DC to AC Conversion (INVERTER ) - ENCON

Power Electronics and Drives (Version 2): Dr.

Zainal Salam, 2002

1

Chapter 4DC to AC Conversion

(INVERTER)• General concept• Basic principles/concepts• Single-phase inverter

– Square wave– Notching– PWM

• Harmonics• Modulation• Three-phase inverter


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

IDC Iac

+

−

VDC Vac

+

−


Zainal Salam, 2002

3

Types of inverter

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

"DC LINK" Iac

+

−

VDC Load Voltage

+

−

L ILOAD

Load CurrentIDC+

−

VDC

C


Zainal Salam, 2002

4

Voltage source inverter (VSI) with variable DC link

DC LINK

+

-

Vs Vo

+

-

C+

-Vin

CHOPPER(Variable DC output)

INVERTER(Switch are turned ON/OFFwith 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.


Zainal Salam, 2002

5

Variable DC link inverter (2)

• Advantages:– simple waveform generation– Reliable

• Disadvantages:– Extra conversion stage– Poor harmonics

T1 T2 t

Vdc1

Vdc2 Higher input voltageHigher frequency

Lower input voltageLower frequency


Zainal Salam, 2002

6

VSI with fixed DC link INVERTER

+

−

Vin

(fixed)Vo

+

−

C

Switch turned ON and OFFwith 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


Zainal Salam, 2002

7

Operation of simple square-wave inverter (1)

• To illustrate the concept of AC waveform generation

VDC

T1

T4

T3

T2

+ VO -

D1

D2

D3

D4

SQUARE-WAVEINVERTERS

S1 S3

S2S4

EQUAVALENTCIRCUIT

IO


Zainal Salam, 2002

8

Operation of simple square-wave inverter (2)

VDC

S1

S4

S3

+ vO −

VDC

S1

S4

S3

S2

+ vO −

VDC

vO

t1 t2

t

S1,S2 ON; S3,S4 OFF for t1 < t < t2

t2 t3

vO

-VDC

t

S3,S4 ON ; S1,S2 OFF for t2 < t < t3

S2


Zainal Salam, 2002

9

Waveforms and harmonics of square-wave inverter

FUNDAMENTAL

3RD HARMONIC

5RD HARMONIC

πDCV4

Vdc

-Vdc

V1

31V

51V

INVERTEROUTPUT


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.

vO 1

+

−

LOAD

L

CvO 2

(LOW PASS) FILTER

+

−

vO 1 vO 2


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 schemecan 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.


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


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 MODULATEDAPPROXIMATION TO SINE WAVE


Zainal Salam, 2002

14

PWM- output voltage and frequency control


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. harmonicat impedance theis

:current harmonic with voltageharmonic thereplacingby obtained becan THDCurrent

number. harmonics theis where

,1

2

2,

,1

2

2,1

2

,1

2

2,

nn

nn

RMS

nRMSn

RMS

nRMSRMS

RMS

nRMSn

ZZVI

I

ITHDi

n

V

VV

V

VTHDv

=

=

−

==

∑

∑∑

∞

=

∞

=

∞

=


Zainal Salam, 2002

16

Fourier Series

• Study of harmonics requires understanding of wave shapes. Fourier Series is a tool to analyse wave shapes.

( )

( )

( )t

nbnaavf

dnvfb

dnvfa

dvfa

nnno

n

n

o

ωθ

θθ

θθπ

θθπ

θπ

π

π

π

=

++=

=

=

=

∑

∫

∫

∫

∞

= where

sincos21)(

Fourier Inverse

sin)(1

cos)(1

)(1

SeriesFourier

1

2

0

2

0

2

0


Zainal Salam, 2002

17

Harmonics of square-wave (1)

Vdc

-Vdc

θ=ωtπ 2π

( ) ( )

( ) ( )

−=

=

−=

=

−+=

∫∫

∫∫

∫∫

π

π

π

π

π

π

π

π

π

θθθθπ

θθθθπ

θθπ

2

0

2

0

2

0

sinsin

0coscos

01

dndnVb

dndnVa

dVdVa

dcn

dcn

dcdco


Zainal Salam, 2002

18

Harmonics of square wave (2)

( ) ( )[ ][ ]

[ ]

[ ]

π

π

π

ππ

πππ

ππππ

θθπ

ππ

π

nVb

n

bn

nnV

nnnV

nnnnV

nnnVb

dcn

n

dc

dc

dc

dcn

41cos odd, isn when

01cos even, isn when

)cos1(2)cos1()cos1(

)cos2(cos)cos0(cos

coscos

Solving,

20

=

=

−=

=

−=

−+−=

−+−=

+−=


Zainal Salam, 2002

19

Spectra of square wave

1 3 5 7 9 11

NormalisedFundamental

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.


