unit-iv inverters 6/8/2015copyright by

UNIT-IV

INVERTERS

04/18/23 Copyright by www.noteshit.com 1

Copyright by www.noteshit.com

Single-Phase Inverters

Half-Bridge Inverter

One of the simplest types of inverter. Produces a square wave output.

04/18/23 2


Single-Phase Inverters (cont’d)

Full Bridge (H-bridge) Inverter

Two half-bridge inverters combined.

Allows for four quadrant operation.

04/18/23 3



Quadrant 1: Positive step-down converter (forward motoring) Q1-On; Q2 - Chopping; D3,Q1 freewheeling

04/18/23 4



Quadrant 2: Positive step-up converter

(forward regeneration)

Q4 - Chopping; D2,D1 freewheeling

04/18/23 5



Quadrant 3: Negative step-down converter (reverse motoring) Q3-On; Q4 - Chopping; D1,Q3 freewheeling

04/18/23 6



Quadrant 4: Negative step-up converter

(reverse regeneration)

Q2 - Chopping; D3,D4 freewheeling

04/18/23 7



Phase-Shift Voltage Control - the output of the H-bridge inverter can be controlled by phase shifting the control of the component half-bridges. See waveforms on next slide.

04/18/23 8



04/18/23 9



The waveform of the output voltage vab is a quasi-square wave of pulse width . The Fourier series of vab is given by:

The value of the fundamental, a1=

The harmonic components as a function of phase angle are shown in the next slide.

1,3,5...

4sin cos

2d

abn

V nv n t

n

4sin / 2dV

04/18/23 10



04/18/23 11


Three-Phase Bridge Inverters

Three-phase bridge inverters are widely used for ac motor drives. Two modes of operation - square wave and six-step. The topology is basically three half-bridge inverters, each phase-shifted by 2/3, driving each of the phase windings.

04/18/23 12


Three-Phase Bridge Inverters (cont’d)

04/18/23 13



04/18/23 14



The three square-wave phase voltages can be expressed in terms of the dc supply voltage, Vd, by Fourier series as:

10

1,3,5...

2( 1) cos( )nd

an

Vv n t

10

1,3,5...

2 2( 1) cos( )

3nd

bn

Vv n t

10

1,3,5...

2 2( 1) cos( )

3nd

cn

Vv n t

04/18/23 15



The line voltages can then be expressed as:

0 01,3,5...

2 3cos( / 2) cos( 2)d

bc b cn

Vv v v t n t

0 01,3,5...

2 3cos( 5 / 6) cos( 5 6)d

ca c an

Vv v v t n t

0 01,3,5...

2 3cos( / 6) cos( 6)d

ab a bn

Vv v v t n t

04/18/23 16



The line voltages are six-step waveforms and have characteristic harmonics of 6n1, where n is an integer. This type of inverter is referred to as a six-step inverter.

The three-phase fundamental and harmonics are balanced with a mutual phase shift of 2/3.

04/18/23 17



If the three-phase load neutral n is isolated from the the center tap of the dc voltage supply (as is normally the case in an ac machine) the equivalent circuit is shown below.

04/18/23 18



In this case the isolated neutral-phase voltages are also six-step waveforms with the fundamental component phase-shifted by /6 from that of the respective line voltage. Also, in this case, the triplen harmonics are suppressed.

04/18/23 19



For a linear and balanced 3 load, the line currents are also balanced. The individual line current components can be obtained from the Fourier series of the line voltage. The total current can be obtained by addition of the individual currents. A typical line current wave with inductive load is shown below.

04/18/23 20



The inverter can operate in the usual inverting or motoring mode. If the phase current wave, ia, is assumed to be perfectly filtered and lags the phase voltage by /3 the voltage and current waveforms are as shown below:

04/18/23 21


Three-Phase Bridge Inverters The inverter can also operate in rectification or regeneration

mode in which power is pushed back to the dc side from the ac side. The waveforms corresponding to this mode of operation with phase angle = 2/3 are shown below:

04/18/23 22



The phase-shift voltage control principle described earlier for the single-phase inverter can be extended to control the output voltage of a three-phase inverter.

04/18/23 23



04/18/23 24



The three waveforms va0,vb0, and vc0 are of amplitude 0.5Vd and are mutually phase-

shifted by 2/3.

The three waveforms ve0,vf0, and vg0 are of

similar but phase shifted by .

04/18/23 25



The transformer’s secondary phase voltages, vA0, vB0, and vc0 may be expressed as follows:

where m is the transformer turns ratio

(= Ns/Np). Note that each of these waves is a function of angle.

