# Power Electronics, From ITC, Exercises for Power Electronics

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One of the first and most widely used application of power electronic devices have been in rectification. Rectification refers to the process of converting an ac voltage or current source to dc voltage and current.
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1

Measurement in Power Electronics

1 Introduction

One of the first and most widely used application of power electronic devices have been in

rectification. Rectification refers to the process of converting an ac voltage or current source to

dc voltage and current. Rectifiers specially refer to power electronic converters where the

electrical power flows from the ac side to the dc side. In many situations the same converter

circuit may carry electrical power from the dc side to the ac side where upon they are referred to

as inverters. In this lesson and subsequent ones the working principle and analysis of several

commonly used rectifier circuits supplying different types of loads (resistive, inductive,

capacitive, back emf type) will be presented. Points of interest in the analysis will be.

 Waveforms and characteristic values (average, RMS etc) of the rectified voltage and

current.

 Influence of the load type on the rectified voltage and current.

 Harmonic content in the output.

 Voltage and current ratings of the power electronic devices used in the rectifier circuit.

 Reaction of the rectifier circuit upon the ac network, reactive power requirement, power

factor, harmonics etc.

 Rectifier control aspects (for controlled rectifiers only)

In the analysis, following simplifying assumptions will be made.

 The internal impedance of the ac source is zero.

 Power electronic devices used in the rectifier are ideal switches.

2 Terminologies

Certain terms will be frequently used in this lesson and subsequent lessons while characterizing

different types of rectifiers. Such commonly used terms are defined in this section.

Let “f(t)” be the instantaneous value of any voltage or current associated with a rectifier circuit,

then the following terms, characterizing the properties of “f”, can be defined.

Peak value of f: As the name suggests over all time.

Average (DC) value of f(Fav): Assuming f to be periodic over the time period T

0

1 ( )

T

avF f t dt T  

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RMS (effective) value of f (FRMS): For f, periodic over the time period T,

2

0

1 ( )

T

RMSF f t dt T

 

Form factor of f(fFF) : Form factor of „f „ is defined as

RMS FF

av

F f

F

Ripple factor of f(fRF) : Ripple factor of f is defined as

2 2

2 1 RMS av

RF FF

av

F F f f

F

   

Ripple factor can be used as a measure of the deviation of the output voltage and current of a

rectifier from ideal dc.

Peak to peak ripple of  ˆppf f

max min ˆ

ppf f f  Over period T

Fourier expression:

   0 , , 1

( ) cos sinA n B n n

f t f f n t f n t  

    

Where

0

1 ( )

T f f t dt

T  

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 

 

,

,

2 ( )cos

2 ( )sin

A n T

B n T

f f t n t dt T

f f t n t dt T

Fundamental component of f(F1): It is the RMS value of the sinusoidal component in the

Fourier series expression of f with frequency 1/T.

 2 21 ,1 ,1 1

2 A BF f f 

Kth harmonic component of f(FK): It is the RMS value of the sinusoidal component in the

Fourier series expression of f with frequency K/T.

 2 2, , 1

2 K A K B KF f f 

Crest factor of f(Cf) : By definition

ˆ

f

RMS

f C

F

Distortion factor of f(DFf) : By definition

1 f

RMS

F DF

F

Total Harmonic Distortion of f(THDf): The amount of distortion in the waveform of f is

quantified by means of the index Total Harmonic Distortion (THD). By definition

22 2

1,

01, 1 1

rms rms k f

Krms K

F F F THD

F F

 

      

  

From which it can be shown that

21 f f

f

DF THD

DF

 

Displacement Factor of a Rectifier (DPF): If vi and ii are the per phase input voltage and input

current of a rectifier respectively, then the Displacement Factor of a rectifier is defined as.

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cos iDPF 

Where i is the phase angle between the fundamental components of vi and ii.

Power factor of a rectifier (PF): As for any other equipment, the definition of the power factor

of a rectifier is

Actual power input to the Rectifier

Apparent power input to the Rectifier PF

if the per phase input voltage and current of a rectifier are vi and ii respectively then

1 1cos

i i i

iRMS

I PF DF DPF

I   

21 ii

DPF PF

THD  

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Single Phase Uncontrolled Rectifier

DEFINITION of Rectifier: Converting AC (from mains or other AC source) to DC power by

using power diodes or by controlling the firing angles of thyristors/controllable switches.

