Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

1

Chapter 4

DC to AC Conversion

(INVERTER)

• General concept
• Basic principles/concepts
• Single-phase inverter

– Square wave
– Notching
– PWM

• Harmonics
• Modulation
• Three-phase inverter

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

2

DC to AC Converter

(Inverter)

• DEFINITION: Converts DC to AC power

by switching the DC input voltage (or

current) in a pre-determined sequence so as

to generate AC voltage (or current) output.

• TYPICAL APPLICATIONS:

– Un-interruptible power supply (UPS), Industrial

(induction motor) drives, Traction, HVDC

• General block diagram

I

DC

I

ac

+

−

V

DC

V

ac

+

−

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

3

Types of inverter

• Voltage Source Inverter (VSI)
• Current Source Inverter (CSI)

"DC LINK"

I

ac

+

−

V

DC

Load Voltage

+

−

L

I

LOAD

Load Current

I

DC

+

−

V

DC

C

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

4

Voltage source inverter (VSI)

with variable DC link

DC LINK

+

-

V

s

V

o

+

-

C

+

-

V

in

CHOPPER

(Variable DC output)

INVERTER

(Switch are turned ON/OFF

with square-wave patterns)

• DC link voltage is varied by a DC-to DC converter

or controlled rectifier.

• Generate “square wave” output voltage.

• Output voltage amplitude is varied as DC link is

varied.

•

Frequency of output voltage is varied by changing

the frequency of the square wave pulses.

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

5

Variable DC link inverter (2)

• Advantages:

– simple waveform generation
– Reliable

• Disadvantages:

– Extra conversion stage
– Poor harmonics

T

1

T

2

t

V

dc1

V

dc2

Higher input voltage
Higher frequency

Lower input voltage
Lower frequency

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

6

VSI with fixed DC link

INVERTER

+

−

V

in

(fixed)

V

o

+

−

C

Switch turned ON and OFF

with PWM pattern

• DC voltage is held constant.

• Output voltage amplitude and frequency

are varied simultaneously using PWM
technique.

• Good harmonic control, but at the expense

of complex waveform generation

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

7

Operation of simple square-

wave inverter (1)

• To illustrate the concept of AC waveform

generation

V

DC

T1

T4

T3

T2

+ V

O

-

D1

D2

D3

D4

SQUARE-WAVE

INVERTERS

S1

S3

S2

S4

EQUAVALENT

CIRCUIT

I

O

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

8

Operation of simple square-

wave inverter (2)

V

DC

S1

S4

S3

+ v

O

−

V

DC

S1

S4

S3

S2

+ v

O

−

V

DC

v

O

t

1

t

2

t

S1,S2 ON; S3,S4 OFF

for t

1

< t < t

2

t

2

t

3

v

O

-V

DC

t

S3,S4 ON ; S1,S2 OFF

for t

2

< t < t

3

S2

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

9

Waveforms and harmonics of

square-wave inverter

FUNDAMENTAL

3

RD

HARMONIC

5

RD

HARMONIC

π

DC

V

4

V

dc

-V

dc

V

1

3

1

V

5

1

V

INVERTER

OUTPUT

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

10

Filtering

• Output of the inverter is “chopped AC

voltage with zero DC component”.In some
applications such as UPS, “high purity” sine
wave output is required.

• An LC section low-pass filter is normally

fitted at the inverter output to reduce the
high frequency harmonics.

• In some applications such as AC motor

drive, filtering is not required.

v

O 1

+

−

LOAD

L

C

v

O 2

(LOW PASS) FILTER

+

−

v

O 1

v

O 2

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

11

Notes on low-pass filters

• In square wave inverters, maximum output voltage

is achievable. However there in NO control in
harmonics and output voltage magnitude.

• The harmonics are always at three, five, seven etc

times the fundamental frequency.

• Hence the cut-off frequency of the low pass filter is

somewhat fixed. The filter size is dictated by the
VA ratings of the inverter.

• To reduce filter size, the PWM switching scheme

can be utilised.

•

In this technique, the harmonics are “pushed” to

higher frequencies. Thus the cut-off frequency of
the filter is increased. Hence the filter components
(I.e. L and C) sizes are reduced.

• The trade off for this flexibility is complexity in

the switching waveforms.

