Ch8 Inverters (converting DC to AC)

8-1 Introduction

․Converting DC to AC․Applications: adjustable-speed AC motor drives. Uninterruptible power supply (UPS).

AC appliances run from an automobile battery.

8-2 The full-bridge converter Fig 8-1 :

S1 and S4 should not be closed at the same time, nor should S2 and S3, otherwise, a short circuit would exist across the dc source

8-3 The square-wave Inverter Fig 8-2. (Waveform)

An inductive load presents some considerations in designing the switches in full – bridge circuit because the switch current must be bidirectional.

TtT

,BeR

Vdc

Tt,Ae

R

Vdc

tititi

Tt

t

nfo

2

20

2

:ti f forced current

:tin natural current R

L

In steady – state.

R

VdcinA

axBeR

VdcTi

inAeR

Vdcoi

oo

oo

Im

Im2

Im

RVdcaxB Im

TtT

,eR

VdcaxIm

R

Vdc

Tt,eR

VdcminI

R

Vdc

ti Tt

t

o

2

20

2

By symmetry ， Imax= － Imin =

2

Tio

2

T

eR

VdcinIm

R

dcV

2

2

1

1T

T

e

e

R

VdcinImaxIm

rms load current :

IsVdcP

ceresisloadRRrmsIP

dteR

Vdcin

R

Vdc

Tdtti

TIrms

dc

L

TtT

o

tan

Im21

2

2

2

00

2

If the switches are ideal , then Ldc PP.

.

Fig 8-3 Full – bridge inverter using BJTs

8-4 Fourier Series Analysis

With no dc component in the output

RIPP

)I

(IIrms

tnwsinIti

tnwsinVtv

rms,nn

nn

n

n

nrms,n

nonn

o

nonn

2

11

2

11

2

1

10

2

.

:

nharmonic

atimpedanceloadZ

Z

VI

n

n

nn

In square wave output

tnwn

Vdctv o

oddno sin

4

8-5 Total harmonic distortion (THD)

A quality of the AC output voltage or current. Assuming no dc component in the output

?THD

.V

VV

V

V

THD

I

rms,

rms,rms

rms,

rms,nn

V

1

21

2

1

2

2

8-7 Amplitude and harmonic control By adjusting the interval , the output voltage can be controlled. Fig.8-4

tnwsinVtv

VdcwtdVdcrmsV

onoddn

o

2

11 2

∵Half – wave symmetry

.cos4

cos4

sin2

1

VdcV

nn

VdctwdtnwVdcV oon

Harmonic content can also be controlled by adjusting α Harmonic n is eliminated if

90n

n90

Amplitude control and harmonic reduction may not be compatible. To control both amplitude and harmonic, it is necessary to have control over the dc input voltage.

Fig 8-5 A graphical representation of the integration in the Fourier series coefficient.

8-8 The half – bridge Inverter Fig 8-7

A square-wave output or bipolar pulse-width-modulated (PWM) output.

22

Vdcor

Vdcvo

The voltage across an open switch is twice the load voltage

VdcVdc

22

8-9 pulse-width-modulated (PWM) output Advantage： Reduced filter requirements to decrease harmonics and the control of the output voltage amplitude can be realized Disadvantage： more complex control circuits for switches and increased loss

es due to more frequent switching.

Sinusoidal PWM requires:(1). a reference (modulating or control) signal-sinusoidal.(2). a carrier (triangular wave) signal that controls the switching fre.

Bipolar switching: Fig 8-8 Vdcvo

triesino

triesino

vv,Vdcv

vv,Vdcv

S1 and S2 are on when Vsine>VtriS3 and S4 are on when Vsine < Vtri

Unipolar Switching：

Vdc,,Vdcvo 0

S1 is on when Vsine>Vtri S2 is on when -Vsine<Vtri S3 is on when -Vsine>Vtri S4 is on when Vsine<Vtri

Fig 8-9One (First)

Another (second) Fig 8-10

S1 is on when Vsine>Vtri (high fre) S4 is on when Vsine<Vtri (high fre) S2 is on when Vsine> 0 (low fre)

S3 is on when Vsine< 0 (low fre)

8-10 PWM definitions and considerations.

