Electrical Drives Lectures

Electrical Drives

Electrical Drives

MEP 1422

2004/2005-02

Module 1

.

Introduction to drives:

Elements in electrical drives, overview of DC and AC drives.

Torque equations,

Components of load torque, torque characteristics. Four-quadrant

operation

Notes on

Introduction to Electromechanical Energy Conversion

Module 2

Converters in electric drive systems:

Controlled rectifier, Linear scheme, Non-linear scheme,

Switched-mode converters - average model and transfer function,

Two-quadrant converters, Four-quadrant converters, Bipolar

switching, Unipolar switching,

Current-controlled converters, Fixed switching frequency control,

Hysteresis control

Example of Simulink file

for 2-Q converter (switching and average

model)

Current ripple

in 4 Q converter

Space Vector Modulation

(SVM)

Module 3

DC motor drives

DC drives

in power point format, in

.pdf

Construction, modeling and transfer function, Converters for DC

drives

– quadrant of operations.

MATLAB

–based controller design method –

here

Linear analysis

in Simulink

Large signal simulation using SIMULINK

–

here

Electrical Drives

Module 4

.

Induction motor drives

Dynamic model of induction machine

Construction and principle of operations,

Speed Control-

constant V/f, Scalar control

– problems at low speed,

current

Simulink

example

on open-loop constant V/Hz using SIMULINK

s-function for IM simulation

Compiled with Borland C -

here

Current controlled and voltage boost, open-loop and closed-loop

control.

Field-oriented control of IM:

Rotor flux orientation

Stator flux orientation

Simulink example on

indirect FOC IM

– requires

imch.dll

PPoint

for principles of direct torque control and in

pdf

Direct Torque Control using SIMULINK

and the required

*.dll files

for the S-function

! " # $ % & ' ( ) ! ) * * & + & µ ( ,

) ) * * - # $ % # . # # - # & & & ! / # 0 # 1 & 2 $ % 0 0 . ) / 0 $ % # # # # * ( 0 0 & ' 0 & $ % 2 & ' 0

! ! -# 0 * * 0 3 # 4567 & # $ % # # # # 4587 2 $ $ & %% # # 9 $/: % $ * % # ) • 0 • # $ % • 0 ; • / # 0 $ % 0 • . 0 0 • # 0 0

" #$ #$ % & ' , / * . <' = 7 $ % * = ) $> % * = , $> % < = ) $ ?_% ω = $ ' % . ω' = 7 * = * / 0 * * @ . ( ! ω ω ω ω )

(

)

dt J d T T m l e ω + =

(

)

dt J d ωm dt d J T T m l e ω + = J 1 ω θ A * *

* ! #$ + , 1 B $% / */ / 0 */ 1 *" " $* = -ω% * * & 0 . * * $ % 2 *2 *2 *2= Cω? . *2 *" $ % * *, *, . . * * * */ ω) ω) */ * ω

, * * ? , - * C * * = + ω+ + ω 2 * Cω? ω + + ω =< * -* _, ! #$ -@ * # % $ .#$ ! ! ' * ω * D * $ % * * E E * . .. ... ." ω ω ω ω ω T * * * *

/ F . -* $ = * ω% ! * F .. * * G D * * * F ... * * * * * F ." * * G D * * *

G.K. Dubey, “Fundamental of Electrical Drives”, Narosa, 1994.

