Switch-Mode DC/AC Inverters - libvolume3.xyzlibvolume3.xyz/electrical/btech/semester6/advancedpowerelectronic… · motor 60 Hz AC filter capacitor switch-mode converter Switch-mode

PWM DC-AC Inverters

NCTU 2005 Power Electronics Course Notes 1

page 1

Switch-Mode DC/AC Inverters

2005年6月7日

鄒應嶼教授

國立交通大學電機與控制工程研究所

Filename: \電力電子 (研究所)\PE-08.PWM DC-AC Inverters.ppt

國立交通大學電力電子晶片設計與DSP控制實驗室Power Electronics IC Design & DSP Control Lab., NCTU, Taiwan

http://powerlab.cn.nctu.edu.tw/

POWERLABNCTU

電力電子晶片設計與DSP控制實驗室Power Electronics IC Design & DSP Control Lab.

台灣新竹交通大學 • 電機與控制工程研究所

page 2

Switch-Mode DC/AC Inverters

IntroductionBasic Concepts of Switch-Mode InverterSquare Wave InverterSinusoidal PWM InverterPWM Switching SchemesSingle-Phase InvertersThree-Phase InvertersEffect of Blanking TimeAdvanced PWM Switching SchemesRectifier Mode of Operation

PWM DC-AC Inverters


page 3

Introduction

What is an Inverter ?

AC/DC: RectifierDC/DC: ChopperDC/AC: InverterAC/AC: Cycloconverter

~=

Inverter

AC OUTPUTDC INPUT

page 4

Applications of DC/AC Inverters

PWM Inverters for AC Motor DrivesPWM Inverters for AC Power SourceUPS & AVRInduction HeatingPower Supply for Fluorescent Lamp

PWM DC-AC Inverters


page 5

Inverter in AC Motor Drive

+

−

vdAC

motor60 Hz

AC

diode-rectifier

filter capacitor

switch-mode inverter

Unidirectional Power Flow

+

−

vdAC

motor60 Hz

AC

filter capacitor

switch-mode converter

Switch-mode converter for motoring and regenerative braking in AC motor drive.

switch-mode converter

Bidirectional Power Flow

page 6

Circuit Architecture of Advanced AC Motor Drives

uεud

• PWM Control• Inverter Control• DTC Vector Control• Sensorless Control• Servo Control• Auto-Tuning

• Power Factor Control

• Regenerative Braking Control

• DC-Link Voltage Regulation

• DC-Link Capacitor Minimization

Cd

to sw itches

Inputconverter

Outputconverter

ud

to sw itches

u’1u’2u’3

N S

PWM DC-AC Inverters


page 7

Classifications of Inverters

Voltage Source Inverter (VCI)

Current Source Inverter (VCI)Square Wave CSI

PWM CSI

PWM InverterCurrent-Controlled PWM Inverter

Variable Voltage Inverter

Processed Energy Source and Load (Voltage Source, Current Source)Topology (Single-Phase, Three-Phase, etc.)PWM Strategy (Square, PWM, Sine PWM, Regular PWM, Space Vector PWM, etc.)Switching Devices (SCR, Power Transistor, Power MOSFET, IGBT, etc.)Switching Schemes (PWM, Resonant, Quasi-Resonant, Soft PWM, etc)Control Schemes (Hysteresis, PID, Dead-beat, Variable Structure, Fuzzy, etc.)Controller Implementation (Analog, Microprocessor, DSP, etc.)

page 8

Three-Phase Inverter Drives

b1

b2

b3

b4

b5

b6

Voltage (Line to Neutral)

Current (Line)

3-PhasePowerSupply

N

S

S

N

b1

b2

b3

b4

b5

b6N

S

S

N

3-PhasePowerSupply

Voltage (Line to Neutral)

Current (Line)

VSI: Voltage Source Inverter

CSI: Current Source Inverter

PWM DC-AC Inverters


page 9

Three-Phase Inverter for AC Power Source

3-phaseload

3-PhaseMotor

3-PhasePowerSupply

page 10

Voltage Source Current-Controlled PWM Inverter

Current

Controller

ias*

i as

b1

b2

b3

b4

b5

b6

3-PhasePowerSupply

--

-

ibs*

ics*

i bsi cs

PWM DC-AC Inverters


page 11

Applications of Inverters in UPS

整流器負載

市電

充電器

升壓器換流器濾波器

蓄電池

控制功能：

• 開機程序控制• 蓄電池充電控制• 直流鏈電壓調節• 功率因數補償

• 輸出交流電壓調節• 自我保護機制• 電源監測• 緊急事件處理

控制器

功率級

輸出

濾波器

輸出

轉換開關

page 12

Basic Concepts of Switch-Mode Inverter

+

−

vd

1-phase switch-mode

inverter + filter

ioid

+

−vo

iovo

4 1 2 3

1 inverter

3 inverter

2 rectifier

4 rectifier

io

vo0

Switch-Phase Switch-Mode Inverter

PWM DC-AC Inverters


page 13

One-Leg Switch-Mode Inverter

DA+TA+

TA- DA-

io

+

−vAN

N

o

+

−2dV

+

−2dV

+

−

vd A

( $ ) . ( )V V VAo d d14

21 273

2= =

π

( $ ) ($ )V VhAo hAo= 1

1.41.21.00.80.60.40.2

00 1 3 5 7 9 11 13 15

x

xx x x x x x

( $ ) / ( )V VAo d1 2

(h: harmonics of f1)

t

Vd2

−Vd2

v Ao

11f

0

Spectrum of Square-Wave PWM

page 14

Analysis of a Step Approximated Inverter for UPS

π ω t

V

2πu21

ππ u

D−

=

)(tv

一個由責任比(D)控制的方波如下圖所示：

1. 計算此方波的諧波。2. 計算此責任比控制方波之基本波(fundamental)與諧波的均方根值。3. 計算此責任比控制方波的均方根值 Vrms = ?4. 當此方波之均方根值與峰值為V的正弦波相同時，責任比D = ?5. 此方波的總諧波失真(THD)為何？

