DC to AC Conversion INVERTER

Power Electronics and Drives (Version 3-2003):

Dr. Zainal Salam UTM-JB

1

Chapter 4DC to AC Conversion

(INVERTER)

• General concept• Single-phase inverter• Harmonics• Modulation• Three-phase inverter



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.

• General block diagram

IDC Iac

�

−

VDC Vac

�

−

• TYPICAL APPLICATIONS:– Un-interruptible power supply (UPS), Industrial

(induction motor) drives, Traction, HVDC



3

Simple square-wave inverter (1)

• To illustrate the concept of AC waveform generation

��

��

��

��

��

��

��

��

��

��

��

��

��

S1 S3

S2S4

��

��



4

AC Waveform Generation

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



5

AC Waveforms

FUNDAMENTAL COMPONENT

3RD HARMONIC

5RD HARMONIC

πDCV4

Vdc

-Vdc

V1

31V

51V

INVERTER OUTPUT VOLTAGE

π 2π



6

Harmonics Filtering

• Output of the inverter is “chopped AC voltage with zero DC component”. It contain harmonics.

• An LC section low-pass filter is normally fitted at the inverter output to reduce the high frequency harmonics.

• In some applications such as UPS, “high purity” sine wave output is required. Good filtering is a must.

• In some applications such as AC motor drive, filtering is not required.

vO 1

+

−

L

CvO 2

(LOW PASS) FILTER

+

−

vO 1vO 2

BEFORE FILTERING AFTER FILTERING

INVERTER LOADDC SUPPLY



7

Variable Voltage Variable Frequency Capability

�

� t

Vdc1

Vdc2 Higher input voltageHigher frequency

Lower input voltageLower frequency

• Output voltage frequency can be varied by “period” of the square-wave pulse.

• Output voltage amplitude can be varied by varying the “magnitude” of the DC input voltage.

• Very useful: e.g. variable speed induction motor drive



8

Output voltage harmonics/ distortion

• Harmonics cause distortion on the output voltage.

• Lower order harmonics (3rd, 5th etc) are very difficult to filter, due to the filter size and high filter order. They can cause serious voltage distortion.

• Why need to consider harmonics?– Sinusoidal 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.



9

Total Harmonics Distortion (THD)

( )

( ) ( )

( )

frequency. harmonicat impedance theis

:THDCurrent

known, is vaveformfor the voltagerms theIf

....

voltage,harmonicth theis If :THD Voltage

,1

2

2,

,1

2

2,1

2

,1

2,2

2,3

2,2

,1

2

2,

n

n

nn

RMS

nRMSn

RMS

nRMSRMS

RMS

RMSRMSRMS

RMS

nRMSn

n

Z

ZV

I

I

I

THDi

V

VV

THDv

V

VVV

V

V

THDv

nV

=

=

−=

+++=

=

�

�

�

∞

=

∞

=

∞

=



10

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

term) sin"(" sin)(1

term) cos"(" cos)(1

term) DC"(" )(1

SeriesFourier

1

2

0

2

0

2

0



11

Harmonics of square-wave (1)

( ) ( )

( ) ( )��

�

��

�−=

=��

�

��

�−=

=��

�

��

�−+=

��

��

��

π

π

π

π

π

π

π

π

π

θθθθπ

θθθθπ

θθπ

2

0

2

0

2

0

sinsin

0coscos

01

dndnV

b

dndnV

a

dVdVa

dcn

dcn

dcdco

Vdc

-Vdc

θ�ω�π 2π



12

Harmonics of square wave (2)

( ) ( )[ ][ ]

[ ]

[ ]

π

π

π

ππ

πππ

ππππ

θθπ

ππ

π

nV

b

n

b

nn

nnV

nnnV

nnnnV

nnnV

b

dcn

n

dc

dc

dc

dcn

41cos odd, isn When

exist)not do harmonicseven i.e.(

01cos even, is When

)cos1(2

)cos1()cos1(

)cos2(cos)cos0(cos

coscos

Solving,

20

=

=

−=

=

−=

−+−=

−+−=

+−=



13

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 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 very difficult.



14

Quasi-square wave (QSW)

( ) [ ]( ) ( )[ ]

( ) ( )

( )[ ]

( )[ ]παπ

απαπ

απαπαπαπαπ

απαπ

θπ

θθπ

απα

απ

α

nnnV

nnnnV

b

nnnnnn

nnn

nnnV

nnV

dnVb

a

dc

dcn

dc

dcdcn

n

cos1cos2

coscoscos2

coscossinsincoscoscoscos

:Expanding

coscos2

cos2

sin1

2

symmetry) wave-half to(due .0 that Note

−=

−=

=+=−=−

−−=

−=��

�

��

�=

=

−−

�

π π2

α α αVdc

-Vdc



15

Harmonics control

( )

( )

n

n

b

b

Note

Vb

nnV

b

b

o

dc

dcn

n

o

3

1

1

90

:if eliminated be will harmonic general,In waveform. thefrom eliminated is harmonic

thirdor the,0then ,30 if exampleFor

:nEliminatio Harmonics

, adjustingby controlled be alsocan Harmonics

� by varying controlled is ,, lfundamenta The:

cos4

:is lfundamenta theof amplitude ,particularIn

cos4

odd, isn If

,0 even, isn If

=

==

=

=

=

α

α

α

απ

απ



16

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



17

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.