E L E C T R O M E C H A N I C A L E N E R G Y C O N V E R S I O N E l e c t r o m e c h a n i c a l e n e r g y c o n v e r s i o n p r o c e s s i n v o l v e s t h r e e f o r m s o f e n e r g y : e l e c t r i c a l , m a g n e t i c f i e l d a n d m e c h a n i c a l . I n r o t a t i n g e l e c t r i c a l m a c h i n e s , e n e r g y i s c o n t i n u o u s l y c o n v e r t e d f r o m e l e c t r i c a l t o m e c h a n i c a l , o r v i c e v e r s a . E l e c t r i c a l m o t o r s c o n v e r t s e l e c t r i c a l e n e r g y t o m e c h a n i c a l e n e r g y a n d i t i s r e v e r s e d i n t h e c a s e o f g e n e r a t o r s . I n b o t h c a s e s , m a g n e t i c f i e l d a c t s a s a m e d i u m i n t h e p r o c e s s o f e l e c t r o m e c h a n i c a l e n e r g y c o n v e r s i o n . W e w i l l l o o k ( o r r e v i e w ) t h e p r o c e s s o f e l e c t r o m e c h a n i c a l e n e r g y c o n v e r s i o n o f a s i m p l e t r a n s l a t i o n a l s y s t e m f o r a n o n -l i n e a r a n d l i n e a r m a g n e t i c s y s t e m . W e w i l l t h e n a p p l y t h i s b a s i c p r i n c i p l e t o a r o t a t i n g m a c h i n e . E x a m p l e o f e l e c t r o m e c h a n i c a l s y s t e m T h e c h a r a c t e r i s t i c o f t h e f l u x l i n k a g e a n d c u r r e n t (λ -i ) o f a s y s t e m s h o w n i n F i g 1 i s d e t e r m i n e d b y t h e B -H c h a r a c t e r i s t i c o f t h e c o r e a n d t h e l e n g t h o f t h e a i r -g a p . W i t h s m a l l a i r -g a p l e n g t h , g , t h e λ -i c h a r a c t e r i s t i c i s d o m i n a t e d b y t h e B -H c h a r a c t e r i s t i c o f t h e c o r e w h i c h h a s a n o n -l i n e a r c h a r a c t e r i s t i c d u e t o t h e c o r e m a g n e t i c s a t u r a t i o n . W i t h l a r g e g , h o w e v e r , t h e l i n e a r m a g n e t i c c h a r a c t e r i s t i c o f t h e a i r -g a p w i l l d o m i n a t e . T h u s f o r l a r g e a i r -g a p s y s t e m t h e λ -i c u r v e o f t h e s y s t e m d i s p l a y s a l i n e a r c h a r a c t e r i s t i c . I f a l i n e a r s y s t e m i s a s s u m e d , a l l o f t h e m m f d r o p s a p p e a r a c r o s s t h e a i r -g a p . I n o t h e r w o r d s , i t i s a s s u m e d t h a t t h e r e l u c t a n c e o f t h e c o r e i s n e g l i g i b l y s m a l l c o m p a r e d t o t h a t o f t h e a i r -g a p ’ s r e l u c t a n c e . T h i s a s s u m p t i o n i s b a s e d o n t h e f a c t t h a t t h e m a g n e t i c p e r m e a b i l i t y o f t h e c o r e i s m u c h l a r g e r t h a n t h e a i r -g a p p e r m e a b i l i t y . T h e λ -i c u r v e s f o r d i f f e r e n t a i r -g a p v a l u e s a r e t h e r e f o r e l i n e a r . F i g . 1 − − n o n -l i n e a r s y s t e m l i n e a r s y s t e m F i g . 2 T h e d i f f e r e n t i a l r e l a t i o n b e t w e e n t h e 3 f o r m s o f e n e r g y e x i s t s i n t h e s y s t e m c a n b e w r i t t e n a s : d W e = d W f + d W m ( 1 ) W h e r e d W e – d i f f e r e n t i a l c h a n g e i n e l e c t r i c a l e n e r g y d W f -d i f f e r e n t i a l c h a n g e i n f i e l d e n e r g y d W m -d i f f e r e n t i a l c h a n g e i n m e c h a n i c a l e n e r g y I f t h e p o s i t i o n o f t h e m o v i n g p a r t i s f i x e d ( a i r -g a p l e n g t h i s f i x e d , t h u s d W m = 0 ) a n d t h e c u r r e n t i n t h e c o i l i s i n c r e a s e d f r o m 0 t o i x_, t h e f i e l d e n e r g y w i l l i n c r e a s e a n d i s g i v e n b y : d W e = e . i d t = d W f ( 2 ) S u b s t i t u t i n g e = dλ / d t , d W f = i dλ ( 3 ) I f t h e f l u x l i n k a g e i n c r e a s e d f r o m 0 t o λ x_, t h e s t o r e d e n e r g y c a n b e w r i t t e n a s : λ λ = x 0 f i d W ( 4 ) λ x i x λ c o -e n e r g y e n e r g y λ λ F i g . 3