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

12

“Notching”of square wave

Vdc

Vdc

−

Vdc

Vdc

−

Notched Square Wave

Fundamental Component

• Notching results in controllable output

voltage magnitude (compare Figures
above).

• Limited degree of harmonics control is

possible

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

13

Pulse-width modulation

(PWM)

• A better square wave notching is shown

below - this is known as PWM technique.

• Both amplitude and frequency can be

controlled independently. Very flexible.

1

1

pwm waveform

desired
sinusoid

SINUSOIDAL PULSE-WITDH MODULATED

APPROXIMATION TO SINE WAVE

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

14

PWM- output voltage and

frequency control

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

15

Output voltage harmonics

• Why need to consider harmonics?

– Waveform quality must match TNB supply.

“Power Quality” issue.

– Harmonics may cause degradation of

equipment. Equipment need to be “de-rated”.

• Total Harmonic Distortion (THD) is a measure to

determine the “quality” of a given waveform.

• DEFINITION of THD (voltage)

(

)

(

)

(

)

(

)

frequency.

harmonic

at

impedance

the

is

:

current

harmonic

with

voltage

harmonic

the

replacing

by

obtained

be

can

THD

Current

number.

harmonics

the

is

where

,

1

2

2

,

,

1

2

2

,

1

2

,

1

2

2

,

n

n

n

n

RMS

n

RMS

n

RMS

n

RMS

RMS

RMS

n

RMS

n

Z

Z

V

I

I

I

THDi

n

V

V

V

V

V

THDv

=

=

−

=

=

∑

∑

∑

∞

=

∞

=

∞

=

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

16

Fourier Series

• Study of harmonics requires understanding

of wave shapes. Fourier Series is a tool to
analyse wave shapes.

( )

( )

(

)

t

n

b

n

a

a

v

f

d

n

v

f

b

d

n

v

f

a

d

v

f

a

n

n

n

o

n

n

o

ω

θ

θ

θ

θ

θ

π

θ

θ

π

θ

π

π

π

π

=

+

+

=

=

=

=

∑

∫

∫

∫

∞

=

where

sin

cos

2

1

)

(

Fourier

Inverse

sin

)

(

1

cos

)

(

1

)

(

1

Series

Fourier

1

2

0

2

0

2

0

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

17

Harmonics of square-wave (1)

V

dc

-V

dc

θ=ωt

π

2π

( )

( )

( )

( )













−

=

=













−

=

=













−

+

=

∫

∫

∫

∫

∫

∫

π

π

π

π

π

π

π

π

π

θ

θ

θ

θ

π

θ

θ

θ

θ

π

θ

θ

π

2

0

2

0

2

0

sin

sin

0

cos

cos

0

1

d

n

d

n

V

b

d

n

d

n

V

a

d

V

d

V

a

dc

n

dc

n

dc

dc

o

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

18

Harmonics of square wave (2)

( )

( )

[

]

[

]

[

]

[

]

π

π

π

π

π

π

π

π

π

π

π

π

θ

θ

π

π

π

π

n

V

b

n

b

n

n

n

V

n

n

n

V

n

n

n

n

V

n

n

n

V

b

dc

n

n

dc

dc

dc

dc

n

4

1

cos

odd,

is

n

when

0

1

cos

even,

is

n

when

)

cos

1

(

2

)

cos

1

(

)

cos

1

(

)

cos

2

(cos

)

cos

0

(cos

cos

cos

Solving,

2

0

=

=

−

=

=

−

=

−

+

−

=

−

+

−

=

+

−

=

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

19

Spectra of square wave

1

3

5

7

9

11

Normalised

Fundamental

3rd (0.33)

5th (0.2)

7th (0.14)

9th (0.11)

11th (0.09)

1st

n

• Spectra (harmonics) characteristics:

– Harmonic decreases as n increases. It decreases

with a factor of (1/n).

– Even harmonics are absent
– Nearest harmonics is the 3rd. If fundamental is

50Hz, then nearest harmonic is 150Hz.

– Due to the small separation between the

fundamental an harmonics, output low-pass
filter design can be quite difficult.