0 0 0( )A ad a dv mv m v v

0 0 0( )B be b ev mv m v v

0 0 0( )C cf c fv mv m v v

04/18/23 26



The output line voltages are given by:

While the component voltage waves va0, vd0, vA0 … etc. all contain triplen harmonics, they are eliminated from the line voltages because they are co-phasal. Thus the line voltages are six-step waveforms with order of harmonics = 6n1 at a phase angle .

0 0AB A Bv v v

0 0BC B Cv v v

0 0CA C Av v v

04/18/23 27



The Fourier series for vA0 and vB0 are given by:

01,3,5...

4sin cos

2d

An

mV nv n t

n

01,3,5...

4sin cos 2 / 3

2d

Bn

mV nv n t

n

04/18/23 28



The Fourier series for vAB is given by:

Note that the triplen harmonics are removed in vAB although they are present in vA0 and vB0.

1,5,7,11...

4 2sin cos cos

2 3d

n

mV nn t n t

n

0 0AB A Bv v v

04/18/23 29


PWM Technique

While the 3 6-step inverter offers simple control and low switching loss, lower order harmonics are relatively high leading to high distortion of the current wave (unless significant filtering is performed).

PWM inverter offers better harmonic control of the output than 6-step inverter.

04/18/23 30


PWM Principle

The dc input to the inverter is “chopped” by switching devices in the inverter. The amplitude and harmonic content of the ac waveform is controlled by the duty cycle of the switches. The fundamental voltage v1 has max. amplitude = 4Vd/ for a square wave output but by creating notches, the amplitude of v1 is reduced (see next slide).

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PWM Principle (cont’d)

04/18/23 32


PWM Techniques

Various PWM techniques, include:

• Sinusoidal PWM (most common)• Selected Harmonic Elimination (SHE)

PWM• Space-Vector PWM• Instantaneous current control PWM• Hysteresis band current control PWM• Sigma-delta modulation

04/18/23 33


Sinusoidal PWM

The most common PWM approach is sinusoidal PWM. In this method a triangular wave is compared to a sinusoidal wave of the desired frequency and the relative levels of the two waves is used to control the switching of devices in each phase leg of the inverter.

04/18/23 34


Sinusoidal PWM (cont’d)

Single-Phase (Half-Bridge) Inverter

Implementation

04/18/23 35



when va0> vT T+ on; T- off; va0 = ½Vd

va0 < vT T- on; T+ off; va0 = -½Vd

04/18/23 36



04/18/23 37



Definition of terms:

Triangle waveform switching freq. = fc (also called carrier freq.)

Control signal freq. = f (also called modulation freq.)

Amplitude modulation ratio, m = Vp

VT

Frequency modulation ratio,

mf (P)= fc / f

Peak amplitudePeak amplitudeof control signalof control signal

Peak amplitudePeak amplitudeof triangle waveof triangle wave

04/18/23 38


Multiple Pulse-Width Modulation

• In multiple-pulse modulation, all pulses are the same width

• Vary the pulse width according to the amplitude of a sine wave evaluated at the center of the same pulse

04/18/23 39


Generate the gating signal

2 Reference Signals, vr, -vr04/18/23 40


Comparing the carrier and reference signals

• Generate g1 signal by comparison with vr

• Generate g4 signal by comparison with -vr

04/18/23 41


Comparing the carrier and reference signals

04/18/23 42


Potential problem if Q1 and Q4 try to turn ON at the same time!

04/18/23 43


If we prevent the problem

Output voltage is low when g1 and g4 are both high

04/18/23 44


This composite signal is difficult to generate

04/18/23 45


Generate the same gate pulses with one sine wave

04/18/23 46


Alternate scheme

04/18/23 47


rms output voltage

• Depends on the modulation index, M

2

1

pm

o S S m

pV V V

Where δm is the width of the mth pulse

04/18/23 48


Fourier coefficients of the output voltage

2

1

4 3sin sin sin

4 4 41,3,5,..

pS m m m

n m mm

V nB n n

nn

04/18/23 49


Harmonic Profile

04/18/23 50


Compare with multiple-pulse case for p=5

Distortion Factor is considerably less04/18/23 51


Series-Resonant Inverter

04/18/23 52


Operation

T1 fired, resonant pulse of current flows through the load. The current falls to zero at t = t1m and T1 is “self – commutated”.

T2 fired, reverse resonant current flows through the load and T2 is also “self-commutated”.

