVtdtvV msrms

mmrms VV

V 5.02

==

Therefore the rms value of the load current Irms is

RV

RV

RV

I mmrmsrms5.0

2===

The output AC power is given by ( ) ( )

RV

RV

VIP mrmsrmsrmsac22 5.0===


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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).

Fig. 9 Single -phase half-wave rectifier: a) circuit and b) waveforms [3]




Period Diode State Diode Representation Output


Diode Voltage (vD)

0 t < t1 OFF (Reverse Biased) OC* E Zero vs - E

t1 t < t2 ON (Forward Biased) SC* vs (vs - E)/ R Zero

t2 t < t4 OFF (Reverse Biased) OC* E Zero vs - E * SC = Short Circuit, and OC = Open Circuit

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

PRV = Vm + E E

Vm


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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, ( ) ( ) sinms VEtv ===

=

mVE1sin

Since the wave form during the first half cycle is symmetric around2 =t . Therefore, the

angle at which the diode stops conducting ( ) is be given by, =


( ) ( )

+=

+= +

+

22

sin21

21 tdEtdtVtdEtdtvV msdc

( ) ( )[ ]EVV mdc ++= 2cos221

The average value of the load current Idc is

( ) ( )[ ] ( ) ( )[ ]

REV

R

EEV

REV

I mm

dcdc

2

2cos22cos2

21

+=

++

==


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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]




Period Conducting Diodes Output Voltage

(vo) Load Current

(io ) Supply Current

(is ) Diode Voltage

(vD)

0 t < D1 & D2 vs vs / R vs / R - vs for D3 & D4 t < 2 D3 & D4 - vs - vs / R vs / R vs for D1 & D2


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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) whereas the supply current is negative (is = vs / R).


( ) ( ) == 00

sin11 tdtdtVtdtvV msdc

mmdc VV

V 6366.02 ==


R

VR

VR

VI mmdcdc

6366.02 ===


( ){ } ( ){ } == 0

2

0

2 sin11 tdtVtdtvV msrms

mmrms VV

V 707.02==


R

VR

VR

VI mmrmsrms

707.02

===

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.


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Therefore, the load time constant RL= is very high and can be considered infinity.

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 [2xxxx]


Period Conducting Diodes Output Voltage

(vo) Load Current


(is ) Diode Voltage

(vD)

0 t < D1 & D2 vs Ia Ia - vs for D3 & D4 t < 2 D3 & D4 - vs Ia - Ia 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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( ) ( ) == 00

sin11 tdtVtdtvV msdc

mmdc VV

V 6366.02 ==

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,

R

VR

VR

VII mmdcadc

6366.02 ====

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

R

EVR

EV

REV

II mm

dcadc

=

=== 22

(provided that E < Vdc)


( ){ } ( ){ } == 0

2

0


mmrms VV

V 707.02==

Since the load current is constant over the studied period, therefore the rms value of the load

current Irms is

adcrms III ==

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.


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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 wont 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]


( ) ( ) ==

tdtVtdtvV msdc sin2

121


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( )( ) cos12 += mdcV

V


( )( ) cos12 +== RV

RV

I mdcdc


Period Thyristor State Thyristor Representation Output


Thyristor Voltage (vT1)

0 t < OFF (Forward Blocking) OC* Zero Zero vs t < ON (Forward Biased) SC* vs vs / R Zero

t < 2 OFF (Reverse Biased) OC* Zero Zero vs * SC = Short Circuit, and OC = Open Circuit

Therefore, the average output voltage can vary from 0 to mV and the average load current will

vary from 0 to R

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. 12-b.


( ){ } ( ){ } ==

tdtVtdtvV msrms

22 sin21

21

( )

+=

22sin1

2

mrms

VV


( )

+==

22sin1

2R

VR

VI mrmsrms

The PRV of the thyristor for this configuration is Vm.


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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.


Period Conducting Thyristors Output

Voltage (vo) Load Current


(is ) Thyristor

Voltage (vT)

t < + T1 & T2 vs Ia Ia - vs for T3 & T4 + t < 2 + T3 & T4 - vs Ia - Ia vs for T1 & T2

Fig. 13 Single -phase full-wave rectifier loaded with highly inductive load [2]


( ) ( ) ++ ==

tdtdtVtdtvV msdc sin

11

( ) cos2 m

dcV

V =


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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,

( )

R

EV

REV

II

m

dcadc

===

cos2

(provided that E < Vdc and Vdc > 0)

In case the load doesnt contain a DC battery E (or a back emf) in addition to the highly

inductive load, the load current will be

( ) cos2

RV

RV

II mdcadc === (provided that Vdc > 0)

Therefore, the average output voltage can vary from mV2 to

mV2 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.