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

20

Quasi-square wave (QSW)

( )

[

]

( )

(

)

[

]

(

)

(

)

α

π

α

π

α

π

α

π

α

π

α

π

α

π

θ

π

θ

θ

π

α

π

α

α

π

α

n

n

n

n

n

n

n

n

n

n

n

n

V

n

n

V

d

n

V

b

a

dc

dc

dc

n

n

cos

cos

sin

sin

cos

cos

cos

cos

Expanding,

cos

cos

2

cos

2

sin

1

2

symmetry,

wave

-

half

to

Due

.

0

that

Note

=

+

=

−

=

−

−

−

=

−

=













=

=

−

−

∫

π

π

2

α

α

α

V

dc

-V

dc

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

21

Harmonics control

( )

[

]

( )

[

]

( )

( )

n

n

b

b

V

b

n

n

V

b

b

n

n

n

V

n

n

n

n

V

b

o

dc

dc

n

n

dc

dc

n

o

3

1

1

90

:

if

eliminated

be

will

harmonic

general,

In

waveform.

the

from

eliminated

is

harmonic

third

or the

,

0

then

,

30

if

example

For

,

adjusting

by

controlled

be

also

can

Harmonics

α

by varying

controlled

is

,

,

l

fundamenta

The

cos

4

:

is

l

fundamenta

the

of

amplitude

,

particular

In

cos

4

odd,

is

n

If

,

0

even,

is

n

If

cos

1

cos

2

cos

cos

cos

2

=

=

=

=

=

⇒

=

⇒

−

=

−

=

⇒

α

α

α

α

π

α

π

π

α

π

α

π

α

π

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

22

Example

degrees

30

with

case

wave

square

-

quasi

for

(c)

and

(b)

Repeat

harmonics

zero

-

non

e

first thre

the

using

by

THDi

the

c)

harmonics

zero

-

non

e

first thre

the

using

by

THDv

the

b)

formula.

exact"

"

the

using

THDv

the

a)

:

Calculate

series.

in

10mH

L

and

10R

R

is

load

The

100V.

is

ge

link volta

DC

The

signals.

wave

square

by

fed

is

inverter

phase

single

bridge

-

full

A

=

=

=

α

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

23

Half-bridge inverter (1)

V

o

R

L

+

−

V

C1

V

C2

+

-

+

-

S

1

S

2

V

dc

2

Vdc

2

Vdc

−

S1 ON
S2 OFF

S1 OFF
S2 ON

t

0

G

• Also known as the “inverter leg”.
• Basic building block for full bridge, three

phase and higher order inverters.

• G is the “centre point”.
• Both capacitors have the same value.

Thus the DC link is equally “spilt”into
two.

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

24

Half-bridge inverter (2)

• The top and bottom switch has to be

“complementary”, i.e. If the top switch is
closed (on), the bottom must be off, and
vice-versa.

• In practical, a dead time as shown below is

required to avoid “shoot-through” faults.

t

d

t

d

"Dead time' = t

d

S

1

signal

(gate)

S

2

signal

(gate)

S1

S2

+

−

V

dc

R

L

G

"Shoot through fault" .

I

short

is very large

I

short

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

25

Single-phase, full-bridge (1)

• Full bridge (single phase) is built from two

half-bridge leg.

• The switching in the second leg is “delayed

by 180 degrees” from the first leg.

S1

S4

S3

S2

+

-

G

+

2

dc

V

2

dc

V

-

2

dc

V

2

dc

V

dc

V

2

dc

V

−

2

dc

V

−

dc

V

−

π

π

π

π

2

π

2

π

2

t

ω

t

ω

t

ω

RG

V

G

R

V

'

o

V

G

R

o

V

V

V

RG

'

−

=

groumd"

virtual

"

is

G

LEG R

LEG R'

R

R'

-

o

V

+

dc

V

+

-

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

26

Three-phase inverter

• Each leg (Red, Yellow, Blue) is delayed by

120 degrees.