10 ≤≤ D

PWM DC-AC Inverters


page 15

Harmonics of Odd-Quad Symmetric Square Waveform

) cos sin(

3cos 2cos cos 3sin 2sin sin) (

10

321

3210

∑∞

=

++=

+++

++++=

nnn tnBtnAA

tBtBtBtAtAtAAtf

ωω

ωωωωωωω

L

L

L+++= tAtAtAtv ωωω 5sin3sinsin)( 531

1. 由於此方波具有奇函數對稱 f(ωt) = -f(-ωt) 且半波對稱 f(ωt) = -f(ωt + π) 的特性，因此稱之為奇四分對稱(odd quad symmetric)，其傅立葉係數(Fourier Series)僅含有正弦函數的奇次項。

∫=π

ωωπ

2

0 cos)(1 tdtntfAn

[ ]

2sin

142

cos14

coscos

)( sin)( sin1

5.025.0

5.05.0

5.0

5.0

5.02

5.0

πππ

ωωπ

ωωωωπ

ππ

π

π π

π

nDVn

unV

n

tntnnV

tdtnVtdtnVA

uu

uu

u

u

u

un

==

+−=

⎥⎦⎤

⎢⎣⎡ −+=

−+

−

− −

+∫ ∫

page 16

Harmonics of Duty-Ratio Controlled Square Wave

1.0

31

71

91

111

131

1VV n

0 n1 3 5 7 9 11 13

51

⎩⎨⎧=

n/nVn

Vn of values oddfor of valueseven for 0

1

) cos991+7sin

715sin

513sin

31(sin

2sin4 L++++= tttttnDVVn ωωωωω

ππ

2sin14 π

πnDV

nVn =

2. 基本波的均方根值：

tDVtV ωππ

sin2

sin4)(1 =

2sin

24

1π

πDVV rms =

PWM DC-AC Inverters


page 17

RMS Value of Harmonics

第n次諧波的均方根值：

tnnDVn

tVn ωπ

πsin

2sin14)( =

2sin

214 π

πnDV

nVnrms =

諧波的均方根值：

21

227

25

23 rmsrmshrms VVVVVV −=+++= L

2cos8)

2(sin8 22

22

uDVDDVVhrms ππ

π−=−= π

π uD −=

page 18

RMS Value of the Square Waveform

ππ

ωπ

ωπ

ππ

uVDV

tdVtdvVD

rms

−==

⎥⎦⎤

⎢⎣⎡== ∫∫

2/1

0

22

0

2 )(1)(21

DVVrms =

3. 由於週期波之平移，不會改變其均方根值，因此：

[ ]2/1

,..5,3,1

22/12

52

32

1 2sin14

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛=+++= ∑

∞ ππ

nDVn

VVVVrms L

2VDVVrms ==

4. 當此方波之均方根值與峰值為V的正弦波相同時：

%50 =∴ D

PWM DC-AC Inverters


page 19

THD of a Square Waveform

rms

rmsrms

rms VVV

VVVV

THD1

21

2

1

27

25

23 −=

+++=

L

5. 此方波之總諧波失真：

2sin

214

)2

(sin8

2sin

24

)2

sin2

4()( 22

22

1

27

25

23

ππ

ππ

ππ

ππ

D

DD

DV

DVDV

VVVV

THDrms

−=

−=

+++=

L

當責任比 D=1 時：

%3.48

214

81

2sin

24

)2

(sin8 22

2=

−=

−=

π

ππ

π

ππ

DV

DDTHD

page 20

Sinusoidal PWM

b1b2b3b4

PWMModulator

L

R

C

v irc

v c

v o

ic

ioiLrL+

−−−

+

+b1

Vdcb2

b3

b4

AB

vcontrol

vcarrier

0 t

vcontrol

vcarrier vcontrolvcarrier

+_

b1 b4

b2 b3

PWM DC-AC Inverters


page 21

Pulse-Width Modulator

+

-Amp comparator

repetitive waveform

switch control signal

vcontrolvo (desired)

vo (actual)

ton toffTs

on on

off off

switch control signal

stV̂vst = sawtooth voltage vcontrol (amplified error)

vcontrol > vst

vcontrol < vst

(switch frequency fs = 1/Ts)

sts Vv

Tt

D ˆcontrolon ==

page 22

Natural Sinusoidal PWM

⎭⎬⎫

⎩⎨⎧ <

+− off : ,on :tricontro

AA

l

TTvv

⎭⎬⎫

⎩⎨⎧ <

−+ off : ,on :tricontrol

AA TTvv

Vd2

−Vd2

1 lfundamenta , )( AoAo vv =uAo

controlV triV

( )1f s

0 t

t0

t = 0

tri

controlˆ

ˆ

VVma =

1ffm sf =

Amplitude Modulation Ratio

Frequency Modulation Ratio

dAoA VvTvv 21 on, istricontrol => +

dAoA VvTvv 21 on, istricontrol −=< −

DA+TA+

TA- DA-

io

+

−vAN

N

o

+

−2

dV

+

−2

dV

+

−

Vd

dAoAN Vvv 21

+= hAohAN VV )ˆ()ˆ( =

A

PWM DC-AC Inverters


page 23

Spectrum of Sinusoidal PWM

x

xx

x

x x x x x x x xx x x xxxx

( $ )/

VV

Ao h

d 2

15 ,8.0 == fa mm

( )m f + 2m f

( )2 1m f +2m f 3m f

( )3 2m f +Harmonics h of f1

00.20.40.60.81.01.2

Vd2

−Vd2

1 lfundamenta , )( AoAo vv =uAo

controlV triV

( )1f s

0 t

t0

page 24

Sinusoidal PWM

controlvtriv

0

Vd2

( )− Vd2

vAo

controlvtriV̂

0 t

triv

t

tricontroltri

control ˆ2ˆ

VvVV

vV dAo ≤=

PWM DC-AC Inverters


page 25

Linear Control of the Sinusoidal PWM Output

tricontrol1controlcontrolˆˆsinˆ VVtVv ≤= ω

0.1for 2

sin2

sinˆˆ

)( 11tri

control1 ≤== adadAo m

VtmVtV

Vv ωω

0.1 2

)ˆ( 1 ≤= adaAo mVmV

which shows that in a sinusoidal PWM, the amplitude of the fundamental-frequency component of the output voltage varies linearly with ma (provided ma ≤ 1.0). Therefore, the range of ma from 0 to 1 is referred to as the linear range.