Block diagram

1. Single phase uncontrolled halfwave rectifier

1.1 Circuit diagram:

1.2 Principle:

When the output voltage at the terminal of secondary winding transformer is positive then diode

is forward bias where the current flow across R

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When the output voltage at the terminal of secondary winding transformer is negative then diode

is reverse bias where there is no current flow across R.

The inverse voltage of diode D

, max 2D inv iV V V 

Where Vmax maximum voltage

Vi roots means square voltage

Output waveform

Output Voltage

Maximum value

,max 2o rmsV V

Averaged value

 

2 max

0 0

max max2 max0

1 1 ( ) sin

cos 0.318

T T

OAV

T

V Vo t dt V tdt T T

V V t V

T

  

 

   

 

Where 2t 

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RMS value

2 2 2 2max 2 ,

0 0

2

max 2

0

2

max

1 ( ) sin

1 cos 2

2

4

T T

O rms o

T

V V V t dt tdt

T T

V t dt

T

V T

T

 

 

 

max ,

2 O rms

V V 

Where 2t 

Output current

Maximum current:

max ,maxo

V I

R

Averaged current:

2 , max max

0 0

max max

1 1 sin sin

2

0.318

T

o avI I tdt I d T

I I

   

 

 

 

rms current:

max ,

2 o rms

I I

1.3 Form factor:

Form factor is defined by

rms value FF

averaged value

Form factor of output voltage:

, max

, max

/ 2

/ 2

o rms

o av

V V FF

V V

   

In particulars for pure resistive loads io voFF FF

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1.4 Ripple Factor:

Ripple factor of f is defined as

2 2

2 1 RMS av

RF FF

av

F F f f

F

   

2 2 , , 2

0,

,

2

1

1 2

o rms o av

RF

o av

V V V FF

V

    

      

1.5 THDv:

Total Harmonics Distortion:

2 2

1,

1,

RMS RMS

V

RMS

V V THD

V

 

1.6 Efficiency:

Efficiency is defined by

100

Output Power x

Input Power  

,

2 2

max , ,

1o av out o av o av

V V P V I

R R

       

 

,

2 2

max , ,

1

2

i rms

in i rms i rms

V V P V I

R R

       

 

2

max 2

2

max

1

2 100 100 100

1

2

out

in

V

P R

P V

R

 

   

            

   

40.6% 

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2. Single phase uncontrolled fullwave rectifier

2.1 Circuit diagram:

Vs

D1

v1 v2

ii

Vo

io

D3

D2D4

R

2.2 Principle:

Vs

D1

v1 v2

ii

Vo

io

D3

D2D4

R

At 0 / 2t T 

2 2,max sin 0V V t 

D1 and D2 are forward bias

2 2,max sinoutV V V t  

1 2 0D DV V  

When the output voltage at the terminal of secondary winding transformer is positive then diode

D1 and D2 are forward bias where the current flow across R

Vs

D1

v1 v2

ii

Vo

io

D3

D2D4

R

At / 2T t T 

2 2,max sin 0V V t 

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D3 and D4 are forward bias

2 2,max sinoutV V V t    

1 2 2 2,max sinD DV V V V t   

When the output voltage at the terminal of secondary winding transformer is negative then diode

D3 and D4 are forward bias where there is current flow across R.

The inverse voltage of diode D

Output waveform

Vo, Io

t 0

Vmax

Iomax

T/2 T 2T

t 0

V2(t)

VD1 t 0

-Vmax

Output Voltage

Averaged value

 

2 max

0 0

max max2 max ,0

1 2 ( ) sin

2 2 cos 0.636 0.9

T T

OAV

T

i rms

V Vo t dt V tdt T T

V V t V V

T

  

 

    

 

RMS value

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2 2 2 2max 2 ,

0 0

2 2

max max2

0

21 ( ) sin

2 21 cos 2

2 4

T T

O rms o

T

V V V t dt tdt

T T

V Vt T dt

T T

 

  

 

max ,

2 O rms

V V 

Output current

Maximum current:

max ,maxo

V I

R

Averaged current:

2 , max max

0 0

max max

2 1 sin sin

2 0.636

T

o avI I tdt I d T

I I

   

 

 