T h e c o -e n e r g y , w h i c h i s u s e d l a t e r t o c a l c u l a t e t h e f o r c e , i n t h i s p a r t i c u l a r e x a m p l e i s d e f i n e d a s : λ = i x 0 f d i 'W ( 5 ) I t s h o u l d b e n o t e d t h a t f o r a l i n e a r s y s t e m , W f = W f_’ I f t h e m o v i n g p a r t i s a l l o w t o m o v e s l o w l y , f r o m x = x 1 t o x = x 2_, s u c h t h a t t h e a i r -g a p i s r e d u c e d , t h e r a t e o f c h a n g e o f f l u x l i n k a g e w i l l b e v e r y s m a l l d u r i n g t h i s m o v e m e n t a n d h e n c e t h e c u r r e n t c a n b e a s s u m e d t o b e c o n s t a n t . F i g . 4 T h e m e c h a n i c a l f o r c e a s s o c i a t e d w i t h t h i s m o v e m e n t c a n b e o b t a i n e d i f t h e c h a n g e i n m e c h a n i c a l e n e r g y i s k n o w n . T h u s , d W m = d W e -d W f ( 6 ) D u r i n g t h e m o t i o n , d W e = e . i d t = i dλ . H e n c e λ λ λ = 2 x 1 x e i d W T h e c h a n g e i n t h e s t o r e d f i e l d e n e r g y c a n b e o b t a i n e d b y c a l c u l a t i n g t h e d i f f e r e n c e i n s t o r e d e n e r g y b e t w e e n t h e t w o p o s i t i o n s . I t c a n b e s h o w n g r a p h i c a l l y t h a t W m i s g i v e n b y t h e s h a d e d a r e a o f F i g . 4 w h i c h e s s e n t i a l l y i s t h e i n c r e a s e i n c o -e n e r g y . T h u s : d W m = d W f_’ S i n c e d W m = f d x , t h e m e c h a n i c a l f o r c e c a n b e c a l c u l a t e d a s : t t a n c o n s i f m x )x ,i (' W f = ∂ ∂ = ( 7 ) I f t h e m o v e m e n t o f t h e m o v i n g p a r t i s v e r y f a s t ( i . e . f o r t h e s a m e d i s p l a c e m e n t b u t f o r a v e r y s h o r t t i m e ) , t h e c h a n g e i n f l u x l i n k a g e c a n b e a s s u m e d n e g l i g i b l e . H o w e v e r , t h e r a t e o f c h a n g e o f t h e f l u x l i n k a g e w i t h t i m e i s f i n i t e a n d h e n c e c a u s e s t h e c u r r e n t t o d e c r e a s e d u r i n g t h i s m o v e m e n t . I t c a n b e g r a p h i c a l l y s h o w n t h a t t h e m e c h a n i c a l e n e r g y i s g i v e n b y t h e s h a d e d λ λ a r e a o f F i g 5 , w h i c h i s a r e d u c t i o n i n f i e l d e n e r g y . T h u s t h e m e c h a n i c a l f o r c e i s g i v e n b y : t t a n c o n s f m x )x ,i ( W f = λ ∂ ∂ − = ( 8 ) I f t h e d i f f e r e n t i a l m o v e m e n t i s s m a l l , t h e s h a d e d a r e a o f F i g 4 a n d F i g 5 i s t h e s a m e . H e n c e t h e f o r c e c a l c u l a t e d u s i n g e q u a t i o n ( 7 ) a n d ( 8 ) w i l l b e t h e s a m e . F i g . 5 L i n e a r s y s t e m F o r l i n e a r s y s t e m , t h e f l u x l i n k a g e i s p r o p o r t i o n a l t o t h e c u r r e n t , w h e r e t h e c o n s t a n t o f p r o p o r t i o n a l i t y i s t h e i n d u c t a n c e o f t h e c o i l . T h e i n d u c t a n c e h o w e v e r d e p e n d s o n t h e p o s i t i o n , x . T h u s , λ = L ( x ) i ( 9 ) T h e c o -e n e r g y i s g i v e n b y : )x (L i 2 1 d i ' W 2 i 0 f = λ = ( 1 0 ) U s i n g e q u a t i o n ( 7 ) , d x )x ( d L i 2 1 x )x ,i (' W f 2 t t a n c o n s i f m = ∂ ∂ = = ( 1 1 ) R o t a t i n g m a c h i n e s F i g 6 s h o w s a g e n e r a l r o t a t i n g m a c h i n e w i t h s a l i e n t s t a t o r a n d s a l i e n t r o t o r . B o t h s t a t o r a n d r o t o r a r e e x i t e d ( d o u b l y – f e d ) . W e a r e i n t e r e s t e d i n o b t a i n i n g t h e e l e c t r o m a g n e t i c t o r q u e e x p r e s s i o n o f t h e s y s t e m . W e c a n d o t h i s b y o b t a i n i n g t h e e x p r e s s i o n f o r t h e c o – e n e r g y ( o r e n e r g y ) a n d d i f f e r e n t i a t e i t w i t h r e s p e c t t o x f o r c o n s t a n t c u r r e n t ( o r c o n s t a n t f l u x ) . λ λ