page 26

Harmonics of the Sinusoidal Modulated PWM Waveforms

1)( fkjmf fh ±=

kmjh f ±= )(

x

xx

x

x x x x x x xx x x x xxxx

( $ )/

VV

Ao h

d 2

15 ,8.0 == fa mm

( )m f + 2m f

( )2 1mf +2m f 3m f

( )3 2m f +

Harmonics h of f1

00.20.40.60.81.01.2

The frequencies at which voltage harmonics occur can be indicated as:

The harmonic order h corresponds to the kth sideband of j times the frequency modulation ratio mf:

where the fundamental frequency correspond to h = 1.

PWM DC-AC Inverters


page 27

Generalized Harmonics of vAo for a Large mf

0.2 0.4 0.6 0.8 1.0

fundamental1

mah

0.2 0.4 0.6 0.8 1.0

mf

mf ± 2

mf ± 4

1.242

0.016

1.15 1.006 0.818 0.601

0.061 0.131 0.220 0.318

0.018

2mf ± 3

2mf ± 5

0.190 0.326 0.370 0.314 0.181

0.024 0.071 0.139 0.212

0.044

2mf ± 1

0.016

3mf ± 2

3mf ± 4

0.335 0.123 0.083 0.171 0.113

0.139 0.203 0.176 0.062

0.157

3mf

0.104

3mf ± 6 0.0440.016

0.044

0.012 0.047

4mf ± 3

4mf ± 5

0.163 0.157 0.008 0.105 0.068

0.070 0.132 0.115 0.009

0.119

4mf ± 1

0.084

4mf ± 7 0.0500.017

0.012

0.034

Generalized Harmonics of vAo for a Large mf.

Note: is tabulated as a function of ma.])2/1/()ˆ([)2/1/()ˆ( dhANdhAo VVVV =

page 28

Example 8.1: Harmonic Analysis

In the following circuit, Vd = 300 V, ma = 0.8, mf = 39, and the fundamental frequency is 47 Hz. Calculate the rms values of the fundamental-frequency voltage and some of the dominant harmonics in vAo using Table 8-1.

Example 8.1: Harmonics Analysis of a One-Leg Switching-Mode Inverter

DA+TA+

TA- DA-

io

+

−vAN

N

o

+

−2dV

+

−2dV

+

−

Vd A

PWM DC-AC Inverters


page 29

Solution of Example 8.1

2/)ˆ(

07.1062/)ˆ(

221

)(d

hAo

d

hAodhAo V

VVVV

V ==

Fundamental: (VAo)1 = 106.07 × 0.8 = 84.86 V at 47 Hz(VAo)37 = 106.07 × 0.22 = 23.33 V at 1739 Hz(VAo)39 = 106.07 × 0.818 = 86.76 V at 1833 Hz(VAo)41 = 106.07 × 0.22 = 23.33 V at 1927 Hz(VAo)77 = 106.07 × 0.314 = 33.31 V at 3619 Hz(VAo)79 = 106.07 × 0.314 = 33.31 V at 3713 Hzetc.

Solution:The rms voltage at any value of h is given as

From Table 8-1 the rms voltages are as follows:

page 30

Selection of Frequency Modulation Ratio

Small mf (mf 21)

1. Synchronous PWM

2. mf should be an odd integer

1. Subharmonics due to asynchronous PWM becomes small as mf is increased.

2. In applications of inverter motor drives, asynchronous PWM should be avoid when operating in low frequency range.

PWM DC-AC Inverters


page 31

Harmonics Due to Overmodulation

1harmonics h

3 5 7 9 11 13 150

)2

/()ˆ( dhAoVV

0.5

1.0

17 19 21 23 25 27

x

x

x x xx x

xx x x x x x

ma = 2.5mf = 15

(mf)

Harmonics due to overmodulation; drawn for ma = 2.5 and mf = 15.

page 32

Voltage Control by Varying ma

0

)2

/()ˆ( 1 dAoVV

1.0

(= 1.278)

ma0 1.0 3.24

linearover-

modulationSquare-wave

π4

(for mf = 15)

24)ˆ(

2 1d

Aod VVV

π

PWM DC-AC Inverters


page 33

Square-Wave Switching Scheme

t0

1

x

h (harmonics of f1)3 5 7 9 11 13 150

00.20.40.60.81.01.21.4

xx x x x x x

V d2

2dV−

11f

v Ao

( $ ) . ( )V V VAo d d14

21 273

2= =

π

( $ ) ($ )V VhAo hAo= 1

( $ ) / ( )V VAo d1 2

page 34

Single-Phase Half-Bridge Inverter

D+T+

T− D−

io

+

−vo

N

o

+

−2dV

+

−2dV

+

−

vd

Half-bridge inverter.

VT = Vd IT = io, peak

The input capacitor C+ and C- act as dc blocking capacitor, there will no dc component flows through io.The peak voltage and current ratings of the switches are:

PWM DC-AC Inverters


page 35

Single-Phase Full-Bridge Inverter

io

+

−vo

o

+

−2dV

+

−2dV

+

−

vd

Single-phase full-bridge inverter.