The series resonant circuit must be underdamped,

R2 < (4L/C)

04/18/23 53


Operation in Mode 1 – Fire T1

11 1

1

1(0)

(0) 0

(0)

C S

C C

diL Ri i dt v Vdt C

i

v V

04/18/23 54


21 1

12 2

2

11

0

1

( ) sin

1

4

( ) sin

2

RtL

r

r

s c

t r

ts cr

r

i t A e t

R

LC L

V VdiA

dt L

V Vi t e t

L

R

L

04/18/23 55


To find the time when the current is maximum, set the first derivative = 0

1

1

1

0

sin cos 0

.....

tan

tan

1tan

2

t ts cr r r

r

rr m

r mr m

rm

r

di

dt

V Ve t e t

L

t

tt

t

04/18/23 56


To find the capacitor voltage, integrate the current

1

1

1

1

1

0

0

1

1 1

1( ) ( )

1( ) sin

...

( ) ( ) ( sin cos ) /

0 ( )

( ) r

t

C c

tts c

C r Cr

tC s C r r r r s

mr

C m C s C s

v t i t dt VC

V Vv t e t dt V

C L

v t V V e t t V

t t

v t V V V e V

The current i1 becomes = 0 @ t=t1m

04/18/23 57

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Operation in Mode 2 – T1, T2 Both OFF

2 1

2 2 1

2

2

( ) 0

( )

( )m

C C

C C C

i t

v t V

v t V V

04/18/23 59


t2m

04/18/23 60


Operation in Mode 3 – Fire T2

3

3 2 1

33 3

3

1(0) 0

(0) 0

(0)

C

C C C

diL Ri i dt vdt C

i

v V V

04/18/23 61


1

3 1

1

3

3

3

0

3

( ) sin

1( )

( sin cos )( )

0 ( )m

C tr

r

t

C C

tC r r r

Cr

r

Vi t e t

L

v t i dt VC

V e t tv t

t t

04/18/23 62


3 3 1

1 1

1

1

3

1

( )

( ) ( )

.

.

1

1

1

r

m

r

m

C C C C

C C S C S

C S z

z

C S z

C S C

v t V V V e

v t V V V e V

V Ve

eV V

eV V V

04/18/23 63


• Space Vector Diagram

1V

0V

3V

2V

4V

5V

6V

j

POO

PPOOPO

OPP

OOP POP

refV

OOOPPP

SECTOR ISECTOR III

SECTOR IV SECTOR VISECTOR V

SECTORII

• Active vectors: to (stationary, not rotating)

• Zero vector:

1V

6V

0V

• Six sectors: I to VI

Space Vector Modulation

04/18/23 64


• Space Vectors

• Three-phase voltages

0)()()( tvtvtv COBOAO

• Two-phase voltages

)(

)(

)(

3

4sin

3

2sin0sin

3

4cos

3

2cos0cos

3

2)(

)(

tv

tv

tv

tv

tv

CO

BO

AO

• Space vector representation)()()( tvjtvtV

(2) (3)

3/43/20 )()()(3

2)( j

CO

j

BO

j

AO etvetvetvtV

where xjxe jx sincos

(3)

(1)

(2)

(4)


04/18/23 65


• Space Vectors (Example)

Switching state [POO] S1, S6 and S2 ON

dBOdAO VtvVtv3

1)(,

3

2)(

dCO Vtv3

1)( and

(5) (4)

(7)

(5)

(6)0

1 3

2 j

d eVV

Similarly,

3)1(

3

2

kj

dk eVV

.6...,,2,1k

1V

0V

3V

2V

4V

5V

6V

j

POO

PPOOPO

OPP

OOP POP

refV

OOOPPP

SECTOR ISECTOR III


SECTORII


04/18/23 66


• Active and Zero Vectors

S p a c e V e c t o r S w it c h in g S t a t e ( T h r e e P h a s e s )

O n - s t a t e S w it c h V e c t o r

D e f in it io n

[ P P P ] 531 ,, SSS Z e r o V e c t o r 0V

[ O O O ] 264 ,, SSS 00 V

1V

[ P O O ] 261 ,, SSS 01 3

2 jd eVV

2V

[ P P O ] 231 ,, SSS 32 3

2

j

d eVV

3V

[ O P O ] 234 ,, SSS 3

2

3 3

2

j

d eVV

4V

[ O P P ] 534 ,, SSS 3

3

4 3

2

j

d eVV

5V

[ O O P ] 564 ,, SSS 3

4

5 3

2

j

d eVV

A c t iv e V e c t o r

6V

[ P O P ] 561 ,, SSS 3

5

6 3

2

j

d eVV

• Active Vector: 6

• Zero Vector: 1

• Redundant switching states: [PPP] and [OOO]