Zainal Salam, 2002

20

Quasi-square wave (QSW)

( ) [ ]( ) ( )[ ]

( ) ( )

απαπαπ

απαπ

απαπ

θπ

θθπ

απα

απ

α

nnnnnn

nnn

nnnV

nnVdnVb

a

dc

dcdcn

n

coscossinsincoscos

coscos

Expanding,

coscos2

cos2sin12

symmetry, wave-half toDue

.0 that Note

=+=

−=−

−−=

−=

=

=

−−

∫

π π2

α α αVdc

-Vdc


Zainal Salam, 2002

21

Harmonics control

( )[ ]

( )[ ]

( )

( )

n

n

b

b

Vb

nnVb

b

nnnV

nnnnVb

o

dc

dcn

n

dc

dcn

o

3

1

1

90:if eliminated be will harmonic general,

In waveform. thefrom eliminated is harmonic thirdor the,0then ,30 if exampleFor , adjustingby controlled be alsocan Harmonics

α by varying controlled is ,, lfundamenta The

cos4:is lfundamenta theof amplitude ,particularIn

cos4 odd, isn If

,0 even, isn If

cos1cos2coscoscos2

=

==

=

=⇒

=⇒

−=

−=⇒

α

αα

απ

απ

παπ

απαπ


Zainal Salam, 2002

22

Example

degrees30 with case wavesquare-quasifor (c) and (b)Repeat

harmonics zero-non efirst thre theusingby THDi thec)harmonics zero-non efirst thre theusingby THDv theb)

formula. exact"" theusing THDv thea):Calculate series.in 10mHL and

10RR is load The 100V. is gelink volta DC The signals. wavesquareby fed isinverter phase single bridge-fullA

=

==

α


Zainal Salam, 2002

23

Half-bridge inverter (1)

Vo

RL

+−

VC1

VC2

+

-

+

-S1

S2

Vdc

2Vdc

2Vdc

−

S1 ONS2 OFF

S1 OFFS2 ON

t0G

• 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.


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.

��

��

��

td td

"Dead time' = td

S1signal(gate)

S2signal(gate)

S1

S2

+

−

Vdc

RL

G

"Shoot through fault" .Ishort is very large

Ishort


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

+

2dcV

2dcV

-

2dcV

2dcV

dcV

2dcV

−

2dcV

−

dcV−

π

π

π

π2

π2

π2

tω

tω

tω

RGV

GRV '

oV

GRo VVVRG '−=

groumd" virtual" is G

LEG R LEG R'

R R'- oV+

dcV

+

-


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

ZYZR ZB

G R Y B

iR iYiB

ia ib

+Vdc

N

S1

S4 S6

S3 S5

S2

+

+

−

−

Vdc/2

Vdc/2


Zainal Salam, 2002

27

Square-wave inverter waveforms

13

2,4

23,54

35

4,6

41,56

51

2,6

61,32

VAD

VB0

VC0

VAB

VAPH

(a) Three phase pole switching waveforms

(b) Line voltage waveform

(c) Phase voltage waveform (six-step)

600 1200

IntervalPositive device(s) on

Negative devise(s) on

2VDC/3VDC/3

-VDC/3-2VDC/3

VDC

-VDC

VDC/2

-VDC/2t

t

t

t

t

Quasi-square wave operation voltage waveforms


Zainal Salam, 2002

28

Three-phase inverter waveform relationship

• VRG, VYG, VBG are known as “pole switching waveform” or “inverter phase voltage”.

• VRY, VRB, VYB 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 VRN ,VYN,VBN. It is also popularly termed as “six-step” waveform


Zainal Salam, 2002

29

MODULATION: Pulse Width Modulation (PWM)

Modulating Waveform Carrier waveform

1M1+

1−

0

2dcV

2dcV

−

00t 1t 2t 3t 4t 5t

• 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.