DA+TA+

TA- DA-

TB+ DB+

TB- DB-

A

B

VT = Vd IT = io, peak

With the same dc voltage, the maximum output voltage of the full-bridge inverter is twice that of the half-bridge inverter. The peak voltage and current ratings of the switches are:

vo = vAo -vBo

page 36

PWM with Bipolar Voltage Switching

0

0

( )1f s

controv triv

vo1

V d

−Vd

vo

t

t

io

+−vo

o

+

−2dV

+

−2dV

+

−

vd

DA+TA+

TA- DA-

TB+ DB+

TB- DB-

A

B

)()( tvtv AoBo −=

)(2)()()( tvtvtvtv AoBoAoo =−=

)0.1( ˆ 1 ≤= adao mVmV

)0.1(4ˆ 1 >

PWM DC-AC Inverters


page 37

Example 8.2: Harmonics of Bipolar PWM

In the full-bridge converter circuit of Fig. 8-11, Vd = 300 V, ma = 0.8, mf = 39, and the fundamental frequency is 47 Hz. Calculate the rms values of the fundamental-frequency voltage and some of the dominant harmonics in the output voltage can be obtained by using Table 8-1, as illustrated by the following example.

2/)ˆ(13.212

2/)ˆ(

22/)ˆ(

22

21)(

d

hAo

d

hAod

d

hAodhAo V

VVVV

VVVV ==⋅⋅=

Fundamental: Vo1 = 212.13 × 0.8 = 169.7 V at 47 Hz(Vo)37 = 212.13 × 0.22 = 46.67 V at 1739 Hz(Vo)39 = 212.13 × 0.818 = 173.52 V at 1833 Hz(Vo)41 = 212.13 × 0.22 = 46.67 V at 1927 Hz(Vo)77 = 212.13 × 0.314 = 66.60 V at 3619 Hz(Vo)79 = 212.13 × 0.314 = 66.60 V at 3713 Hzetc.

Example 8.2: Harmonics of Full-Bridge Bipolar PWM Inverter

Solution:

page 38

DC-Side Current id

filter

+

−eo

+

−

vd

*di

Lf1

Cf1

idfilter

Lf2

Cf2+

−

vo

ioload

Lload


fs → ∞Lf1, Cf1 → 0

fs → ∞Lf2, Cf2 → 0

fs

Inverter with “fictitious” filters at the input dc side and the output ac side.

tVvv ooo 11 sin2 ω==

)sin(2 1 φω −= tIi oo

PWM DC-AC Inverters


page 39

Analysis of DC-Side Current id

)sin(2sin2)()()( 11* φωω −== tItVtitvtiV oooodd

)2cos(2

)2cos(cos)(

12

21*

φω

φωφ

−−=

+=−−=

tII

iItV

IVV

IVti

dd

ddd

oo

d

ood

φcosd

ood V

IVI =

d

ood V

IVI2

12 =

On the dc side, the LC filter will filter the high-frequency components in id, and idwould only consist of the low-frequency and dc components.

*

Therefore,

where

page 40

Waveforms of DC-Side Current id

0

0

ω1t

ω1t

vo (filtered)

idioid2

Vd

-Vd

Id

φ

The DC-side current in a single-phase inverter with PWM bipolar voltage switching.

PWM DC-AC Inverters


page 41

Voltage Ripples at DC Link

The voltage ripples at the dc link are caused by two reasons:Rectification of the line voltageSecond harmonic current component occurs (of the fundamental frequency at the inverter output) at the dc link due to the sinusoidal output.

page 42

Unipolar PWM

io

+−vo

o

+

−2dV

+

−2dV

+

−

vd

DA+TA+

TA- DA-

TB+ DB+

TB- DB-

A

B

vcontrol > vtri : TA+ on and vAN = Vdvcontrol < vtri : TA- on and vAN = 0

(-vcontrol) > vtri : TB+ on and vBN = Vd(-vcontrol) < vtri : TB- on and vBN = 0

leg A switches:

leg B switches:

vtri vcontrol

t0

(-vcontrol)TA+on

TB+on

-vcontrol > vtrivcontrol > vtri

Vd

Vd

t

t

0

0

vAN

vBN

PWM DC-AC Inverters


page 43

PWM with Unipolar Voltage Switching (Single Phase)

x

d

hAo

VV )ˆ(

00.20.40.60.81.0

vtri vcontrol

t0

(-vcontrol)TA+on

TB+on

-vcontrol > vtri vcontrol > vtri

xx x

x

Vd

Vd

t

t

0

0

vAN

vBN

Vd

-Vd

0

vo(= vAN - vBN)

vo1

t

1x x x x x x

(2mf - 1) (2mf + 1)2mfmf 3mf 4mf

h

(harmonics of f1)

page 44

PWM with Unipolar Voltage Switching

TA+, TB- on : vAN = Vd, vBN= 0; vo= Vd

TA-, TB+ on : vAN = 0, vBN= Vd ; vo= -Vd

TA+, TB+ on : vAN = Vd, vBN= Vd ; vo= 0

TA-, TB- on : vAN = 0, vBN= 0 ; vo= 0

Vd

-Vd

0

vo(= vAN - vBN)

vo1

t

3-level switching

PWM DC-AC Inverters


page 45

Unipolar vs. Bipolar PWM

Compared with the bipolar PWM, the unipolar PWM has the following advantages:

“effectively” doubling the switching frequency Reduced voltage jumps in the output voltage

Lower current ripples and harmonic distortion!

page 46

Harmonics of the Unipolar PWM

)0.1(ˆ 1 ≤= adao mVmV

)0.1(4ˆ 1 >

PWM DC-AC Inverters


page 47

Example 8.3: Analysis of Unipolar PWM

In Example 8-2, suppose that a PWM with unipolar voltage-switching scheme is used, with mf = 38. Calculate the rms values of the fundamental-frequency voltage and some of the dominant harmonics in the output voltage.