• A three-phase inverter with star connected

load is shown below

Z

Y

Z

R

Z

B

G

R

Y

B

i

R

i

Y

i

B

i

a

i

b

+V

dc

N

S1

S4

S6

S3

S5

S2

+

+

−

−

V

dc

/2

V

dc

/2

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

27

Square-wave inverter

waveforms

1
3

2,4

2

3,5

4

3
5

4,6

4

1,5

6

5
1

2,6

6

1,3

2

V

AD

V

B0

V

C0

V

AB

V

APH

(a) Three phase pole switching waveforms

(b) Line voltage waveform

(c) Phase voltage waveform (six-step)

60

0

120

0

Interval

Positive device(s) on

Negative devise(s) on

2V

DC

/3

V

DC

/3

-V

DC

/3

-2V

DC

/3

V

DC

-V

DC

V

DC

/2

-V

DC

/2

t

t

t

t

t

Quasi-square wave operation voltage waveforms

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

28

Three-phase inverter

waveform relationship

• V

RG

, V

YG

, V

BG

are known as “pole

switching waveform” or “inverter phase
voltage”.

• V

RY

, V

RB

, V

YB

are known as “line to line

voltage” or simply “line voltage”.

• For a three-phase star-connected load, the

load phase voltage with respect to the “N”
(star-point) potential is known as V

RN

,V

YN

,

V

BN

. It is also popularly termed as “six-

step” waveform

Power Electronics and

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Zainal Salam, 2002

29

MODULATION: Pulse Width

Modulation (PWM)

Modulating Waveform

Carrier waveform

1

M

1

+

1

−

0

2

dc

V

2

dc

V

−

0

0

t

1

t

2

t

3

t

4

t

5

t

• Triangulation method (Natural sampling)

– Amplitudes of the triangular wave (carrier) and

sine wave (modulating) are compared to obtain
PWM waveform. Simple analogue comparator
can be used.

– Basically an analogue method. Its digital

version, known as REGULAR sampling is
widely used in industry.

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

30

PWM types

• Natural (sinusoidal) sampling (as shown

on previous slide)

– Problems with analogue circuitry, e.g. Drift,

sensitivity etc.

• Regular sampling

– simplified version of natural sampling that

results in simple digital implementation

• Optimised PWM

– PWM waveform are constructed based on

certain performance criteria, e.g. THD.

• Harmonic elimination/minimisation PWM

– PWM waveforms are constructed to eliminate

some undesirable harmonics from the output
waveform spectra.

– Highly mathematical in nature

• Space-vector modulation (SVM)

– A simple technique based on volt-second that is

normally used with three-phase inverter motor-
drive

Power Electronics and

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Zainal Salam, 2002

31

Natural/Regular sampling

( )

(1,2,3...)

integer

an

is

and

signal

modulating

the

of

frequency

the

is

where

M

:

at

located

normally

are

harmonics

The

.

frequency"

harmonic

"

the

to

related

is

M

waveform

modulating

the

of

Frequency

veform

carrier wa

the

of

Frequency

M

)

(

M

RATIO

MODULATION

ly.

respective

voltage,

(DC)

input

and

voltage

output

the

of

l

fundamenta

are

,

where

M

:

holds

ip

relationsh

linear

the

1,

M

0

If

versa.

vice

and

high

is

output

wave

sine

the

then

high,

is

M

If

magnitude.

tage

output vol

wave)

(sine

l

fundamenta

the

to

related

is

M

veform

carrier wa

the

of

Amplitude

waveform

modulating

the

of

Amplitude

M

:

M

INDEX

MODULATION

R

R

R

R

1

I

1

I

I

I

I

I

k

f

f

k

f

p

p

V

V

V

V

m

m

in

in

=

=

=

=

=

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

=

<

<

=

=

Power Electronics and

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Zainal Salam, 2002

32

Asymmetric and symmetric

regular sampling

T

sample

point

t

M

m

ω

sin

1

1

+

1

−

4

T

4

3T

4

5T

4

π

2

dc

V

2

dc

V

−

0

t

1

t

2

t

3

t

t

asymmetric

sampling

symmetric

sampling

t

Generating of PWM waveform regular sampling

Power Electronics and

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Zainal Salam, 2002

33

Bipolar and unipolar PWM

switching scheme

• In many books, the term “bipolar” and

“unipolar” PWM switching are often
mentioned.

• The difference is in the way the sinusoidal

(modulating) waveform is compared with
the triangular.

• In general, unipolar switching scheme

produces better harmonics. But it is more
difficult to implement.

• In this class only bipolar PWM is

considered.