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


Zainal Salam, 2002

31

Natural/Regular sampling

( )

(1,2,3...)integer an is and signal modulating theoffrequency theis where

M

:at locatednormally are harmonics The .frequency" harmonic" the torelated is M

waveformmodulating theofFrequency veformcarrier wa theofFrequency M

)(MRATIO MODULATION

ly.respective voltage,(DC)input and voltageoutput theof lfundamenta are , where

M

:holds iprelationshlinear the1, M0 If

versa. viceandhigh isoutput wavesine the thenhigh, isM If magnitude. tageoutput vol

wave)(sine lfundamenta the torelated is M

veformcarrier wa theof Amplitude waveformmodulating theof AmplitudeM

:MINDEX MODULATION

R

R

R

R

1

I1

I

II

I

I

kf

fkf

p

p

VV

VV

m

m

in

in

=

==

==−−−−−−−−−−−−−−−−−−−−−−−−−−−−

=

<<

=

=


Zainal Salam, 2002

32

Asymmetric and symmetric regular sampling

T

samplepoint

tM mωsin11+

1−

4T

43T

45T

4π

2dcV

2dcV

−

0t 1t 2t 3tt

asymmetric sampling

symmetricsampling

t

Generating of PWM waveform regular sampling


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.


Zainal Salam, 2002

34

Bipolar PWM switching

��

��

��

k1δk2δ

kα

∆4∆

=δ

π π2

carrierwaveform

modulatingwaveform

pulsekth

π π2


Zainal Salam, 2002

35

Pulse width relationships

��

��

��

k1δk2δ

kα

∆4∆

=δ

π π2

carrierwaveform

modulatingwaveform

pulsekth

π π2


Zainal Salam, 2002

36

Characterisation of PWM pulses for bipolar switching

pulse PWMkth The

∆

0δ 0δ 0δ 0δ

k1δk2δ

2SV+

2SV

−

kα


Zainal Salam, 2002

37

Determination of switching angles for kth PWM pulse (1)

v Vmsin θ( )

��

��

��

��

��

��

Ap2Ap1

2dcV+

2dcV

−

AS2

AS1

22

11

second,- volt theEquating

ps

ps

AA

AA

=

=


Zainal Salam, 2002

38

PWM Switching angles (2)

[ ]

)sin(sin2

cos)2cos(sin

sinusoid, by the supplied second- voltThe

where; 2

Similarly,

where

22

2)2(

2

:asgiven is pulse PWM theof cycle halfeach during voltageaverage The

21

2222

11

11

111

okom

kokmms

o

okk

dckk

o

okk

sk

o

okdc

o

kokdck

V

VdVA

VV

VV

VV

k

ok

δαδ

αδαθθ

δδδ

ββ

δδδ

β

βδ

δδδ

δδδ

α

δα

−=

−−==

−=

=

−=

=

−

=

−−

=

∫−


Zainal Salam, 2002

39

Switching angles (3)

)sin()2(

)sin(222

edge leading for the Hence,

;strategy, modulation thederive To

22

;22

, waveformsPWM theof seconds- voltThe

)sin(2Similarly,

)sin(2, smallfor sin

Since,

1

1

2211

21211

2

1

okdc

mk

okmoodc

k

spsp

odc

kpodc

kp

okmos

okmosooo

VV

VV

AAAA

VAVA

VA

VA

δαβ

δαδδβ

δβδβ

δαδ

δαδδδδ

−=⇒

−=

==

=

=

+=

−=→


Zainal Salam, 2002

40

PWM switching angles (4)

[ ]

[ ])sin(1 and

)sin(1

width,-pulse for the solve tongSubstituti

)sin(

:derived becan edge trailing themethod,similar Using

)sin(

Thus,

1. to0 from It varies depth.or index

modulation asknown is 2

ratio, voltageThe

2

1

11

2

1

okIok

okIoko

okk

okIk

okIk

dc

mI

M

M

M

M

)(VVM

δαδδ

δαδδδ

δδβ

δαβ

δαβ

++=

−+=⇒

−=

−=

−=

=


Zainal Salam, 2002

41

PWM Pulse width

[ ]kIok

kk

kk

M αδδ

δδδδδ

δα

δα

sin1

,Modulation SymmetricFor different. are andi.e ,Modulation

Asymmetricfor validisequation above The

:edge Trailing

:edge Leading

:is pulsekth theof angles switching theThus

k 2k 1k 2k 1k

1

1

+=⇒

==

+

−


Zainal Salam, 2002

42

Example• For the PWM shown below, calculate the switching

angles for all the pulses.