2/)(13.212)(

d

hAoho V

VV =

h = j(2mf) ± k

Fundamental: Vo1 = 0.8 × 212.13 = 169.7 V at fundamental or 47 Hz(Vo)75 = 0.314 × 212.13 = 66.60 V at h = 2mf − 1 = 75 or 3525 Hz(Vo)77 = 0.314 × 212.13 V at h = 2mf + 1 = 77 or 3619 Hzetc.

Example 8.3: Analysis of the Full-Bridge Unipolar PWM Inverter

Solution:

page 48

DC-Side Current id

0 t

id io

(-io)

The DC-side current in a single-phase inverter with PWM unipolar voltage switching.

Vd

-Vd

0

vovo1

t

PWM DC-AC Inverters


page 49

Square Wave Operation

do VV π4ˆ

1 =

t0

1

x

h (harmonics of f1)3 5 7 9 11 13 150

0

0.20.4

0.60.81.01.21.4

xx x x x x x

V d2

2dV−

11f

vAo

( $ ) . ( )V V VAo d d14

21 273

2= =

π

( $ ) ($ )V VhAo hAo= 1

( $ ) / ( )V VAo d1 2

page 50

Single-Phase, Full-Bridge InverterControl by Voltage Cancellation

io

+−vo

+

−

vd

DA+TA+

TA- DA-

TB+ DB+

TB- DB-

A

B

N

id

00

0.2

0.4

0.6

0.8

1.0

60 120 1807th

5th 3rd

total harmonic distortion

fundamental

α

vAN

vBN

vo

0

0

0

α

Vd

Vd

Vd

(180- α )°

(180- α )°

α

β -Vd

180°

180°

°−=°−= )2

90(2

)180( ααβ

(a) power circuit

(b) waveform(c) normalized fundamental and harmonic voltage output and total harmonic distortion as a function of α.

PWM DC-AC Inverters


page 51

Pulsewidth Control

∫∫ −− ==β

β

π

πθθ

πθθ

πdhVdhvV doho )cos(

2)cos(2)ˆ(2/

2/

)sin(4)ˆ( βπ

hVh

V dho =∴

vo

0

Vd

(180- α )°

(180- α )°

α

β-Vd

°−=°−= )2

90(2

)180( ααβ

where β = 90o - 0.5α and h is an odd integer.

page 52

Application in Step Wave UPS

io

+

−vo

o

+

−2dV

+

−2dV

+

−

vd

DA+TA+

TA- DA-

TB+ DB+

TB- DB-

A

B

vo = vAo -vBo

π ω t

V

2πu21

ππ u

D−

=

)(tv

10 ≤≤ D

PWM DC-AC Inverters


page 53

Ripple in Single-Phase Inverter Output

+

−eo(t)

+

−

Vd

induction-motor load

L


(a) circuit

+

−

vo

io+ −

~tEe oo 1sin2 ω=

vL = vL1 + vripple

+

−

L

io1 + −

~ tEe oo 1sin2 ω=

vL1

+

−~vo1

L+ −

+

−vripple

vrippleiripple

(b) fundamental-frequency components

(c) ripple frequency components

vL = vL1 + vripple

iL = iL1 + iripple

Th ripple in a periodic waveform refers to the difference between the instantaneous values of the waveform and its fundamental-frequency component.

page 54

Current Ripple Analysis

Vo1 = Eo + VL1 = Eo +jω1LIo1

vripple(t) = vo - vo1

kvL

tit

+= ∫0 rippleripple )(1)( ζ

Vo1

j(ω1L)Io1 = VL1 Eo

Io1

(d) fundamental-frequency phasor diagram

Th output current ripple can be expressed as

where k is a constant and ζ is a variable of integration.

PWM DC-AC Inverters


page 55

Ripple in the Inverter Output

vo

vripple = vo - vo1

iripple

0

vo1

t

t

t t

t

t

0

0

0

0

0

vo

vripple = vo - vo1

iripple

vo1

(a) square-wave switching (b) PWM bipolar voltage switching

page 56

Push-Pull Inverters

)0.1(ˆ 1 ≤= adao mnVmV

)0.1(4ˆ 1 >

PWM DC-AC Inverters


page 57

Comments on Push-Pull Inverters

In control of the push-pull inverter, it should avoid the dc saturation in the primary side of the transformer.Very good magnetic coupling between the two primary windings is required to reduce the energy associated with the leakage inductance.

D2T2T1D1

+

−

Vd

id

n : 1

+

−vo

Require very good coupling!

page 58

Comments on Push-Pull Inverters

D2T2T1D1

+

−

Vd

id

n : 1

+

−vo

Number of turns will be high for sinusoidal output!

In a PWM push-pull inverter for producing sinusoidal output (unlike those used in switch-mode dc power supplies), the transformer must be designed for the fundamental output frequency. This requires a high number of turns.A higher number of of turns will result in a high transformer leakage inductance, which is proportional to the square of the number of turns.

i

Nv+

_

L N Al

dBdH

N Alm m

= =2 2µ

lm

PWM DC-AC Inverters


page 59

Switch Utilization in Single-Phase Inverters

TT

oo

IqVIV

max,1 ratio nutilizatio Switch =

22

422 max,max,1max,

max, ==== qnV

Vn

IIVV do

oTdT π

16.021 ratio nutilizatio switch Maximum ≅=∴π

n = turn ratio

where q is the number of switches in an inverter, VT and IT as the peak voltage and current ratings of a switch, Vo1 is the rms value of the output fundament voltage, Io,max is the rms value of the output current. (The output current is assumed as sinusoidal)

The utilization factor of all the switches in an inverter is defined as:

Single-Phase Inverter: Square-wave mode at maximum rated output

Push-Pull Inverter

page 60

Switch Utilization in Single-Phase Inverters

222

422 max,max,1max,max, ==== qV

VIIVV dooTdT π


42

422 max,max,1max,max, ==== qVVIIVV dooTdT π


Half-Bridge Inverter

Full-Bridge Inverter

16.021 ratio nutilizatio switch Maximum ≅=π

Conclusion: all the single-phase inverter has the same switch utilization ratio

PWM DC-AC Inverters


page 61

Comments on Switch Utilization Ratio

aa mm 81

421 ratio nutilizatio switch Maximum ≅= ππ

)0.1 PWM,( ≤am

Switch ratings are chosen conservatively to provide safety margins.In determining the switch current rating in a PWM inverter, one would have to take into account the variations in the input dc voltage available.The output ripple current would influence the switch current rating.