Power Electronics and

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Zainal Salam, 2002

34

Bipolar PWM switching

k

1

δ

k

2

δ

k

α

∆

4

∆

=

δ

π

π

2

carrier

waveform

modulating

waveform

pulse

kth

π

π

2

Power Electronics and

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Zainal Salam, 2002

35

Pulse width relationships

k

1

δ

k

2

δ

k

α

∆

4

∆

=

δ

π

π

2

carrier

waveform

modulating

waveform

pulse

kth

π

π

2

Power Electronics and

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Zainal Salam, 2002

36

Characterisation of PWM

pulses for bipolar switching

pulse

PWM

kth

The

∆

0

δ

0

δ

0

δ

0

δ

k

1

δ

k

2

δ

2

S

V

+

2

S

V

−

k

α

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

37

Determination of switching

angles for kth PWM pulse (1)

v

Vmsin θ

( )

A

p2

A

p1

2

dc

V

+

2

dc

V

−

A

S2

A

S1

2

2

1

1

second,

-

volt

the

Equating

p

s

p

s

A

A

A

A

=

=

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

38

PWM Switching angles (2)

[

]

)

sin(

sin

2

cos

)

2

cos(

sin

sinusoid,

by the

supplied

second

-

volt

The

where

;

2

Similarly,

where

2

2

2

)

2

(

2

:

as

given

is

pulse

PWM

the

of

cycle

half

each

during

voltage

average

The

2

1

2

2

2

2

1

1

1

1

1

1

1

o

k

o

m

k

o

k

m

m

s

o

o

k

k

dc

k

k

o

o

k

k

s

k

o

o

k

dc

o

k

o

k

dc

k

V

V

d

V

A

V

V

V

V

V

V

k

o

k

δ

α

δ

α

δ

α

θ

θ

δ

δ

δ

β

β

δ

δ

δ

β

β

δ

δ

δ

δ

δ

δ

δ

α

δ

α

−

=

−

−

=

=













−

=













=













−

=













=













−













=













−

−













=

∫

−

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

39

Switching angles (3)

)

sin(

)

2

(

)

sin(

2

2

2

edge

leading

for the

Hence,

;

strategy,

modulation

the

derive

To

2

2

;

2

2

,

waveforms

PWM

the

of

seconds

-

volt

The

)

sin(

2

Similarly,

)

sin(

2

,

small

for

sin

Since,

1

1

2

2

1

1

21

2

1

1

2

1

o

k

dc

m

k

o

k

m

o

o

dc

k

s

p

s

p

o

dc

k

p

o

dc

k

p

o

k

m

o

s

o

k

m

o

s

o

o

o

V

V

V

V

A

A

A

A

V

A

V

A

V

A

V

A

δ

α

β

δ

α

δ

δ

β

δ

β

δ

β

δ

α

δ

δ

α

δ

δ

δ

δ

−

=

⇒

−

=













=

=













=













=

+

=

−

=

→

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

40

PWM switching angles (4)

[

]

[

]

)

sin(

1

and

)

sin(

1

width,

-

pulse

for the

solve

to

ng

Substituti

)

sin(

:

derived

be

can

edge

trailing

the

method,

similar

Using

)

sin(

Thus,

1.

to

0

from

It varies

depth.

or

index

modulation

as

known

is

2

ratio,

voltage

The

2

1

1

1

2

1

o

k

I

o

k

o

k

I

o

k

o

o

k

k

o

k

I

k

o

k

I

k

dc

m

I

M

M

M

M

)

(V

V

M

δ

α

δ

δ

δ

α

δ

δ

δ

δ

δ

β

δ

α

β

δ

α

β

+

+

=

−

+

=

⇒

−

=

−

=

−

=

=

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

41

PWM Pulse width

[

]

k

I

o

k

k

k

k

k

M

α

δ

δ

δ

δ

δ

δ

δ

δ

α

δ

α

sin

1

,

Modulation

Symmetric

For

different.

are

and

i.e

,

Modulation

Asymmetric

for

valid

is

equation

above

The

:

edge

Trailing

:

edge

Leading

:

is

pulse

kth

the

of

angles

switching

the

Thus

k

2k

1k

2k

1k

1

1

+

=

⇒

=

=

+

−

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

42

Example

• For the PWM shown below, calculate the switching

angles for all the pulses.

V

5

.