( ){ } ( ){ } ++ ==

tdtVtdtvV msrms

22 sin11

mmrms VV

V 707.02==

Since the load current is constant over the studied period, therefore the rms value of the load

current Irms is

adcrms III ==

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 used


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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.


( ) ( ) ==

tdtVtdtvV msdc sin

11

( )( ) cos1+= mdcV

V

Fig. 14 Single -phase semiconverter loaded with highly inductive load [2]


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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 & D2 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 & D1 - vs Ia - Ia 0 vs for T1 & D2 & Dm


( )( )

R

EV

REV

I

m

dcdc

+==

cos12 (provided that E < Vdc)

Therefore, the average output voltage can vary from 0 to mV 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.


( ){ } ( ){ } ==

tdtVtdtvV msrms

22 sin11

( )

+=

22sin1

2

mrms

VV

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.


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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


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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,

skT

sdc VktdVTV ==

0

1

where Vs is the DC supply voltage, T is the total period of operation, and k is the duty

cycle given by T

tk on= , where ton is the period at which the chopper is ON.

Table 9 Operation modes

Currents Period Mode Chopper State Diode State

Output Voltage

(vo) (i ) (is ) (iDm ) 0 t < kT 1 ON (SC*) OFF (OC*) Vs i1 i1 0 kT t < T 2 OFF (OC*) ON (SC*) 0 i2 0 i2

* SC = Short Circuit, and OC = Open Circuit


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For (L / R) >> T, the load current is continuous and can be expressed by,

( )( ) ( )( ) ( )

=

+==

TktTtTkeREeIti

TkteR

EVeIti

tiLRtLRt

LRtsLRt

)1('0,1'

01

/'/'22

//11

where t is a new reference related to t by t = t - kT

where RE

ee

RV

Iz

zks

=11

1 , RE

ee

RV

Iz

zks

=

11

2 , and LRT

z = .

For discontinuous mode of operation (I1 = 0), the load current can be expressed by,

( )( ) ( )

( ) ( )

=

==

TktTtTkeREeIti

TkteR

EVti

tiLRtLRt

LRts

)1('0,1'

01

/'/'22

/1

For the load current to be continuous, I1 should be greater than or equal to zero. Therefore,

011

1

=

RE

ee

RV

Iz

kzs

011

sz

kz

VE

ee

11

z

kz

s ee

VE

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.


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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,

( ) sT

kTsdc VktdVT

V == 11 where Vs is the DC supply voltage, T is the total period of operation, and k is the duty

cycle given by T

tk on= , where ton is the period at which the chopper is ON.


Currents Period Chopper State Diode State Load Voltage (vo)

(iL ) (is ) (ich ) 0 t < kT ON (SC*) OFF (OC*) 0 i1 0 i1 kT t < T OFF (OC*) ON (SC*) Vs i2 i2 0

* SC = Short Circuit, and OC = Open Circuit

For continuous load current operation, the load current is can be expressed by,

( )( )

( ) ( ) ( )

+=

+=

TktTtTkeR

VEeIti

TkteREeI

tiLRtsLRt

LRtLRt

L

1'0,1'

01

/'/'22

//1

where t is a new reference related to t by t = t kT

where ( )

RE

ee

RV

Iz

zks

=

11 1

1 , RE

eee

RV

Iz

zzks

=

12 , and L

RTz = .


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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


Mode Conducting switch Load Current (iL) Load Voltage (vL)

1 S1 + ve Vs 2 S4 - ve 0 3 D1 - ve Vs 4 D4 + ve 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


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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



1 S2 + ve 0 2 S3 - ve - Vs 3 D2 - ve 0 4 D3 + ve - 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, 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 first quadrant by controlling S1

and keeping S2 switched ON, S3 and S4 switched OFF. Moreover, it can operate in the second

quadrant by controlling S4 and keeping S2, S3 and S1 switched OFF. S2 and S3 (as well as S1 and

S4) 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


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Table 13 Four quadrant operation modes


1 S1 and S2 + ve Vs 2 D3 and D4 + ve - Vs 3 S3 and S4 - ve - Vs 4 D1 and D2 - ve Vs

Diodes D3 and D4 are activated once S1 and S2 are switched off (and S3 and S4 are triggered)

while diodes D1 and D2 are activated once S3 and S4 are switched off (and S1 and S2 are

triggered).

