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T3GEE,2010-2011

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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 ** ˆ*ppf f *

max min ˆ

*ppf f f* Over period T

**Fourier expression:
**

0 , , 1

( ) cos sin*A 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 2*D inv iV V V*

Where Vmax maximum voltage

Vi roots means square voltage

**Output waveform
**

**Output Voltage
**

** Maximum value
**

,max 2*o 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 **2*t*

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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 **2*t*

**Output current
**

Maximum current:

max
,max*o
*

*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 , ,

1*o 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 / 2*t T*

2 2,max sin 0*V V t*

D1 and D2 are forward bias

2 2,max sin*outV V V t*

1 2 0*D 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 / 2*T t T*

2 2,max sin 0*V V t*

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D3 and D4 are forward bias

2 2,max sin*outV V V t*

1 2 2 2,max sin*D 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
,max*o
*

*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
0.9*o 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

0*Dv *

0 2*v v*

0 *ii i*

for β ≤ ωt ≤ 2π

0 0*v *

0 0*ii i*

2 0 2*Dv 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 sin*i rms
dio
*

*v V t L Ri
dt
*

0 00 0*i 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 2*Z 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 sin*e
*

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

**4. Single phase uncontrolled fullwave rectifier RL Load
**

**2.7 Circuit diagram 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 0*v *

2*ov v*

0*ii i*

For T/2 ≤ t ≤ T 2 0*v *

2*ov v*

0*ii 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

0.90*i
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 0*i iV V t*

0*oV *

*T iV V*

For *t*

ˆ sin 0*i iV V t*

*o iV V*

0*TV *

At 2*t*

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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 0*i iV V t*

Thyristor T turns OFF

0*oV*

*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 **2*t*

** 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 0*i iV V t*

*o iV V*

1 0*TV *

Vs

T1

v<0

ii

Vo

io

T3

T2T4

VT1 R

L

**T3 and T4 are conducted
**

For 2*t*

ˆ sin 0*i iV V t*

*o iV V*

1*T iV V*

**Output waveform
**

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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 **2*t*

** RMS value **