V5.1V2

π π2

1 2 3 4 5 6 7 8 9

t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12t13

t14t15

t16t17

t18 π2π

1α

carrierwaveform

modulatingwaveform


Zainal Salam, 2002

43

Harmonics of bipolar PWM

{

})2(cos)(cos )(cos)(cos )(cos)2(cos

: toreduced becan Which

sin2

2

sin2

2

sin2

2

sin)(12

:as computed becan pulse PWM (kth)each ofcontent harmonicsymmetry,

wave-half is waveformPWM theAssuming

212

1

2

2

0

2

2

1

1

okkkkkkk

kkokdc

nk

dc

dc

dc

T

nk

nnnnnn

nV b

dnV

dnV

dnV

dnvfb

ok

kk

kk

kk

kk

ok

δαδαδαδα

δαδαπ

θθπ

θθπ

θθπ

θθπ

δα

δα

δα

δα

δα

δα

+−++−−++

−−−−=

−+

+

−=

=

∫

∫

∫

∫

+

+

+

−

−

−


Zainal Salam, 2002

44

Harmonics of PWM

[]

equation. thisofn computatio

theshows pagenext on the slide The

:i.e. period, oneover pulses for the of sum isthe waveformPWM

for the coefficentFourier ly.Theproductive simplified becannot equation This

2coscos2 )2(cos)(cos2

Yeilding,

1

11

∑=

=

+

−−−=

p

knkn

nk

ok

kkkkdc

nk

bb

pb

nnnn

nVb

δα

αδαπ


Zainal Salam, 2002

45

PWM Spectra

p p2 p3 p40.1=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 FORSINUSOIDAL PULSE-WITDH MODULATION

Depth ofModulation


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: V1= MIVin

• The harmonics appear in “clusters” with main components at frequencies of : f = kp (fm); k=1,2,3.... where fm 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 MI. 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.


Zainal Salam, 2002

47

Bipolar PWM Harmonics

h MI

0.2 0.4 0.6 0.8 1.0

1 0.2 0.4 0.6 0.8 1.0

MR 1.242 1.15 1.006 0.818 0.601

MR +2 0.016 0.061 0.131 0.220 0.318

MR +4 0.018

2MR +1 0.190 0.326 0.370 0.314 0.181

2MR +3 0.024 0.071 0.139 0.212

2MR +5 0.013 0.033

3MR 0.335 0.123 0.083 0.171 0.113

3MR +2 0.044 0.139 0.203 0.716 0.062

3MR +4 0.012 0.047 0.104 0.157

3MR +6 0.016 0.044

4MR +1 0.163 0.157 0.008 0.105 0.068

4MR +3 0.012 0.070 0.132 0.115 0.009

4MR+5 0.034 0.084 0.1194MR +7 0.017 0.050


Zainal Salam, 2002

48

Bipolar PWM harmonics calculation example

( )

harmonics.dominant theof some and voltagefrequency -lfundamenta theof valuestheCalculate 47Hz. is lfrequencyfundamenta

The 39.M 0.8,M 100V,V inverter, PWM phase single bridge-full In the

:Example

M offunction a as2

ˆ:from computed are harmonics The

2

PWM,bipolar phase-single bridge fullfor :Note

RIDC

I

',

===

=−==

DCnRG

RGGRRGRRo

VV

vvvvv


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.


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.


Zainal Salam, 2002

51

Waveform: effect of “triplens”2dcV

2dcV

−

2dcV

2dcV

−

2dcV

−

2dcV

−

2dcV

2dcV

dcV

dcV

dcV−

dcV−

π π2

RGV

RGV

RYV

RYV

YGV

YGV

6.0,8 == Mp

6.0,9 == MpILLUSTRATION OF BENEFITS OF USING A FREQUENCY RATIOTHAT IS A MULTIPLE OF THREE IN A THREE PHASE INVERTER


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 38.0

Fundamental

41 4339

3745

472319

21 63

6159

5765

6769 77

798183 85

8789

91

19 2343

4741

3761

5965

6783

7985

89

COMPARISON OF INVERTER PHASE VOLTAGE (A) & INVERTER LINE VOLTAGE(B) HARMONIC (P=21, M=0.8)

A

B

Harmonic Order


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(fm), wherefm 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.


Zainal Salam, 2002

54

ExampleThe amplitudes of the pole switching waveform harmonics of the redphase of a three-phase inverter is shown in Table below. The inverter uses a symmetric regular sampling PWM scheme. The carrier frequencyis 1050Hz and the modulating frequency is 50Hz. The modulationindex 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

DC to AC Conversion (INVERTER ) - ENCON

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