 

rms current:

max ,

2 o rms

I I

2.3 Form factor:

rms value FF

averaged value

Form factor of output voltage:

, max

, max

/ 2

2 / 2 2

o rms

o av

V V FF

V V

   

In particulars for pure resistive loads io voFF FF

2.4 Ripple Factor:

2 2

2 1 RMS av

RF FF

av

F F f f

F

   

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2 2

, , 2

0,

,

2 2

1

8 1

82 2

o rms o av

RF

o av

V V V FF

V

 

    

       

2.5 THDv:

Total Harmonics Distortion:

2 2

1,

1,

RMS RMS

V

RMS

V V THD

V

 

2.6 Efficiency:

Efficiency is defined by

100

Output Power x

Input Power  

 , 2

2

, , ,

1 0.9o avout o av o av i rms

V P V I V

R R    

,

2

, ,

i rms

in i rms i rms

V P V I

R   

 

   

2

, 2

2

,

1 0.9

100 100 0.9 100 1

i rms out

in i rms

V P R

P V

R

      

81% 

Problem

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Single Phase Uncontrolled Rectifier RL Load

3. Single phase uncontrolled halfwave rectifier

1.7 Circuit diagram:

Vs

D1

v1 v2

ii

Vo

io

L

R

1.8 Principle:

Vs

D1

v1 v2

ii

Vo

io

L

R

When the output voltage at the terminal of secondary winding transformer is positive then diode

is forward bias where the current flow across RL

Vs v1 v2<0

ii

Vo

io

L

R

When the output voltage at the terminal of secondary winding transformer is negative then diode

is reverse bias where there is no current flow across RL.

Output waveform

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0

v2(t)

t

0

t

0

t

vo, io

VD

TT/2 2T

TT/2 2T

β

From the preceding discussion

For 0 ≤ ωt ≤ β

0Dv

0 2v v

0 ii i

for β ≤ ωt ≤ 2π

0 0v

0 0ii i 

2 0 2Dv v v v  

2

0 0 2 0 0

1 1 ˆ sin 2 2

AVV v d t V td t  

    

  

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Or 2,2

0

ˆ 21 cos 1 cos

2 2

rms

AV

VV V

 

 

             

2 2

0 2 0

1 ˆ sin 2

RMSV V td t

  

 

2 2,2 2

ˆ ˆ1 2 sin 2 2 sin 2 sin 2

4 2 2 2 22

rmsVV V        

       

 

Form factor of the voltage waveform is

  0

0 2

0

2 sin 2

2 1 cos

RMS FF

AV

V v

V

    

  

Ripple factor.

 

  2

0 2

2 sin 2 1 1

2 1 cos FF OFFv v

  

 

    

All these quantities are functions of β which can be found as follows.

For 0 ≤ ωt ≤ β

2, 02 sini rms dio

v V t L Ri dt

  

   0 00 0i t i t     

The solution is given by

 2,tan0 0 2

sin

t

rmsV i I e t

Z

    

Where tan L

R

  

And 2 2 2Z R L 

Putting the initial conditions of

L

R

Z

f

Im

Re

j? L

R

Z

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 2, tan0 2

sin sin

t

rmsV i e t

Z

    

      

   2, tan0 2

sin sin 0 rmsV

i t e Z

      

        

Or  tansin sine

   

β as a function of φ can be obtained by solving equation

4. Single phase uncontrolled fullwave rectifier RL Load

Vs

D1

v1 v2

ii

Vo

io

D3

D2D4

L

R

Above figure shows the circuit diagram of a single phase supply, uncontrolled full wave rectifier

supplying an R – L load.

2.8 Waveform

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0

v2(t)

t

0 t

0 t

vo

TT/2 2T

TT/2 2T

0 t

TT/2 2T

io

ii

For 0 ≤ t ≤ T/2 2 0v

2ov v

0ii i

For T/2 ≤ t ≤ T 2 0v

2ov v 

0ii i 

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2.9 Large inductance

Vs

D1

v1 v2

ii

Vo

io

D3

D2D4

L

R

Large inductance

Waveform

V2

T/2 T 2T

T/2 T 2T

T/2 T 2T

t

t

t

t

0

0

0

0

Vo

io

ii

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2.10 Harmonic in input current

The Fourier expression of input current is given

4 sin3 sin5 sin 7 ( ) sin

3 5 7

o i

I t t t i t t

   