F i g . 6 W i t h n o r o t a t i o n ( r o t o r n o t m o v i n g ) , t h e s t o r e d f i e l d e n e r g y c a n b e c a l c u l a t e d a s : d W f = e s_i s d t + e r_i r d t ( 1 2 ) S u b s t i t u t i n g e s = dλ s_/ d t a n d e r = dλ r_/ d t , d W f = i s dλ s + i r dλ r ( 1 3 ) T h e f l u x l i n k a g e o f t h e s t a t o r w i n d i n g c a n b e e x p r e s s e d i n t e r m s o f s t a t o r s e l f i n d u c t a n c e a n d m u t u a l i n d u c t a n c e : λ s = L s s_i s + L s r_i r ( 1 4 ) T h e f i r s t t e r m o f ( 1 4 ) i s t h e f l u x l i n k a g e o f t h e s t a t o r w i n d i n g c a u s e d b y t h e s t a t o r c u r r e n t w h e r e a s t h e s e c o n d t e r m i s c a u s e d b y t h e r o t o r c u r r e n t . S i m i l a r l y , t h e f l u x l i n k a g e o f t h e r o t o r w i n d i n g c a n b e e x p r e s s e d a s , λ r = L r r_i r + L s r_i s ( 1 5 ) S u b s t i t u t i n g ( 1 4 ) a n d ( 1 5 ) i n t o ( 1 3 ) , d W f = L s s_i s_d i s + L r r_i r_d i r + L s r d ( i s_i r₎ ( 1 6 ) F o r a l i n e a r s y s t e m , W f = W f_’ . I t c a n b e s h o w n t h a t f o r r o t a t i o n a l s y s t e m s , = θ∂ θ ∂ = ( 1 7 ) T h u s t h e t o r q u e i s g i v e n b y : θ + θ + θ = d dL i i 2 1 d dL i 2 1 d dL i 2 1 T s r r s r r 2 r s s 2 s ( 1 8 ) B a s e d o n e q u a t i o n ( 1 8 ) , t w o t y p e s o f t o r q u e c a n b e c l a s s i f i e d : θ i ) R e l u c t a n c e t o r q u e ( t h e f i r s t t w o t e r m s o f e q u a t i o n ( 1 8 ) ) . I t i s c a u s e d b y a t e n d e n c y o f t h e i n d u c e d p o l e t o a l i g n w i t h t h e e x c i t e d p o l e s u c h t h a t m i n i m u m r e l u c t a n c e i s p r o d u c e d . T h e t o r q u e o n l y e x i s t s i f t h e s t a t o r o r r o t o r ( o r b o t h ) s e l f i n d u c t a n c e s d e p e n d s o n t h e r o t o r p o s i t i o n . T h i s c a n e x i s t s i f : 1 ) b o t h s t a t o r a n d r o t o r a r e s a l i e n t , 2 ) e i t h e r s t a t o r o r r o t o r i s s a l i e n t . I n o t h e r w o r d s , i n a c y l i n d r i c a l m a c h i n e ( w h e r e b y b o t h s t a t o r a n d r o t o r a r e n o n -s a l i e n t ) r e l u c t a n c e t o r q u e w i l l n o t e x i s t . F u r t h e r i t c a n b e s e e n t h a t b o t h s t a t o r a n d r o t o r n e e d n o t t o b e e x c i t e d a t t h e s a m e t i m e . i i ) A l i g n m e n t t o r q u e ( t h e t h i r d t e r m o f e q u a t i o n ( 1 8 ) ) . I t i s c a u s e d b y a t e n d e n c y o f t h e e x c i t e d r o t o r t o a l i g n w i t h e x c i t e d s t a t o r . B o t h w i n d i n g s m u s t b e e x c i t e d . T h e m u t u a l i n d u c t a n c e d e p e n d s o n r o t o r p o s i t i o n r e g a r d l e s s o f w h e t h e r t h e s t a t o r o r r o t o r i s s a l i e n t o r n o t . I n o t h e r w o r d s , t h e a l i g n m e n t t o r q u e e x i s t s e v e n i f b o t h s t a t o r a n d r o t o r i s n o t s a l i e n t . I n i n d u c t i o n m a c h i n e s , r o t o r c u r r e n t i s p r o d u c e d t h r o u g h i n d u c t i o n r a t h e r t h a n e x c i t a t i o n b y e x t e r n a l c i r c u i t , a s i n t h e c a s e o f c y l i n d r i c a l s y n c h r o n o u s m a c h i n e s . S t a t o r -n o n -s a l i e n t R o t o r – s a l i e n t - S t a t o r s e l f i n d u c t a n c e d e p e n d s o n r o t o r p o s i t i o n - R o t o r s e l f i n d u c t a n c e d o e s n o t d e p e n d o n r o t o r p o s i t i o n S t a t o r -s a l i e n t R o t o r – s a l i e n t - S t a t o r s e l f i n d u c t a n c e d e p e n d s o n r o t o r p o s i t i o n - R o t o r s e l f i n d u c t a n c e d e p e n d s o n r o t o r p o s i t i o n S t a t o r -s a l i e n t R o t o r – n o n -s a l i e n t - S t a t o r s e l f i n d u c t a n c e d o e s n o t d e p e n d o n r o t o r p o s i t i o n - R o t o r s e l f i n d u c t a n c e d e p e n d s o n r o t o r p o s i t i o n S t a t o r -s a l i e n t R o t o r – n o n -s a l i e n t - S t a t o r s e l f i n d u c t a n c e d o e s n o t d e p e n d o n r o t o r p o s i t i o n - R o t o r s e l f i n d u c t a n c e d o e s n o t d e p e n d o n r o t o r p o s i t i o n