1S

2S

3S 5S

4S 6S

B

C

P

N

dV

A


04/18/23 67


(8)

• Reference Vector Vref

• Definition

1V

0V

3V

2V

4V

5V

6V

j

POO

PPOOPO

OPP

OOP POP

refV

OOOPPP

SECTOR ISECTOR III


SECTORII

• Angular displacement

t

dtt0

)( (9)

jrefref eVV

• Rotating in space at ω


f 2

04/18/23 68


• Relationship Between Vref and VAB

• Vref is approximated by two active and a zero vectors

• Vref rotates one revolution, VAB completes one cycle

• Length of Vref corresponds to magnitude of VAB

1V

2V

refV

1VT

T

s

a

2VT

T

s

b

SECTOR I

Q


04/18/23 69


• Dwell Time Calculation• Volt-Second Balancing

0

0021

TTTT

TVTVTVTV

bas

basref

(10)

• Ta, Tb and T0 – dwell times for and , 21 VV

0V

• Ts – sampling period

• Space vectors

d

j

refref VVeVV3

2, 1

3

2 3

2 j

d eVV

00 V

, and

(11) (10)

bdsref

bdadsref

TVTV

TVTVTV

3

1)(sin

3

1

3

2)(cos

:Im

:Re

(11)

(12)

1V

2V

refV

1VT

T

s

a

2VT

T

s

b

SECTOR I

Q


04/18/23 70


• Dwell Times

Solve (12)

bas

d

refs

b

d

refs

a

TTTT

V

VTT

V

VTT

0

sin3

)3

(sin3

3/0 (13)


04/18/23 71


• Vref Location versus Dwell Times

refV Location 0

60

6

36

3

Dwell Times 0

0

b

a

T

T baTT baTT baTT 0

0

b

a

T

T

1V

2V

refV

1VT

T

s

a

2VT

T

s

b

SECTOR I

Q


04/18/23 72


• Modulation Index

cbs

asb

asa

TTTT

mTT

mTT

0

sin

)3

(sin

(15)

d

ref

a V

Vm

3 (16)


04/18/23 73


• Modulation Range

• Vref,max

32

3

3

2max,

ddref

VVV (17)

1V

0V

3V

2V

4V

5V

6V

j

POO

PPOOPO

OPP

OOP POP

refV

OOOPPP

SECTOR ISECTOR III


SECTORII

(17) (16)

• ma,max = 1

• Modulation range: 0 ma 1 (18)


04/18/23 74


• Switching Sequence Design

• Basic Requirement:

Minimize the number of switchings per

sampling period Ts

• Implementation:

Transition from one switching state to

the next involves only two switches in

the same inverter leg.


04/18/23 75


• Seven-segment Switching Sequence

dV

20T

2aT

2bT

2aT

BNv

ANv

CNv

0

1V

1V

2V

0V

2V

POOOOO PPO PPP PPO POO OOO

dV

dV

40T

40T

2bT

sT

0

0

0V

0V

• Total number of switchings: 6

• Selected vectors: V0, V1 and V2

• Dwell times: Ts = T0 + Ta + Tb


04/18/23 76


• Undesirable Switching Sequence

• Vectors V1 and V2 swapped

dV

20T

2aT

2bT

2aT

BNv

ANv

CNv

0

1V

1V

2V

2V

POOOOO PPO PPP PPOPOO OOO

dV

dV

40T

40T

2bT

sT

0

0

0V

0V

0V

• Total number of switchings: 10


04/18/23 77


• Switching Sequence Summary (7–segments)

Sector Switching Sequence

0V 1V

2V

0V

2V

1V

0V

I OOO POO PPO PPP PPO POO OOO

0V 3V

2V

0V

2V

3V

0V

II OOO OPO PPO PPP PPO OPO OOO

0V 3V

4V

0V

4V

3V

0V

III OOO OPO OPP PPP OPP OPO OOO

0V 5V

4V

0V

4V

5V

0V

IV OOO OOP OPP PPP OPP OOP OOO

0V 5V

6V

0V

6V

5V

0V

V OOO OOP POP PPP POP OOP OOO

0V 1V

6V

0V

6V

1V

0V

VI OOO POO POP PPP POP POO OOO

Note: The switching sequences for the odd and ever sectors are different.