(1) Frequency modulation ratio ：esin

tri

reference

carrierf f

f

f

fm

The Fourier series of the PWM output voltage has a fundamental fre. which is the same as that of the reference signal. Harmonic frequencies exist at and around multiple of the switching fre. A simple low-pass filter can be effective in removing those (harmonics). Higher losses in switches

(2) Amplitude modulation ratio ：tri,m

esin,m

carrior,m

reference,ma V

V

V

Vm

If , then , (linearly).1am VdcmV a1

If , the amplitude of the output increases with ,but not linearly.

1am am

(3). Swithes： carrying current in either direction. Feedback.diode allowing for switching times in the control.

(4). Reference voltage： sinusoidal, non-sinusoidal

8-11 PWM harmonics Bipolar switching： (Fig 8-8)

If mf =odd integer, the PWM output then exhibits odd symmetry

tnwsinVtv onn

o

1

For the k-th pulse of the PWM output. (Fig 8-11)

/2.ToverpulsepV

TtwdtnwtvT

V

nnnn

Vdc

twdtnwVdctwdtnwVdcV

pnk

p

k

ooo

T

n

kkkk

oo

k

kkoo

kk

knk

:

2,sin4

)](cos2cos[cos2

sinsin2

1

2

0

1

1

,

Normalized frequency spectrum ： Fig 8-12 Normalized Fourier coefficients ： Table 8-3

dc

nV

V

Unipolar switching： (Fig 8-9) If mf =even integer, some harmonics that were in the spectrum for the bipolar scheme are absent. (seeing Fig 8-13 and Table 8-5)

8-13 Three-phase inverters

Six-step Inverter： Fig 8-17

CABCAB v,v,v 為 Vdcor,,Vdc 0

00

00

00

ACCA

CBBC

BAAB

vvv

,vvv

,vvv

Because of the six steps in the output waveform for the line-to-neutralvoltage resulting from the six switching transitions per period.

)3(

)2(

)1(

00

00

00

NNCC

NBNB

NANA

vvv

vvv

vvv

000 CBA v,v,v 為 +Vdc or 0

(1)+(2)+(3) CoBoAoNo vvvv 3

1

)vvv( CNBNAN 0

BoAoCoNoCoCN

AoCoBoNoBoBN

CoBoAoNoAoAN

vvvvvv

vvvvvv

vvvvvv

3

1

3

23

1

3

23

1

3

2

3

2

32

3

2

6

4

ncosncosn

VdcV

ncosn

VdcV

NL,n

LL,n

Where n=1, 6k , k=1 、 2…

1

(fundamental)

(harmonics)

%31VTHD for line-to-line and line-to-neutral voltages.

ITHD is load dependent and is smaller for a R-L load.

Output fre. can be controlled by changing the switching fre..Output voltage can be controlled by adjusting the DC input voltage.

PWM three-phase Inverters Fig 8-18

S1 is on when triref,A vv

S3 is on when triref,B vv

S5 is on when triref,C vv

S2 is on when triref,C vv

S4 is on when triref,A vv

S6 is on when triref,B vv

Harmonics will be minimized if the carrier fre. is chosed to be an odd triple multiple of the reference fre. (that is 3,9,15,…times the reference).

For line-to-line voltage,

3sin

2sin

3sin

2sin

3

3

23

233

nnVB

nnVA

BAV

nn

nn

nnn

….. 三相

nk

P

kn VV

1

( 參考 P.313) …… 單相

Table 8-8 Significant amplitude coefficients

8-15 Induction motor speed control

Synchronous speed ： PS

2

S1ip ： S

rSS

ω ： electric fre. P ： number of poles ωr ： rotor speed

If the applied electrical fre. is changed, the motor speed will change proportionally.

To avoid the magnetic flux in the air gap saturated, constant should be held .

f

V

Fig 8-19

The six-step inverter can be used for this application if the dc input is adjustable

Fig 8-20

If the DC source is not controllable, a DC-DC converter may be inserted between the DC source and the inverter. The PWM inverter is also another selection for this application.