In practice, the switch utilization ratio would be much smaller than 0.16, due to

Single-Phase Inverter: PWM mode

Conclusion: The theoretical maximum switch utilization ratio in a PWM switching is only 0.125 at ma = 1, as compared with 0.16 in a square-wave inverter.

page 62

Example 8.4: Calculation of Switch Utilization Ratio

In a single-phase full-bridge PWM inverter, Vd varies in a range of 295-325V. The output voltage is required to be constant at 200 V (rms), and the switch utilization ratio (under these idealized conditions, not accounting for any overcurrent capabilities).

V325max, == dT VV

14.141022 =×== oT II

4switches of no. ==q

VA200010200max,1 =×=oo IV

11.014.143254

2000 ratio nutilizatio Switch max,1 ≅××

==TT

oo

IqVIV

Example 8.4: Calculation of Switch Utilization Ratio

Solution:

PWM DC-AC Inverters


page 63

Three-Phase Inverter

DA+TA+

TA- DA-

id

A

o

+

−2dV

+

−2dV

+

−

vd

DB+TB+

TB- DB-

B

DC+TC+

TC- DC-

C

N

page 64

Three-Phase PWM InverterDefinition of Voltages and Currents

o : neutral of the invertern : neutral of the loadVdc : voltage of the dc-linkS1 ... S6 : switches of the invertervao, vbo, vco : pole voltage of the invertervan, vbn, vcn : phase voltage of the loadias, vbs, vcs : phase current of the loadvab, vbc, vca : line-to-line voltage

180o180o

0.5Vd

S1S2 0.5Vd

S1

S3S4

S3S4

vac

vba

vca

0ωt

ωt

ωtS6S5

S6S5

(a)

(b)

(c)vab

ωtVd

vbc

ωt

(d)

(e)

vcaωt

(f)

van

ωt

Vd23

ias

120o120o

Vd

Six-Step Voltage and Current Waveforms

(g)

ias

ibsics

S1

S2

S3

S4

S5

S6

o n

vasvbs

vcs

Vdc

A

C

B

vao = van + vnovbo = vbn + vnovco = vcn + vno

vno =(1/3) (vao + vbo + vco)van = vao - vnovbn = vbo - vnovcn = vco - vnovao = 0.5Vdc[ds1 (k)- ds2 (k)]

d k d ks s1 2 1( ) ( )+ ≤

where ds1 (k), ds2 (k) are the on-duty ratio of S1 and S2during the k-th switching interval.

vno

PWM DC-AC Inverters


page 65

Considerations in 3-Phase PWM Inverters

For low values of mf (mf 1.0), regardless of the value of mf, the conditions pertinent to a small mf should be observed.

page 66

PWM in Three-Phase Voltage Source Inverters

vtri vcontrol, A vcontrol, B vcontrol, C

t0

Vdt0

vAN

Vd

vBN

0 t 00.20.40.60.81.0

1

x

(2mf+1)

d

ho

VV )ˆ(

15 ,8.0 == fa mm

mf 2mf 3mf

x x xx

x x x x xx x x x x h

Harmonics of f1

(mf+2) (3mf+2)

Vdt0

fundamental vLL1

Cancel out!

PWM DC-AC Inverters


page 67

Linear Modulation (ma

PWM DC-AC Inverters


page 69

Overmodulation

0

square-wave

d

LL

VV )rms(1

1

linear

overmodulationsquare-wave

78.06 =π

612.0223

=

3.24ma

(for mf = 15)

Three-phase inverter; VLL1(rms)/Vd as a function of ma.

page 70

Square-Wave Operation

00.20.40.60.81.0

d

LL

VV

h

ˆ

TA+0

vAN

ω 1tVd TA

180°

180°vBN

TB-TB+

0 ω 1tvCN

0 ω 1tTC+ TC-

TC+

vAB

Vd0 ω 1t

1LLv

1.2

harmonics of f1

1 3 5 7 9 11 13

DA +

TA+

TA- DA-

id

A

+

−

Vd

DB +

TB+

TB- DB-

B

DC +TC+

TC- DC-

C

N

A B C

x

xx x x

Square-wave inverter (three phase).

PWM DC-AC Inverters


page 71

Harmonics of the Square-Wave Inverter

dLLh VhV 78.0=

ddd

LL VVV

V 78.06

24

23

)rms(1 ≅== ππ

h = 6n ± 1 (n = 1,2,3, ...)

The fundamental-frequency line-to-line rms voltage in the output of the one-leg inverter operating in a square-wave mode:

00.20.40.60.81.0

d

LL

VV

h

ˆ

1.2

harmonics of f11 3 5 7 9 11 13

x

x x x x

The line-to-line voltage waveform does not depend on the load and contains harmonics (6n +/- 1; n = 1, 2, …), whose amplitudes decrease inversely proportional to their harmonic order.

page 72

Switch Utilization in Three-Phase Inverters

Assume a pure sinusoidal output current with an rms value of Io,max (both in the PWM and square-wave modes) at maximum loading.

Each switch has the following peak ratings:

In the square-wave mode, this ratio is 1/2π ≅ 0.16 compared to a maximum of 0.125 for a PWM linear region with ma = 1.0.

axoTdT IIVV m,max, 2==

max,max,max,

max,phase3 11

621

263

6)VA(

ratio nutilizatio Switchd

LL

od

oLL

TT VV

IVIV

IV=== −

)0.1(81

223

621(PWM)ratio nutilizatio switch Maximum ≤== aaa mmm

PWM DC-AC Inverters


page 73

Ripple in the Inverter Output

+

−

VdThree-phase

three-leg inverter

L

(load neutral)

~

id

A

iA

B

C

iB

iC

R = 0

eA(t)+

−n

3-phaseac motor load IA1

VAN1

φ EA j(ω1L)IA1

Three-phase inverter: (a) circuit diagram; (b) phasor diagram (fundamental frequency).