       

  

Distortion factor

1,

4

2

o rms

I F

  And rms oF I

1, 4 4 0.90

2 2

rms o i

RMS o

F I DF

F I      

Total Harmonic Distortion of f(THDf): The amount of distortion in the waveform of ii is

quantified by means of the index Total Harmonic Distortion:

2 21 1 0.90 0.4843

0.90i i

i

i

DF THD

DF

    

Or 48.43% ii

THD

Problem

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Lecture 4

Single Phase Controlled Rectifier

5. Single phase uncontrolled halfwave rectifier

1.9 Circuit diagram:

Vs v1 Vi

ii

Vo

io

R

VT

T

1.10 Principle:

When the Voltage Vi is positive then the fired angle is excited, Thyristor T is forward bias where

the current flow across R

Vs v1 Vi

ii

Vo

io

R

VT

T

For 0 t  

ˆ sin 0i iV V t 

0oV

T iV V

For t   

ˆ sin 0i iV V t 

o iV V

0TV

At 2t   

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Vs v1 Vi

ii

Vo

io

R

VT

T

When the output voltage at the terminal of secondary winding transformer is negative then diode

is reverse bias where there is no current flow across R.

ˆ sin 0i iV V t 

Thyristor T turns OFF

0oV 

T iV V 

Output waveform

Vi

Vo, Io

T/2 T 3T/2 2T

T/2 T 3T/2 2T

T/2 T 3T/2 2T 0

t

0 t

0 t

VT

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Output Voltage

Averaged value

     

0

1 1 ˆ( ) sin 2

ˆ ˆ cos 1 cos 1 cos

2 2 2

T

OAV i

i i rms

V Vo t dt V d T

V V V

  

     

 

     

 

Where 2t 

RMS value

2

2

0 0 0

1 ( )

2 RMSV V d

  

 

2 21 2 sin 2

rmsV d

  

  

 

2

1 cos 2 2

rmsV d

  

  

2 sin 2

2 2

rmsV   

     

 

1

2sin 2 1

22

rmsV  

 

     

 

Form factor:

Form factor is defined by

rms value FF

averaged value

Form factor of output voltage:

 

1

2

,

,

sin 2 sin 2 1 1

22 2

1 cos 1 cos

2

rms

o rms

rmso av

V

V FF

VV

    

   

 

          

 

In particulars for pure resistive loads io voFF FF

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1.11 Efficiency:

Efficiency is defined by

100

Output Power x

Input Power  

 

 

,

2 2

, ,

22

1 1 cos

2

1 cos

2

o av rms out o av o av

rms

V V P V I

R R

V

R

 

       

 

      

,

2 1

2 2

, ,

2

sin 2 1 1

22

sin 2 1 1

2 2

i rms rms in i rms i rms

rms

V V P V I

R R

V

R

 

 

 

 

            

 

     

 

   

 

 

22

2

2 2

2

1 cos

1 cos2

sin 2sin 2 11

22 2

2 1 cos

2 2 sin 2

rms

out

in rms

V

P R

P V

R

 

   

  

   

        

           

 

 

6. Single phase controlled full wave rectifier

2.11 Circuit diagram:

Vs

T1

v

ii

Vo

io

T3

T2T4

RVT1

L

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2.12 Principle:

Vs

T1

v>0

ii

Vo

io

T3

T2T4

VT1 R

L

T1 and T2 are conducted

When the outputvoltage at the terminal of secondary winding transformer is positive with the

fired angle then diode T1 and T2 are conducted the current flow across R and L.

For t     

ˆ sin 0i iV V t 

o iV V

1 0TV

Vs

T1

v<0

ii

Vo

io

T3

T2T4

VT1 R

L

T3 and T4 are conducted

For 2t       

ˆ sin 0i iV V t 

o iV V 

1T iV V

Output waveform

T3GEE,2010-2011

25

0

0

t

t

t

t

t

0

0

0

V

Vo

io

ii

vT

T/2 T 2T

T/2 T 2T

T/2 T 2T

Averaged value

   

0

1 1 ˆ( ) sin

ˆ ˆ cos cos cos( ) cos

2

T

OAV i

i i rms

V Vo t dt V d T

V V V

 

 

  

       

 

     

 

Where 2t 

RMS value