C O N V E R T E R S I N E L E C T R I C D R I V E S Y S T E M S C O N T R O L L E D R E C T I F I E R W e h a v e s e e n i n p r e v i o u s c o u r s e ( u n d e r g r a d u a t e c o u r s e ) t h a t a r e l a t i o n b e t w e e n t h e a v e r a g e v o l t a g e a n d t h e f i r i n g a n g l e ( o r d e l a y a n g l e ) o f a s i n g l e -p h a s e c o n t r o l l e d r e c t i f i e r i s g i v e n b y : w h e r e α i s t h e d e l a y a n g l e , V m i s t h e p e a k i n p u t v o l t a g e a n d V a i s t h e a v e r a g e v o l t a g e . N o t e t h a t t h i s r e l a t i o n i s o n l y v a l i d f o r c o n t i n u o u s c u r r e n t m o d e . I t d e s c r i b e s t h e ‘ a v e r a g e ’ b e h a v i o r o f t h e r e c t i f i e r o v e r a p e r i o d o f t h e o u t p u t v o l t a g e . T h e d y n a m i c c h a r a c t e r i s t i c o f t h e c o n t r o l l e d r e c t i f i e r i s h o w e v e r v e r y n o n – l i n e a r w h i c h c a n b e d e s c r i b e d b y n o n – l i n e a r d i f f e r e n t i a l e q u a t i o n s . I n o r d e r t o s i m p l i f y t h e d e s i g n e d o f t h e c o n t r o l l e r c o n t a i n i n g c o n t r o l l e d – r e c t i f i e r c i r c u i t , a n a p p r o x i m a t i o n u s i n g t h e a v e r a g e v a l u e i s n o r m a l l y u s e d . T h i s a p p r o x i m a t i o n i s h o w e v e r v a l i d p r o v i d e d t h a t t h e b a n d w i d t h o f t h e c o n t r o l l o o p i s m a i n t a i n e d w e l l b e l o w h a l f o f t h e m a x i m u m t i m e f o r t h e a v e r a g e v o l t a g e t o c h a n g e . F o r i n s t a n c e , i f a 3 -p h a s e s y s t e m , 5 0 H z s y s t e m i s u s e d a s t h e i n p u t t o t h e f u l l -w a v e c o n t r o l l e d r e c t i f i e r , t h e n , t h e t i m e t a k e n f o r t h e a v e r a g e v o l t a g e t o c h a n g e v a r i e s b e t w e e n 0 t o 3 . 3 3 m s ( ( 1 / 5 0 ) / 6 ) . T h e a v e r a g e t i m e o f 3 . 3 3 m s / 2 = 1 . 6 7 m s i s t a k e n . I f t h i s d e l a y i s n o t t o b e u s e d i n t h e m o d e l , t h e b a n d w i d t h o f t h e d r i v e m u s t b e m a d e m u c h s m a l l e r t h a n 6 0 0 H z . T h e S C R s a r e n o r m a l l y t r i g g e r e d b a s e d o n t h e c o n t r o l s i g n a l g e n e r a t e d , f o r e x a m p l e , b y a c u r r e n t c o n t r o l l e r . D e p e n d i n g o n t h e f i r i n g c i r c u i t u s e d , a l i n e a r o r a n o n – l i n e a r r e l a t i o n b e t w e e n v c a n d V a c a n b e o b t a i n e d . α π = α α = = α π = α = π = M O D E L I N G O F S W I T C H -M O D E C O N V E R T E R S I N E L E C T R I C D R I V E S I n t r o d u c t i o n M o d e l i n g i s a s i m p l i f i e d r e p r e s e n t a t i o n o f a p h y s i c a l s y s t e m . I n e l e c t r i c a l e n g i n e e r i n g , p h y s i c a l s y s t e m s a r e n o r m a l l y m o d e l e d u s i n g m a t h e m a t i c a l e q u a t i o n s . T h e c o m p l e x i t y o f t h e d e v e l o p e d m o d e l o f p o w e r e l e c t r o n i c c o n v e r t e r s w i l l d e p e n d o n t h e a p p l i c a t i o n s o f t h e m o d e l . F o r i n s t a n c e , a m o d e l f o r a s w i t c h i n g d e v i c e u s e d t o a n a l y z e i t s s w i t c h i n g c h a r a c t e r i s t i c o r s w i t c h i n g l o s s e s i s d i f f e r e n t f r o m a m o d e l d e v e l o p u s e d t o s t u d y t h e f u n d a m e n t a l b e h a v i o r o f a c o n v e r t e r c o n t a i n i n g t h a t p a r t i c u l a r s w i t c h i n g d e v i c e . H e r e w e w i l l l o o k o n h o w s w i t c h -m o d e c o n v e r t e r s u s e d i n D C d r i v e s a r e m o d e l e d . T h e a p p l i c a t i o n o f o u r m o d e l i s i n t h e d e s i g n i n g o f l i n e a r c o n t r o l l e r s f o r d r i v e s y s t e m s u s i n g l i n e a r c o n t r o l s y s t e m t h e o r y . W e t h e r e f o r e n e e d t o o b t a i n t h e l i n e a r m o d e l s o f t h e c o n v e r t e r s , i . e . w e n e e d t o e s t a b l i s h a l i n e a r r e l a t i o n b e t w e e n t h e c o n t r o l s i g n a l a n d t h e a v e r a g e o u t p u t v o l t a g e . T w o t y p i c a l s w i t c h -m o d e c o n v e r t e r s u s e d i n D C d r i v e s a r e t h e 2 -q u a d r a n t a n d 4 -q u a d r a n t c o n v e r t e r s s h o w n b e l o w . W e w i l l a s s u m e t h a t t h e c o n v e r t e r s o b t a i n e d t h e s w i t c h i n g s i g n a l s f r o m a c o m p a r i s o n b e t w e e n c o n t r o l s i g n a l v c a n d a t r i a n g u l a r w a v e f o r m s . T w o -q u a d r a n t c o n v e r t e r A s i n a l l o t h e r c o n v e r t e r s , t h e s t a t u s o f t h e u p p e r a n d l o w e r s w i t c h e s i n a l e g , m u s t a l w a y s c o m p l e m e n t , i . e . i f t h e u p p e r s w i t c h i s o n , t h e l o w e r s w i t c h m u s t b e o f f o r v i c e v e r s a -t h u s o n l y o n e c o n t r o l s i g n a l i s r e q u i r e d t o c o n t r o l a l e g o f a t w o -q u a d r a n t c o n v e r t e r . I f t h e u p p e r s w i t c h i s O N , t h e o u t p u t v o l t a g e , v o e q u a l s V d c a n d i f t h e l o w e r s w i t c h i s O N v o = 0 . T h e i n s t a n t a n e o u s o u t p u t v o l t a g e w i l l s w i n g b e t w e e n V d c a n d 0 , h o w e v e r i t s a v e r a g e v a l u e d e p e n d s o n h o w l o n g t h e s w i t c h u p p e r ( o r l o w e r ) s w i t c h i s O N . ω F o u r -q u a d r a n t + v a – T w o -q u a d r a n t + V a