(a) (b)

Back emf

),,( CBAkvvv nNkNkn =−=

),,( CBAkedtdiLv knkkn =+=

Under balanced operating conditions, it is possible to express the inverter phase output voltage vAN in terms of the inverter output voltage.

Each phase voltage can be written as

N

page 74

Phase-to-Neutral Voltage

0=++ CBA iii

0)( =++ CBA iiidtd

0=++ CBA eee

0=++ CnBnAn vvv

)(31

CNBNANnN vvvv ++=

)(31

32

CNBNAnAn vvvv +−=

111 AAAn Lj IEV ω+=

In a three-phase, three-wire load

and

Phase-to-neutral voltage for phase is

+

−

VdThree-phase

three-leg inverter

L

~

id

A

iA

B

C

iB

iC

R = 0

eA(t)+

−n

N

nNv

PWM DC-AC Inverters


page 75

Phase-to-Neutral Variables of a Three-Phase Inverter

vAn1vAn

iripple, A

peak

dV32

dV31

t0

0 t

vAn

dV32

dV31

0vAn1

iripple, A

0 t

peak

Phase-to-neutral variables of a three-phase inverter.

(a) square wave (b) PWM

)(31

32

CNBNAnAn vvvv +−=

page 76

DC-Side Current id

)()()()()()(111

* titvtitvtitviV CnCBBnAAndd ++=

)quantity dca (cos3)]1201cos()1201cos(

]120cos()120cos()cos([cos2 1111*

dd

oo

d

ood

IV

IVtt

ttttV

IVi

==

−°+°++

−°−°−+−=

φ

φωω

φωωφωω

id

Id = id*0 t

Input DC current in a three-phase inverter.

PWM DC-AC Inverters


page 77

Conduction of Switches in 3-Phase Inverters

vAn

0 t

vAn1

vAn

iAiA1 (fundamental)

0 t

DA+ TA+ DA- TA-

Square-wave inverter: phase A waveforms.

Square-Wave Operation:

page 78

Conduction of Switches in 3-Phase InvertersvAN

iA

0 t

φ = 30°vBN

iB

0 t

vCN

iC0 t

(TA- , TB- , DC-) conducting (TA+ , TB + , DC +) conducting

PWM inverter waveforms: load power factor angle = 30° (lag).

PWM Operation:

PWM DC-AC Inverters


page 79

Short-Circuit in a 3-Phase PWM Inverter

iA+

−

vd

id = 0

iB

iCLoad

iA+

−

vd

id = 0

iB

iCLoad

page 80

Effect of Blanking Time t∆

TA+

TA-

+

−

Vd

N

A

DA+

DA-

t

t

t

t

t

t

t

0

0

0

0

0

0

0

vtri vcontrol

)ideal(cont +ATv

)ideal(cont −ATv

+ATvcont

−ATvcont ∆t

∆t

vAN

loss

ideal actual

actualideal

gainTs(iA > 0)

(iA < 0)

(a)

(b)

(c)

(e)

(d)

PWM DC-AC Inverters


page 81

Voltage Drop ∆id Induced by the Blanking Time

actualideal )()( ANAN vvv −=ε

⎪⎪⎩

⎪⎪⎨

⎧

+=∆

∆

∆

0

0

Ads

Ads

ANiV

Tt

iVTt

V

⎪⎪⎩

⎪⎪⎨

⎧

−=∆

∆

∆

0

0

Ads

Ads

BNiV

Tt

iVTt

V

⎪⎪⎩

⎪⎪⎨

⎧

+=∆−∆=∆

∆

∆

02

02

Ads

Ads

BNAN

oiV

Tt

iVTtVV

V

t0vAN ideal actual

Ts

(iA > 0)

t0vBN

ideal actual

Ts

t0

page 82

Effect of t∆ on Vo

TA+

TA-

+

−

VdA

DA+

DA-

(a)

TB+

TB-

B

DB+

DB-

N

iA

io+ −vo

iB

0

Vo

vcontrol

∆Vo

∆Vo

io < 0

io > 0

No blanking time(independent of io)

(b)

Effect of t∆ on Vo, where ∆Vo is defined as a voltage drop if positive.

PWM DC-AC Inverters


page 83

Effect of t∆ on the Sinusoidal Output

0

0

t

t

io

Vo(t)

ideal

actual

page 84

Programmed Harmonic Elimination Switching

-1.0

0

1.0

2/dAo

VV

Notch1 Notch2 Notch3 Fundamental

α2α1 α3 (π - α3) (π - α1)(π - α2)

(π + α1)

2ππ

2π

23π

ω1t

Notch4 Notch5 Notch60 20 40 60 80 1000

10

20

30

40

50

60

α3

α2

α1 22.06°

16.24°

Fundamental as a percentage of maximum fundamental voltage

Not

ch a

ngles

in d

egre

es

Programmed harmonic elimination of fifth and seventh harmonics.

273.142/)ˆ( 1 ==

πdAo

VV 188.1

2/)ˆ( max,1 =

d

Ao

VV

PWM DC-AC Inverters


page 85

Hysteresis Current Controlled PWM Inverter

TA+

TA-

+

−

Vd A

DA+

DA-

N

iA

0 t

TA-: on TA+: on

vAN

actual current iA

reference current iA*nt

iA*

iA

iε ABC

comparatortolerance band

Switch-mode

inverterΣ

Tolerance band current control.

page 86

Fixed-Frequency Current Controlled PWM Inverter

Σ ΣABC

Switch-mode

inverter

comparator

PI controller

0 t

iA*

iA+−

iε vcontrol

Feed forward

+

+

−

Fixed-frequency current control.