• 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.



18

Shoot through fault and “Dead-time”

• In practical, a dead time as shown below is required to avoid “shoot-through” faults, i.e. short circuit across the DC rail.

• Dead time creates “low frequency envelope”. Low frequency harmonics emerged.

• This is the main source of distortion for high-quality sine wave inverter.

td td

"Dead time' = td

S1signal(gate)

S2signal(gate)

S1

S2

+

−−−−

Vdc

RL

G

"Shoot through fault" .Ishort is very large

Ishort



19

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

+

-



20

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



21

Three phase inverter waveforms

13

2,4

23,54

35

4,6

41,56

51

2,6

61,32

Inverter PhaseVoltage

(or pole switchingwaveform)

VRG

2400

IntervalPositive device(s) on

Negative device(s) on

2VDC/3

VDC/3

-VDC/3

-2VDC/3

VDC

-VDC

VDC/2

-VDC/2

Quasi-square wave operation voltage waveforms

1200

VDC/2

VDC/2

-VDC/2

-VDC/2

VYG

VBG

lIne-to -ineVoltage

VRY

Six-stepWaveform

VRN



22

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.



23

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



24

Modulation Index, Ratio

waveformmodulating theofFrequency veformcarrier wa theofFrequency

M

)(MRatio) (Frequency Ratio Modulation

veformcarrier wa theof Amplitude waveformmodulating theof Amplitude

M

:MDepth)n (ModulatioIndex Modulation

R

R

I

I

==

==

=

=

p

p


1M1+

1−

0

2dcV

2dcV−

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



25

( )

(1,2,3...)integer an is and

signal modulating theoffrequency theis where

M:at locatednormally are harmonics The

spectra. in the harmonics of(location)incident thedetermines ratiodulation M

ly.respective voltage,(DC)input and voltage

output theof lfundamenta are , where

M

1, M0 If

component lfundamenta voltageoutput thesdeterrmineIndex Modulation

R

1

I1

I

k

f

fkf

o

VV

VV

m

m

in

in

=

−−−−−−−−−−−−−−−−−−−−−−−−−−−−

=<<

Modulation Index, Ratio



26

Regular sampling

Regular sampling PWM

Sinusoidal modulatingwaveform, vm(t)

Carrier, vc(t)t1 t2

t'1 t'2

t

t

ππ2

)(tvs

pwmv

Regular sampling waveform,



27

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



28

Bipolar Switching


1M1+

1−

0

2dcV

2dcV−

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



29

Unipolar switching

Unipolar switching scheme

A BCarrier waveform

(a)

(b)

(c)

(d)

1S

3S

pwmV



30

Bipolar PWM switching: Pulse-width characterization

k1δk2δ

kα

∆4∆=δ

π π2

carrierwaveform

modulatingwaveform

pulse

kth

π π2



31

The kth Pulse

pulse PWMkth The

∆

0δ 0δ 0δ 0δ

k1δk2δ

2dcV+

kα2

dcV+



32

Determination of switching angles for kth PWM pulse (1)

22

11

second,- volt theEquating

ps

ps

AA

AA

=

=

v Vmsin θ( )

Ap2Ap1

2dcV+

2dcV

−

AS2

AS1



33

PWM Switching angles (2)

( ) ( )

( )( )

( ) ( )

( )( )

[ ]

)sin(2Similarly,

)sin(sin2

cos)2cos(sin

sinusoid, by the supplied second- voltThe

222

half, second for theSimilarly

222

:asgiven is pulse PWM theof cycle halffirst theduring second-Volt The

2

21

2

222

1

111

okmos

okom

kokmms

okdc

kodc

kdc

p

okdc

kodc

kdc

p

VA

V

VdVA

V

VVA

V

VVA

k

ok

δαδ

δαδ

αδαθθ

δδ

δδδ

δδ

δδδ

α

δα

+=

−=

−−==

−=

−�

� �

�−�

� �

�=

−=

−�

� �

�−�

� �

�=

�−



34

Switching angles (3)

( )( )

( ) ( )