-W e w i l l a s s u m e t h e c o n t r o l s i g n a l s f o r t h e s w i t c h e s a r e o b t a i n e d a s a r e s u l t o f c o m p a r i s o n b e t w e e n t h e c o n t r o l s i g n a l a n d a t r i a n g u l a r T h e o u t p u t o f t h e c o m p a r a t o r i s o b t a i n e d a s f o l l o w s : w h e n v c > v t r i_, u p p e r s w i t c h O N ( 1 ) w h e n v c < v t r i_, l o w e r s w i t c h O N O b v i o u s l y , t h e w a v e f o r m o f v a w i l l f o l l o w t h a t o f q . T h e i n s t a n t a n e o u s v a l u e o f v a i s g i v e n b y : v a = q ( V d c₎ T h e a v e r a g e v a l u e o f v a w i l l d e p e n d o n t h e d u t y r a t i o o f q a n d t h e d u t y r a t i o o f q i n t u r n d e p e n d s o n t h e c o n t r o l s i g n a l v c_. W e c a n o b t a i n t h e r e l a t i o n b e t w e e n t h e a v e r a g e v o l t a g e V a a n d t h e d u t y r a t i o d b y c a l c u l a t i n g t h e a v e r a g e v a l u e o f v a i n t e r m s o f d . W h e r e d = t o n_/ T ( 2 ) d i s i n f a c t a n a v e r a g e v a l u e o f q o v e r a c y c l e a n d t h e r e f o r e h a v e a r a n g e o f b e t w e e n 0 a n d 1 , t h u s , ( 3 ) = 0 1 q dc dT 0 dc a dV dt V T 1 V s = = dt q T ₁ d triT t t tri + = ! I f t h e t r i a n g u l a r f r e q u e n c y i s h i g h a n d t h e r e f o r e i s m u c h l a r g e r t h a n t h e c o n t r o l s i g n a l , d c a n b e a s s u m e d c o n t i n u o u s . H o w e v e r w h e n s e l e c t i n g t h e b a n d w i d t h o f t h e c l o s e d -l o o p s y s t e m , t h e d i s c r e t e v a l u e s o f d m u s t b e t a k e n i n t o a c c o u n t , i . e . t h e b a n d w i d t h m u s t b e l i m i t e d t o o n e o r t w o o r d e r l o w e r t h a n t h e t r i a n g u l a r f r e q u e n c y . T h e r e l a t i o n b e t w e e n d a n d v c i s o b t a i n e d a s f o l l o w s : W h e n v c = V t r i , p , d = 1 , w h e n v c = -V t r i , p_, d = 0 . A s s u m i n g d i s c o n t i n u o u s , t h e r e l a t i o n b e t w e e n d a n d v c i s o b t a i n e d a s : ( 4 ) T h e r e l a t i o n b e t w e e n v c a n d V a c a n b e o b t a i n e d b y s u b s t i t u t i n g ( 4 ) i n t o ( 2 ) , ( 5 ) I f w e w a n t t o i n c l u d e t h e c o n v e r t e r i n t o o u r c l o s e d -l o o p m o d e l o f a D C d r i v e s y s t e m , w e n e e d t o o b t a i n t h e s m a l l s i g n a l t r a n s f e r f u n c t i o n b e t w e e n v c a n d V a_. T h i s i s d o n e b y i n t r o d u c i n g s m a l l s i g n a l p e r t u r b a t i o n i n V a a n d v c_. ( 6 ) S e p a r a t i n g t h e d c a n d a c c o m p o n e n t s , ! p, triV2 cv 5. 0 d + = c p, tri dc dc a v V2 V V5 .0 V + = ( ) ( ) c c p, tri dc dc a a v ~ v V2 V V5 .0 v ~ V + + = +

" D C : ( 7 ) A C : ( 8 ) B y t a k i n g L a p l a c e t r a n s f o r m o f e q u a t i o n ( 8 ) , t h e s m a l l s i g n a l t r a n s f e r f u n c t i o n b e t w e e n v c a n d V A c a n b e o b t a i n e d . F o u r -q u a d r a n t c o n v e r t e r T h e m o d e l d e v e l o p e d f o r t h e t w o -q u a d r a n t c o n v e r t e r c a n b e u s e d a s a b u i l d i n g b l o c k i n d e v e l o p i n g t h e m o d e l f o r t h e f o u r -q u a d r a n t c o n v e r t e r . A s i l l u s t r a t e d i n t h e f i g u r e b e l o w , t h e 4 -q u a d r a n t c o n v e r t e r i s c o m p o s e d o f t w o l e g s , w i t h e a c h l e g s i m i l a r t o t h a t o f t h e 2 -q u a d r a n t c o n v e r t e r . W e w i l l c o n s i d e r t w o s w i t c h i n g s c h e m e s n o r m a l l y e m p l o y e d : ( 1 ) B i p o l a r s w i t c h i n g s c h e m e ( 2 ) u n i p o l a r s w i t c h i n g s c h e m e . T h e i n s t a n t a n e o u s v o l t a g e v a c a n b e m a d e e i t h e r e q u a l s V d c , -V d c o r 0 . V a = V d c w h e n Q 1 a n d Q 2 a r e O N v a = -V d c w h e n Q 3 a n d Q 4 a r e O N v a = 0 w h e n c u r r e n t f r e e w h e e l s t h r o u g h Q a n d D T h e r e f o r e t h e o u t p u t v o l t a g e v a c a n s w i n g b e t w e e n V d c a n d – V d c_, V d c a n d 0 o r 0 a n d V d c_, w h i c h i s d e t e r m i n e d b y t h e s w i t c h i n g s c h e m e c h o s e n : c p, tri dc dc a v V2 V V5 .0 V + = c p, tri dc a v ~ V2 V v ~ = p, triV2 dcV # $ # $ % & + v a – ' ' ' ' " ( ( B i p o l a r s w i t c h i n g L e g A a n d L e g B o b t a i n e d t h e s w i t c h i n g s i g n a l s f r o m t h e s a m e c o n t r o l s i g n a l . T h i s i m p l i e s t h a t s w i t c h i n g o f L e g A a n d L e g B a r e a l w a y s c o m p l e m e n t s . I n a f o r w a r d b r e a k i n g m o d e w h e r e t h e a v e r a g e v o l t a g e V a i s p o s i t i v e a n d s m a l l e r t h a n t h e b a c k e m f o f t h e a r m a t u r e , c u r r e n t w i l l f l o w t h r o u g h D 1 a n d D 2 w h e n v a = V d c a n d w i l l f l o w t h r o u g h Q 3 a n d Q 4 w h e n v a = -V d c U s i n g t h e c o m p a r i s o n b e t w e e n t h e c o n t r o l s i g n a l a n d t r i a n g u l a r w a v e f o r m a s s h o w n i n F i g u r e 7 , t h e r e s u l t a n t q a n d q i s a s b e l o w : ) % & * ! !