PWM DC-AC Inverters


page 87

Rectifier Mode of Operation

eA+

− n

+

−

Vd A

N

iA

~

VAn

j(ω1L)IA

(IA)q(IA)pIA

EAδ

δEA

j(ω1L)IAVAn(IA)q

(IA)p

IA

VAn

j(ω1L)IA

EAIA

(a)

(b)

(c)

(d)

Operation modes: (a) circuit; (b) inverter mode; (c) rectifier mode; (d) constant IA.

page 88

Summary of PWM Inverters

Switch-mode, voltage source dc-to-ac inverters are described that accept dc voltage source as input and produce either single-phase or three-phase sinusoidal output voltages at a low frequency relative to the switching frequency.

These dc-to-ac inverters can make a smooth transition into the rectification mode, where the flow of power reverses to be from the ac side to the dc side. This occurs, for example, during braking of an induction motor supplied through such an inverter.

PWM DC-AC Inverters


page 89


The sinusoidal PWM switching scheme allows control of the magnitude and the frequency of the output voltage. Therefore, the input to the PWM inverters is an uncontrolled, essentially constant dc voltage source. This switching scheme results in harmonic voltages in the range of the switching frequency and higher, which can be easily filtered out.

page 90


The square-wave switching scheme controls only the frequency of the inverter output. Therefore, the output magnitude must be controlled by controlling the magnitude of the input dc voltage source.

The square-wave output voltage contains low-order harmonics. A variation of the square-wave switching scheme, called the voltage cancellation technique, can be used to control both the frequency and the magnitude of the single-phase (but not three-phase) inverter output..

PWM DC-AC Inverters


page 91


Due to the harmonics in the inverter output voltage, the ripple in the output current (current ripples) does not depend on the level of power transfer at the fundamental frequency; instead the ripple depends inversely on the load inductance, which is more effective at higher frequencies.

In practice. If a switch turns off in an inverter leg, the turn-on of the others switch must be delayed by a blanking time, which introduces low-order harmonics in the inverter output.

page 92


There are many other switching schemes (PWM strategies) in addition to the sinusoidal PWM. For example, the programmed harmonic elimination switching technique can be easily implemented with the help of VLSI circuits to eliminate specific harmonics from the inverter output.

The current-regulated (current-mode) modulation allows the inverter output currents to be controlled directly by comparing the measured actual current with the reference current and using the error to control the inverter switched.

PWM DC-AC Inverters


page 93

Control of Inverter Output Voltage

triˆ1

Vma

Vd

k(ma) Vd invertervcontrolvo1 = k(ma)Vd

(rms, line-line)

for ma ≤ 1.0 k(ma) = 0.707 ma 1-phase= 0.621 ma 3-phase

k = 0.9 1-phase= 0.78 3-phase

vo2 = kVd(rms, line-to-line)

(a) (b)

Summary of inverter output voltage: (a) PWM operation (ma ≤ 1); (b) square-wave operation.

The relationship between the control input and the full-bridge inverter output magnitude can be summarized as shown in Fig. (a), assuming a sinusoidal PWM in the linear range of ma ≤ 1.0. For a square-wave switching, the inverter does not control the magnitude of the inverter output, and the relationship between the dc input voltage and the output magnitude is summarized in Fig. (b).

page 94

References

[1] J. Holtz, “Pulsewidth modulation – a survey,” IEEE Trans. on Ind. Electron., vol. 39, no. 5, pp. 410-420, Dec. 1992. [2] J. Holtz, "Pulsewidth modulation for electronic power conversion," Proc. of IEEE, vol. 82, no. 8, pp. 1194-1214,

Aug. 1994. [3] J. W. A. Wilson and J. A. Yeamans, “Intrinsic harmonics of idealized inverter PWM systems,” IEEE IAS Annual

Meeting Conf. Rec., pp. 967-973, 1976.[4] G. S. Buja, “Optimum output waveform in PWM inverters,” IEEE Trans. Ind. Appl., vol. 16, pp. 830-836, 1980. [5] S. R. Bowes and A. Midoun, "Microprocessor implementation of new optimal PWM switching strategies,"

IEE proceedings, vol. 13J, Pt.B., no5, Sep. 1988. [6] I. J. Pitel, S. N. Talukdar, and P. Wood, “Characterization of programmed-waveform pulse width modulation,” IEEE

Trans. Indus. Appli., vol. 16, no. 5, pp. 707-715, 1980.[7] T. Ohnishi and H. Okitsu, “A novel PWM technique for 3-phase inverter/converter,” IPEC Conf. Rec., Tokyo, pp.

384-395, 1983.[8] M. Boost and P. D. Ziogas, “State-of-the-art PWM techniques: a critical evaluation,” IEEE PESC Conf. Rec., pp.

425-433, 1986.[9] I. Rosa, “The harmonic spectrum of DC link currents in inverters,” Proceedings of the Fourth International PCI

Conference on Power Conversion, Intertec Communications, Oxnard, CA, pp. 38-52.[10] EPRI Report, “AC/DC power converter for batteries and fuel cells,” Project 841-1, Final Report, EPRI, Palo Alto,

CA, Sept. 1981.[11] T. G. Habetler, "A space vector-based rectifier regulator for AC/DC/AC converters," IEEE Trans. on Power

Electron., vol. 8, no. 1, pp. 30-36, Jan. 1993. [12] S. Vadivel, G. Bhuvaneswari, and G. Sridhara Rao, A unified approach to the real-time implementation of

microprocessor-based PWM waveforms," IEEE Trans. on Power Electron., vol. 6, no. 4, pp. 565-575, Oct. 1991.

Documents

Switch-Mode DC/AC Inverters - libvolume3.xyzlibvolume3.xyz/electrical/btech/semester6/advancedpowerelectronic… · motor 60 Hz AC filter capacitor switch-mode converter Switch-mode