[ ])sin(1:bygiven is waveformPWM theof

cycle halffirst for the width pulse theThus,

modulation asknown is 2

Ratio, Modulation the,definitionBy

sin(2

)sin(2pulse, PWM of cycle halffirst thefor the Hence,

;strategy, modulation thederive To

)sin(2

)sin(2

, sin

angle smallFor

1

1

1

22

11

2

1

okIok

dc

mI

okodc

mok

okmookdc

sp

sp

okmos

okmos

oo

o

M

)(VV

M

VV

VV

AA

AA

VA

VA

δαδδ

δαδδδ

δαδδδ

δαδδαδ

δδδ

−+=

=

−=−

−=−

=

=

−=

−=

→



35

PWM switching angles (4)

[ ]

[ ]kIok

kk

okIok

kk

M

M

αδδ

δδδ

δδ

δα

δαδδ

δα

sin1 Hence

,Modulation SymmetricFor

different. are andi.e ,Modulation Asymmetricfor validisequation above The

: angle edge trailing theAnd

)sin(1: waveformPWM of cycle half second

theof width pulse method,similar Using

:is pulsekth the

of angle switching edge leading theThus

k 2k 1k

2k 1k

2

2

1

+=

==

+

++=

−



36

Example• For the PWM shown below, calculate the switching

angles pulses no. 2.

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



37

Harmonics of bipolar PWM

��

��

��

��

�

� �

�−+

��

��

��

��

�

� �

�+

��

��

��

��

�

� �

�−=

�

�

�

�=

�

�

�

�

+

+

+

−

−

−

ok

kk

kk

kk

kk

ok

dnV

dnV

dnV

dnvfb

dc

dc

dc

T

nk

δα

δα

δα

δα

δα

δα

θθπ

θθπ

θθπ

θθπ

2

2

0

2

2

1

1

sin2

2

sin2

2

sin2

2

sin)(1

2

:as computed becan pulse PWM (kth)

each ofcontent harmonic

symmetry, wavehalf is waveform

PWM theAssuming∆

0δ 0δ 0δ 0δ

k1δk2δ

2dcV+

kα2

dcV+



38

Harmonics of Bipolar PWM

{

}

[]

equation. thisofn computatio theshows slideNext

:i.e. period, oneover pulses

for the of sum isthe waveformPWMfor the coefficentFourier ly.Theproductive

simplified becannot equation This

2coscos2

)2(cos)(cos2

Yeilding,

)2(cos)(cos

)(cos)(cos

)(cos)2(cos

: toreduced becan Which

1

11

2

12

1

�=

=

+

−−−=

+−++−−++

−−−−=

p

knkn

nk

ok

kkkkdc

nk

okkk

kkkk

kkokdc

nk

bb

pb

nn

nnnV

b

nn

nn

nnnV

b

δα

αδαπ

δαδαδαδα

δαδαπ



39

PWM Spectra

p p2 p3 p40.1=IM

8.0=IM

6.0=IM

4.0=IM

2.0=IM

Amplitude

Fundamental

0

2.0

4.0

6.0

8.0

0.1

NORMALISED HARMONIC AMPLITUDES FORSINUSOIDAL PULSE-WITDH MODULATION

ModulationIndex



40

PWM spectra observations

• The harmonics appear in “clusters” at multiple of the carrier frequencies .

• Main harmonics located at : f = kp (fm); k=1,2,3....

where fm is the frequency of the modulation (sine) waveform.

• There also exist “side-bands” around the main harmonic frequencies.

• Amplitude of the fundamental is proportional to the modulation index. The relation ship is given as:

V1= MIVin

• 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. For lower values of p, the side-bands clusters overlap-normalised results no longer apply.



41

Tabulated Bipolar PWM Harmonics

n 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 0.119 4MR +7 0.017 0.050



42

Three-phase harmonics

• For three-phase inverters, there is significant advantage if MR is chosen to be:

– Odd: All even harmonic will be eliminated from the pole-switching waveform.

– triplens (multiple of three (e.g. 3,9,15,21, 27..):All triplens harmonics will be eliminated from the line-to-line output voltage.

• By observing the waveform, it can be seen that with odd MR, the line-to-line voltage shape looks more “sinusoidal”.

• As can be noted from the spectra, 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



43

Effect of odd and “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



44

Spectra: 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

47

2319

21 63

6159

57

6567

69 7779

8183 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



45

Comments on PWM scheme

• It is desirable to have MR as large as possible.

• This will push the harmonic at higher frequencies on the spectrum. Thus filtering requirement is reduced.

• Although the voltage THD improvement is not significant, but the current THD will improve greatly because the load normally has some current filtering effect.

• However, higher MR has side effects:– Higher switching frequency: More losses.– Pulse width may be too small to be constructed.

“Pulse dropping” may be required.



46

Example

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

The 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.

DC to AC Conversion INVERTER

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