Chapter 4 DC to AC Conversion (INVERTER) General concept Single-phase inverter Harmonics Modulation Three-phase inverter.

Chapter 4DC to AC Conversion

(INVERTER)

• General concept

• Single-phase inverter

• Harmonics

• Modulation

• Three-phase inverter

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

IDCIac

+

VDC Vac

+

• TYPICAL APPLICATIONS:

– Un-interruptible power supply (UPS), Industrial (induction motor) drives, Traction, HVDC

Simple square-wave inverter (1)

• To illustrate the concept of AC waveform generation

VDC

T1

T4

T3

T2

+ VO -

D1

D2

D3

D4

SQUARE-WAVEINVERTER

IO

S1 S3

S2S4

EQUIVALENTCIRCUIT

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

AC Waveforms

FUNDAMENTAL COMPONENT

3RD HARMONIC

5RD HARMONIC

DCV4

Vdc

-Vdc

V1

3

1V

5

1V

INVERTER OUTPUT VOLTAGE

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

Variable Voltage Variable Frequency Capability

T1 T2 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

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. – “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.

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

where

sincos21

)(

Fourier Inverse

term) sin"(" sin)(1

term) cos"(" cos)(1

term) DC"(" )(1

SeriesFourier

1

2

0

2

0

2

0

Harmonics of square-wave (1)

2

0

2

0

2

0

sinsin

0coscos

01

dndnV

b

dndnV

a

dVdVa

dcn

dcn

dcdco

Vdc

-Vdc

=t

Harmonics of square wave (2)

n

Vb

n

b

nn

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V

nnn

V

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V

nnn

Vb

dcn

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dc

dc

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)cos1(2

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)cos2(cos)cos0(cos

coscos

Solving,

20

Quasi-square wave (QSW)

nnn

V

nnnn

Vb

nnnnnn

nnn

nnn

V

nn

VdnVb

a

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dcn

dc

dcdcn

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coscoscos2

coscossinsincoscos

coscos

:Expanding

coscos2

cos2

sin1

2

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

2

Vdc

-Vdc

Harmonics control

n

n

b

b

Note

Vb

nn

Vb

b

o

dc

dcn

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:

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Half-bridge inverter (1)

Vo

RL

+

VC1

VC2

+

-

+

-S1

S2

Vdc

2

Vdc

2

Vdc

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.

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

2

2

2

t

t

t

RGV

GRV '

oV

GRo VVVRG '

groumd" virtual" is G

LEG R LEG R'

R R'- oV

dcV

+

-

Three-phase inverter

• Each leg (Red, Yellow, Blue) is delayed by 120 degrees.

• A three-phase inverter with star connected load is shown below

ZYZRZB

G R Y B

iR iYiB

ia ib

+Vdc

N

S1

S4 S6

S3 S5

S2

+

+

Vdc/2

Vdc/2

I. Voltage Source Inverter (VSI)A. Six-Step VSI (1)

3

Six-Step three-phase Voltage Source Inverter

Fig. 1 Three-phase voltage source inverter.

4

I. Voltage Source Inverter (VSI)A. Six-Step VSI (2)

Fig. 2 Waveforms of gating signals, switching sequence, line to negative voltagesfor six-step voltage source inverter.

Gating signals, switching sequence and line to negative voltages

I. Voltage Source Inverter (VSI)A. Six-Step VSI (3)

where, 561 means that S5, S6 and S1 are switched on

Fig. 3 Six inverter voltage vectors for six-step voltage source inverter.

Switching Sequence:

561 (V1) 612 (V2) 123 (V3) 234 (V4)

345 (V5) 456 (V6) 561 (V1)

5

I. Voltage Source Inverter (VSI)A. Six-Step VSI (4)

Fig. 4 Waveforms of line to neutral (phase) voltages and line to line voltagesfor six-step voltage source inverter.

Line to line voltages (Vab, Vbc, Vca) and line to neutral voltages (Van, Vbn, Vcn)

Vab = VaN - VbN

Vbc = VbN - VcN

Vca = VcN - VaN

Line to line voltages

Van = 2/3VaN - 1/3VbN - 1/3VcN

Phase voltages

Vbn = -1/3VaN + 2/3VbN - 1/3VcN

Vcn = -1/3VaN - 1/3VbN + 2/3VcN

6

Three phase inverter waveforms

I. Voltage Source Inverter (VSI)A. Six-Step VSI (5)

Amplitude of line to line voltages (Vab, Vbc, Vca) Fundamental Frequency Component (Vab)1