* F r o m p r e v i o u s a n a l y s i s , t h e a v e r a g e v o l t a g e f o r L e g A a n d L e g B i s g i v e n b y : V A O = d A₍ V d c₎ a n d V B O = d B₍ V d c₎ = ( 1 -d A₎ ( V d c₎ ( 9 ) S i m i l a r l y r e l a t i o n b e t w e e n v c a n d d A a n d d B c a n b e w r i t t e n a s : F o r L e g A ( 1 0 ) F o r L e g B ( 1 1 ) W e a r e i n t e r e s t e d i n t h e v o l t a g e a c r o s s t h e a r m a t u r e c i r c u i t , V A B V A B = V A O – V B O = ( d A – ( 1 -d A₎ ) V d c = ( 2 d A -1 ) V d c ( 1 2 ) S u b s t i t u t i n g d A f r o m ( 1 0 ) i n t o ( 1 2 ) g i v e s , ( 1 4 ) B y t a k i n g t h e L a p l a c e t r a n s f o r m o f t h e a c c o m p o n e n t s i n ( 1 4 ) , t h e t r a n s f e r f u n c t i o n b e t w e e n t h e v A B₍ s ) a n d v c₍ s ) i s o b t a i n e d : ( 1 5 ) ! + ! + , p, tri c A V2 v 5. 0 d + = p, tri c B V2 v 5. 0 d − = c p, tri dc A B v V V V = )s ( v V _V )s ( v c p, tri dc A B = -U n i p o l a r s w i t c h i n g T h e s w i t c h i n g s i g n a l s f o r L e g B i s o b t a i n e d f r o m t h e i n v e r s e o f c o n t r o l s i g n a l f o r L e g A . T h i s i s i l l u s t r a t e d i n F i g u r e 1 0 . A c c o r d i n g t o o u r p r e v i o u s a n a l y s i s , t h e c o n t i n u o u s d u t y r a t i o f o r L e g A , d A_, i s g i v e n b y : ( 1 6 ) S i n c e L e g B u s e s t h e i n v e r s e c o n t r o l s i g n a l , a c c o r d i n g l y t h e c o n t i n u o u s d u t y r a t i o f o r L e g B i s g i v e n b y : ( 1 7 ) T h i s g i v e s a n d a v e r a g e a r m a t u r e v o l t a g e a s , V A B = ( d A – d B₎ V d c = ( 1 8 ) T h e t r a n s f e r f u n c t i o n o b t a i n e d f o r u n i p o l a r s w i t c h i n g s c h e m e i s t h e r e f o r e s i m i l a r t o t h e b i p o l a r s w i t c h i n g s c h e m e . p, tri dc V V # $ # $ -p, tri c A V2 v 5. 0 d + = p, tri c B V2 v 5. 0 d − = c p, tri dc v V _V ! , ! . .