1

V

2

π

π

2

1

2

3

4

5

6

7

8

9

t1

t2

t3 t4 t5 t6 t7 t8 t9 t10 t11 t12

t13

t14

t15

t16

t17

t18 π

2

π

1

α

carrier

waveform

modulating

waveform

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

43

Harmonics of bipolar PWM

{

}

)

2

(

cos

)

(

cos

)

(

cos

)

(

cos

)

(

cos

)

2

(

cos

:

to

reduced

be

can

Which

sin

2

2

sin

2

2

sin

2

2

sin

)

(

1

2

:

as

computed

be

can

pulse

PWM

(kth)

each

of

content

harmonic

symmetry,

wave

-

half

is

waveform

PWM

the

Assuming

2

1

2

1

2

2

0

2

2

1

1

o

k

k

k

k

k

k

k

k

k

o

k

dc

nk

dc

dc

dc

T

nk

n

n

n

n

n

n

n

V

b

d

n

V

d

n

V

d

n

V

d

n

v

f

b

o

k

k

k

k

k

k

k

k

k

o

k

δ

α

δ

α

δ

α

δ

α

δ

α

δ

α

π

θ

θ

π

θ

θ

π

θ

θ

π

θ

θ

π

δ

α

δ

α

δ

α

δ

α

δ

α

δ

α

+

−

+

+

−

−

+

+

−

−

−

−

=



























−

+





























+



























−

=















=

∫

∫

∫

∫

+

+

+

−

−

−

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

44

Harmonics of PWM

[

]

equation.

this

of

n

computatio

the

shows

page

next

on the

slide

The

:

i.e.

period,

one

over

pulses

for the

of

sum

isthe

waveform

PWM

for the

coefficent

Fourier

ly.The

productive

simplified

be

cannot

equation

This

2

cos

cos

2

)

2

(

cos

)

(

cos

2

Yeilding,

1

1

1

∑

=

=

+

−

−

−

=

p

k

nk

n

nk

o

k

k

k

k

k

dc

nk

b

b

p

b

n

n

n

n

n

V

b

δ

α

α

δ

α

π

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

45

PWM Spectra

p

p

2

p

3

p

4

0

.

1

=

M

8

.

0

=

M

6

.

0

=

M

4

.

0

=

M

2

.

0

=

M

Amplitude

Fundamental

0

2

.

0

4

.

0

6

.

0

8

.

0

0

.

1

NORMALISED HARMONIC AMPLITUDES FOR

SINUSOIDAL PULSE-WITDH MODULATION

Depth of

Modulation

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

46

PWM spectra observations

• The amplitude of the fundamental decreases or

increases linearly in proportion to the depth of

modulation (modulation index). The relation ship is

given as: V

1

= M

I

V

in

• The harmonics appear in “clusters” with main

components at frequencies of :

f = kp (f

m

);

k=1,2,3....

where f

m

is the frequency of the modulation (sine)

waveform. This also equal to the multiple of the

carrier frequencies. There also exist “side-bands”

around the main harmonic frequencies.

• The amplitude of the harmonic changes with M

I

. Its

incidence (location on spectra) is not.

• When p>10, or so, the harmonics can be normalised

as shown in the Figure. For lower values of p, the

side-bands clusters overlap, and the normalised

results no longer apply.

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

47

Bipolar PWM Harmonics

h
M

I

0.2

0.4

0.6

0.8

1.0

1

0.2

0.4

0.6

0.8

1.0

M

R

1.242

1.15

1.006

0.818

0.601

M

R

+2

0.016

0.061

0.131

0.220

0.318

M

R

+4

0.018

2M

R

+1

0.190

0.326

0.370

0.314

0.181

2M

R

+3

0.024

0.071

0.139

0.212

2M

R

+5

0.013

0.033

3M

R

0.335

0.123

0.083

0.171

0.113

3M

R

+2

0.044

0.139

0.203

0.716

0.062

3M

R

+4

0.012

0.047

0.104

0.157

3M

R

+6

0.016

0.044

4M

R

+1

0.163

0.157

0.008

0.105

0.068

4M

R

+3

0.012

0.070

0.132

0.115

0.009

4M

R

+5

0.034

0.084

0.119

4M

R

+7

0.017

0.050

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

48

Bipolar PWM harmonics

calculation example

( )

harmonics.

dominant

the

of

some

and

voltage

frequency

-

l

fundamenta

the

of

values

the

Calculate

47Hz.

is

lfrequency

fundamenta

The

39.