Harmonic Frequency Components (Vab)h

: amplitudes of harmonics decrease inversely proportional to their harmonic order

dcdcdc V78.0V

6

2

V4

2

3

(rms))(V 1ab

3,.....)2,1,(n16nhwhere,

V78.0

dcab

h

(rms))(V h

7

8

I. Voltage Source Inverter (VSI)A. Six-Step VSI (6)

Characteristics of Six-step VSI

It is called “six-step inverter” because of the presence of six “steps”

in the line to neutral (phase) voltage waveform

Harmonics of order three and multiples of three are absent from

both the line to line and the line to neutral voltages

and consequently absent from the currents

Output amplitude in a three-phase inverter can be controlled

by only change of DC-link voltage (Vdc)

9

I. Voltage Source Inverter (VSI)B. Pulse-Width Modulated VSI (1)

Objective of PWM

Disadvantages of PWM Increase of switching losses due to high PWM frequency

Reduction of available voltage

EMI problems due to high-order harmonics

Control of inverter output voltage

Reduction of harmonics

I. Voltage Source Inverter (VSI)B. Pulse-Width Modulated VSI (2)

Pulse-Width Modulation (PWM)

Fig. 5 Pulse-width modulation.

10

I. Voltage Source Inverter (VSI)B. Pulse-Width Modulated VSI (3)

Inverter output voltage

When vcontrol > vtri, VA0 = Vdc/2

When vcontrol < vtri, VA0 = -Vdc/2

A01A0

10

Vofcomponentfrequecnylfundamenta:)(Vwhere,

,2/

)(

dc

A

tri

control

V

Vofpeak

v

vm

Modulation Index (m)

Control of inverter output voltage

Amplitude is controlled by the peak value of vcontrol

Fundamental frequency is controlled by the frequency of vcontrol

PWM frequency is the same as the frequency of vtri

11

II. PWM METHODSA. Sine PWM (1)

Fig. 6 Three-phase Sine PWM inverter.

Three-phase inverter

12

II. PWM METHODSA. Sine PWM (2)

VA

0V

B0

VC

0V

AB

VB

CV

CA

t

Fig. 7 Waveforms of three-phase sine PWM inverter.

Three-phase sine PWM waveformsvtri vcontrol_A vcontrol_B vcontrol_C

where, VAB = VA0 – VB0

VBC = VB0 – VC0

VCA = VC0 – VA0

When vcontrol > vtri, VA0 = Vdc/2

When vcontrol < vtri, VA0 = -Vdc/2

Frequency of vtri = fs

Frequency of vcontrol = f1

Frequency of vtri and vcontrol

where, fs = PWM frequency

f1 = Fundamental frequency

Inverter output voltage

13

II. PWM METHODSA. Sine PWM (3)

Amplitude modulation ratio (ma)

A01A0

10

Vofcomponentfrequecnylfundamenta:)(Vwhere,

,2/

)(

dc

A

tri

controla V

Vofvaluepeak

vofamplitude

vofamplitudepeakm

Frequency modulation ratio (mf)

frequencylfundamentafandfrequencyPWMfwhere,, 1s1

f

fm s

f

mf should be an odd integer

if mf is not an integer, there may exist sunhamonics at output voltage

if mf is not odd, DC component may exist and even harmonics are present at output voltage

mf should be a multiple of 3 for three-phase PWM inverter

An odd multiple of 3 and even harmonics are suppressed

14

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.

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

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

Modulating Waveform Carrier waveform

1M1

1

0

2dcV

2dcV

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

(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 voltage

output thesdeterrmineIndex Modulation

R

1

I1

I

k

f

fkf

o

VV

VV

m

m

in

in

Modulation Index, Ratio

Regular samplingh x( ) if k x( ) c x( ) 1 if k x( ) c x( ) 1 0( )( )

1

Regular sampling PWM

Sinusoidal modulatingwaveform, vm(t)

Carrier, vc(t)t1 t2

t'1 t'2

t

t

2

)(tvs

pwmv

Regular sampling waveform,

Asymmetric and symmetric regular sampling

T

samplepoint

tM msin11

1

4

T

4

3T

4

5T4

2dcV

2dcV

0t 1t 2t 3tt

asymmetric sampling

symmetricsampling

t

Generating of PWM waveform regular sampling

Bipolar Switching

Modulating Waveform Carrier waveform

1M1

1

0

2dcV

2dcV

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

Unipolar switching1

Unipolar switching scheme

A BCarrier waveform

(a)

(b)

(c)

(d)

1S

3S

pwmV

Bipolar PWM switching: Pulse-width characterization

k1k2

k

4

2

carrierwaveform

modulatingwaveform

pulse

kth

2

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

Effect of odd and “triplens”

2dcV

2dcV

2dcV

2dcV

2dcV

2dcV

2dcV

2dcV

dcV

dcV

dcV

dcV

2

RGV

RGV

RYV

RYV

YGV

YGV

6.0,8 Mp

6.0,9 Mp

ILLUSTRATION OF BENEFITS OF USING A FREQUENCY RATIOTHAT IS A MULTIPLE OF THREE IN A THREE PHASE INVERTER

Three phase inverter with RL load

• 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.