04/18/23 78


• Simulated Waveforms

ABv

AOv

0

0

0

Ai

dV

3/2 dV

2 3

2 3

VIVISector

III

IIIIV

V

III

IIIIV

V

f1 = 60Hz, fsw = 900Hz, ma = 0.696, Ts = 1.1ms


04/18/23 79


• Waveforms and FFT

0

0.1

0.2

n

0

0

ABv

AOv

Ai

THD =80.2%

THD =80.2%

THD =8.37%

THD =80.2%

dV

3/2 dV

2

dVVAB 566.01

1 5 10 15 20 25 30 35 40 45 50 55 60

dn VVAB /

0 2 3


04/18/23 80


• Waveforms and FFT (Measured)

23

AOv

ABv d

n

V

VAB

0.2

0.1

0

(a) Waveforms 2ms/div (b) Spectrum (500Hz/div)

14

34

10 4758

THD = 80.3%

8

16 29 43


04/18/23 81


0 0.2 0.4 0.6 0.80

0.05

0.10

0.15

10 16 20

dnAB VV /

am

1n

2n 4 8

14

(a) Even order harmonics

57111317n = 19

0.05

0.10

0.15

dnAB VV /

0 0.2 0.4 0.6 0.8 am

1n

(b) Odd order harmonics

0

100

200

300

0

THD(%)

THD

• Waveforms and FFT (Measured)

Hz601 f sec720/1sT ( and )


04/18/23 82


• Even-Order Harmonic Elimination

BNv

ANv

CNv

0

5V

4V

0V

OOPOOO OPP PPP OPP OOP OOO

0

0

0V

0V

ABv0

dV

4V

5V

dV

dV

dV

Type-A sequence (starts and ends with [OOO])

BNv

ANv

CNv

0

0

0

ABv0

dV

5V

OOP4V

OPP

dV

dV

dV

4V

OPP5V

OOPPPP0V

0V

OOO PPP0V

Type-B sequence (starts and ends with [PPP])


04/18/23 83



1V

3V

2V

4V

5V

6V

SECTOR ISECTOR III

SECTOR IV SECTOR VI

SECTOR V

SECTOR II

a

ba

a

a

aa

b

b

bb

b

Type-A sequence

Type-B sequence

30

30

Space vector Diagram


04/18/23 84



(a) Waveforms 2ms/div

AOv

ABv

0.2

0.1

0

d

n

V

VAB

(b) Spectrum (500Hz/div)

23

13 47

35

7

17

65

THD = 80.5%

41

5

• Measured waveforms and FFT


04/18/23 85



17

1913

75

11

100

200

300

0

THD(%)

0 0.2 0.4 0.6 0.8 am0

0.1

0.2

0.3

dnAB VV /

1nTHD

Hz601 f sec720/1sT ( and )


04/18/23 86


• Five-segment SVM

dV

2aT

bT2aT

BNv

ANv

CNv

0

1V

1V

2V

0V

POOOOO PPO POO OOO

dV

sT

0

0

0V

20T

20T

dV

aT

1V

2V

0V

PPP PPO POO PPP

dV

sT

0V

20T

20T

(a) Sequence A

2V

PPO

dV

2bT

2bT

(b) Sequence B


04/18/23 87


• Switching Sequence ( 5-segment)

S ector S w itch in g S eq u en ce (A )

0V

1V

2V

1V

0V

I

O O O P O O P P O P O O O O O 0CNv

0V

3V

2V

3V

0V

II

O O O O P O P P O O P O O O O 0CNv

0V

3V

4V

3V

0V

III

O O O O P O O P P O P O O O O 0ANv

0V

5V

4V

5V

0V

IV

O O O O O P O P P O O P O O O 0ANv

0V

5V

6V

5V

0V

V

O O O O O P P O P O O P O O O 0BNv

0V

1V

6V

1V

0V

V I

O O O P O O P O P P O O O O O 0BNv


04/18/23 88


• Simulated Waveforms ( 5-segment)

dV

2 4

1gv

3gv

5gv

ABv

0

0

Ai

3/2

2 4

2 4

• No switching for a 120° period per cycle. • Low switching frequency but high harmonic distortion

• f1 = 60Hz, fsw = 600Hz, ma = 0.696, Ts = 1.1ms


04/18/23 89

unit-iv inverters 6/8/2015copyright by

Documents

phase bridge inverters

inverters contd quadrant

halfbridge inverters

phase fundamental

bridge hbridge inverter

phase windings

squarewave phase voltages

mutual phase shift