M

0.8,

M

100V,

V

inverter,

PWM

phase

single

bridge

-

full

In the

:

Example

M

of

function

a

as

2

ˆ

:

from

computed

are

harmonics

The

2

PWM,

bipolar

phase

-

single

bridge

full

for

:

Note

R

I

DC

I

'

,

=

=

=













=

−

=

=

DC

n

RG

RG

G

R

RG

RR

o

V

V

v

v

v

v

v

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

49

Three-phase harmonics:

“Effect of odd triplens”

• For three-phase inverters, there is

significant advantage if p is chosen to be:

– odd and multiple of three (triplens) (e.g.

3,9,15,21, 27..)

– the waveform and harmonics and shown on the

next two slides. Notice the difference?

• By observing the waveform, it can be seen

that with odd p, the line voltage shape
looks more “sinusoidal”.

• The even harmonics are all absent in the

phase voltage (pole switching waveform).
This is due to the p chosen to be odd.

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

50

Spectra observations

• Note the absence of harmonics no. 21, 63

in the inverter line voltage. This is due to p
which is multiple of three.

• In overall, the spectra of the line voltage is

more “clean”. This implies that the THD is
less and the line voltage is more sinusoidal.

• It is important to recall that it is the line

voltage that is of the most interest.

• Also can be noted from the spectra that the

phase voltage amplitude is 0.8
(normalised). This is because the
modulation index is 0.8. The line voltage
amplitude is square root three of phase
voltage due to the three-phase relationship.

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

51

Waveform: effect of “triplens”

2

dc

V

2

dc

V

−

2

dc

V

2

dc

V

−

2

dc

V

−

2

dc

V

−

2

dc

V

2

dc

V

dc

V

dc

V

dc

V

−

dc

V

−

π

π

2

RG

V

RG

V

RY

V

RY

V

YG

V

YG

V

6

.

0

,

8

=

= M

p

6

.

0

,

9

=

= M

p

ILLUSTRATION OF BENEFITS OF USING A FREQUENCY RATIO

THAT IS A MULTIPLE OF THREE IN A THREE PHASE INVERTER

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

52

Harmonics: effect of

“triplens”

0

2

.

0

4

.

0

6

.

0

8

.

0

0

.

1

2

.

1

4

.

1

6

.

1

8

.

1

Amplitude

voltage)

line

to

(Line

3

8

.

0

Fundamental

41

43

39

37

45

47

23

19

21

63

61

59

57

65

67

69 77

79

81

83

85

87

89

91

19

23

43

47

41

37

61

59

65

67

83

79

85

89

COMPARISON OF INVERTER PHASE VOLTAGE (A) & INVERTER LINE VOLTAGE

(B) HARMONIC (P=21, M=0.8)

A

B

Harmonic Order

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

53

Comments on PWM scheme

• It is desirable to push p to as large as

possible.

• The main impetus for that when p is high,

then the harmonics will be at higher
frequencies because frequencies of
harmonics are related to: f = kp(f

m

), where

f

m

is the frequency of the modulating

signal.

• Although the voltage THD improvement is

not significant, but the current THD will
improve greatly because the load normally
has some current filtering effect.

• In any case, if a low pass filter is to be

fitted at the inverter output to improve the
voltage THD, higher harmonic frequencies
is desirable because it makes smaller filter
component.

Power Electronics and

Drives (Version 2): Dr.

Zainal Salam, 2002

54

Example

The amplitudes of the pole switching waveform harmonics of the red
phase of a three-phase inverter is shown in Table below. The inverter
uses a symmetric regular sampling PWM scheme. The carrier frequency

is 1050Hz and the modulating frequency is 50Hz. The modulation
index is 0.8. Calculate the harmonic amplitudes of the line-to-voltage
(i.e. red to blue phase) and complete the table.

Harmonic

number

Amplitude (pole switching

waveform)

Amplitude (line-to

line voltage)

1

1

19

0.3

21

0.8

23

0.3

37

0.1

39

0.2

41

0.25

43

0.25

45

0.2

47

0.1

57

0.05

59

0.1

61

0.15

63

0.2

65

0.15

67

0.1

69

0.05