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 (switchs 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


28

4.1 PWM for DC Output

They 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]


29

They are generated by comparing a triangular wave with a variable DC signal to generate

a sinusoidal variable duty cycle tracking the following function,

( ) ( )tDDtd odc sinmax+= where ( )td is the required duty cycle signal to be generated, dcD is the normal duty cycle with no sinusoidal modulation, maxD is the maximum modulation

constant, and o is the modulation frequency.

The generated DC output voltage in this case can be represented by

( ) ( )( ) ( )tVDVDVtDDVtdv odcdcdcdcodcdco sinsin maxmax +=+== 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]


30

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.


31

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,


32

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,

( ) ( ) == 00

sin21

21 tdtVtdtvV msdc

VxVV mdc 517.491102 ===


Ax

RV

RV

I mdcdc 981.1251102 ====

2. The rms value of the load voltage Vrms can be calculated as follows,


33

( ){ } ( ){ } == 0

2

0

2 sin21

21 tdtVtdtvV msrms

VxV

V mrms 782.7721102

2===


AR

VI rmsrms 111.325

782.77 ===

3. The average power delivered to the load is

( ) ( ) ( )[ ] ( )R

Vtd

Rtv

tdtitvP rmssooavr2

0

2

021

21 === ( ) WPavr 002.24225

782.77 2 ==

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,

( ) ( ) ( )sso RRRtvtv +=

Therefore the value of the average power delivered to the load is

( ) ( ) ( )[ ] ( ) ( ) ( )22

02

2

021

21

srms

ssooavr

RRRVtd

RRRtvtdtitvP +=+==

( ) ( ) WPavr 934.20602525782.77

22 =+=

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 .


34

Solution: For full-wave uncontrolled rectifier circuit shown in Fig. 30.

Fig. 30 Single-phase, full-wave rectifier [2]


( ) ( ) == 00

sin11 tdtdtVtdtvV msdc

VxV

V mdc 035.99110222 ===


AR

VI dcdc 961.325

035.99 ===

2. The rms value of the load voltage Vrms can be calculated as follows,

( ){ } ( ){ } == 0

2

0


VxVV mrms 11021102

2===



35

AR

VI rmsrms 4.425

110 ===

3. The average power delivered to the load is

( ) ( ) ( )[ ] ( )R

Vtd

Rtv

tdtitvP rmssooavr2

0

2

0

11 === ( ) WPavr 48425110 2 ==

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: Vs = 220 V, E = 0 V, f = 1 kHz, L = 7.5 mH, R = 5 , and k = 0.5.

Solution: For the first quadrant DC chopper shown in Fig. 31.

Fig. 31 First quadrant DC chopper [2]


VxVktdVT

V skT

sdc 1102205.01

0

====


36

2. Since the back emf (battery voltage) is zero. Then, the load current will satisfy the

continuity equation 11

z

kz

s ee

VE . Therefore, the load current is continuous and the

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

RE

ee

RV

Iz

zks

=11

1 ,

RE

ee

RV

Iz

zks

=

11

2 , and

LRT

z =

using the circuit parameters and substitute in the above equation yield to,

667.0105.71000

53==== xxLf

RLRTz ,

Axe

eIx

38.181948.11396.1440

11

5220

667.0

667.05.0

1 =

=

= , and


37

Axe

eIx

659.251513.01716.0440

11

5220

667.0

667.05.0

2 =

=

=

3. The peak-to-peak load current ripple I can be calculated using, AIII 279.738.18659.2512 ===

4. The approximated average value of the load current can be calculated assuming

that the current waveforms are linear and using the 50 % duty cycle as follows,

AII

I dc 02.22238.18659.25

212 =+=+=

References

[1] Issa Batarseh, Power Electronic Circuits John Wiley & Sons Inc., USA, 2004.

[2] Muhammad H. Rashid, Power Electronics, Third Edition, Pearson Prentice Hall, NJ, USA, 2004.

[3] Theodore Wildi, "Electrical Machines Drives, and Power Systems," Prentice Hall, Ohio, 2006.

[4] Timothy L. Skvarenina, The Power Electronics Handbook, CRC Press, 2001.

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