-C U R R E N T -C O N T R O L L E D C O N V E R T E R D C a n d A C i n d u s t r i a l d r i v e s n o r m a l l y e m p l o y c a s c a d e c o n t r o l s t r u c t u r e . I t c o n s i s t s o f m u l t i p l e l o o p s : w i t h i n n e r m o s t l o o p b e i n g t h e f a s t e s t . T y p i c a l l y , t h e i n n e r m o s t l o o p i s t h e t o r q u e l o o p , f o l l o w e d b y s p e e d l o o p a n d p o s i t i o n l o o p – t h i s i s s h o w n i n F i g u r e 1 2 b e l o w . F i g u r e 1 C a s c a d e c o n t r o l s t r u c t u r e T w o m a i n f e a t u r e s o r a d v a n t a g e s o f c a s c a d e c o n t r o l s t r u c t u r e i s : ! ! , / ! 0 0 , , , 1 θ 2 2 ω 2 3 a ) T h e c o n t r o l v a r i a b l e o f i n n e r l o o p ( e . g . t o r q u e ) c a n b e l i m i t e d b y l i m i t i n g i t s r e f e r e n c e v a l u e b ) I t i s f l e x i b l e – o u t e r l o o p c a n b e r e a d i l y a d d e d o r r e m o v e d d e p e n d i n g o n t h e c o n t r o l r e q u i r e m e n t s I m p l e m e n t i n g c a s c a d e c o n t r o l s t r u c t u r e r e q u i r e s t h e t o r q u e a n d h e n c e t h e c u r r e n t t o b e c o n t r o l l e d . G o o d c u r r e n t c o n t r o l l e d s c h e m e s s h o u l d p r o d u c e l o w c u r r e n t r i p p l e , g o o d t r a c k i n g c a p a b i l i t y w i t h z e r o s t e a d y s t a t e e r r o r , c o n s t a n t s w i t c h i n g f r e q u e n c y r e g a r d l e s s o f o p e r a t i n g c o n d i t i o n s , a n d f a s t d y n a m i c r e s p o n s e . T h e r e a r e t w o w e l l k n o w n m e t h o d s n o r m a l l y u s e d t o c o n t r o l t h e c u r r e n t , i ) f i x e d s w i t c h i n g f r e q u e n c y c o n t r o l – l i n e a r c o n t r o l l e r i i ) h y s t e r e s i s ( o r b a n g -b a n g ) c o n t r o l – n o n – l i n e a r c o n t r o l l e r F i x e d s w i t c h i n g f r e q u e n c y c o n t r o l T h e r e f e r e n c e c u r r e n t i s c o m p a r e d w i t h t h e a c t u a l c u r r e n t a n d t h e e r r o r i s f e d t o t h e P I c o n t r o l l e r . T h e o u t p u t o f t h e P I c o n t r o l l e r i s c o m p a r e d w i t h t h e t r i a n g u l a r w a v e f o r m t o d e t e r m i n e t h e d u t y r a t i o o f t h e s w i t c h e s – e i t h e r t o i n c r e a s e o r r e d u c e t h e c u r r e n t . T h i s m e t h o d r e s u l t e d i n t h e i n v e r t e r s w i t c h e s a t f i x e d f r e q u e n c y r e g a r d l e s s o f o p e r a t i n g c o n d i t i o n s . H o w e v e r t h e b a n d w i d t h o f t h e c u r r e n t l o o p i s l i m i t e d b y t h e t r i a n g u l a r w a v e f o r m . T h e b a n d w i d t h o f t h e c l o s e d – l o o p s y s t e m i s n o r m a l l y s e t t o a t l e a s t a n o r d e r l o w e r t h a n t h e t r i a n g u l a r f r e q u e n c y . F i g u r e 2 F i x e d f r e q u e n c y c u r r e n t – c o n t r o l l e d F o r t h r e e -p h a s e i n d u c t i o n m o t o r w i t h i s o l a t e d n e u t r a l , t h e 3 -p h a s e c u r r e n t s a r e n o t c o m p l e t e l y i n d e p e n d e n t – i . e . o n l y t w o p h a s e s a r e i n d e p e n d e n t , t h e t h i r d p h a s e c u r r e n t c a n b e c o n s t r u c t e d f r o m t h e o t h e r t w o p h a s e s . I n o t h e r w o r d s , o n l y t w o c o n t r o l l e r s a r e r e q u i r e d . T h i s p r o b l e m c a n b e e l i m i n a t e d i f t h e c o n t r o l i s p e r f o r m e d i n d -q a x i s w h e r e b y o n l y t w o c o n t r o l l e r s a r e r e q u i r e d . T w o v a r i a t i o n s h a v e b e e n p r o p o s e d f o r t h i s t e c h n i q u e : s t a t i o n a r y r e f e r e n c e f r a m e a n d s y n c h r o n o u s r e f e r e n c e f r a m e . T r a c k i n g p r o b l e m w i l l p r e s e n t i f t h e c u r r e n t c o n t r o l i s p e r f o r m e d i n s t a t i o n a r y r e f e r e n c e f r a m e . T h i s w i l l r e s u l t s i n t h e a c t u a l c u r r e n t w a v e f o r m t h a t w i l l a l w a y s l a g t h e r e f e r e n c e c u r r e n t . T h e t r a c k i n g p r o b l e m c a n b e a v o i d e d i f t h e 4 5 6

s y n c h r o n o u s f r a m e i s u s e d , h o w e v e r e x t r a w o r k i s r e q u i r e d t o t r a n s f o r m t h e c u r r e n t f r o m t h e s t a t i o n a r y t o t h e s y n c h r o n o u s f r a m e s a n d v i c e v e r s a . I n a d d i t i o n , e x p l i c i t k n o w l e d g e o f s y n c h r o n o u s f r e q u e n c y i s r e q u i r e d t o p e r f o r m t h e s e t r a n s f o r m a t i o n s . T h e r e f e r e n c e v o l t a g e c a n b e i m p l e m e n t e d u s i n g t h e w e l l -k n o w n m o d u l a t i o n t e c h n i q u e s s u c h a s S i n u s o i d a l P u l s e W i d t h M o d u l a t i o n ( S P W M ) o r S p a c e V e c t o r M o d u l a t i o n ( S V M ) . H y s t e r e s i s c o n t r o l T h e r e f e r e n c e c u r r e n t i s c o m p a r e d w i t h t h e a c t u a l t o r q u e u s i n g h y s t e r e s i s c o m p a r a t o r . T h e o u t p u t o f t h e h y s t r e s i s c o m p a r a t o r w i l l d e t e r m i n e w h e t h e r t h e c u r r e n t n e e d t o b e i n c r e a s e d o r d e c r e a s e d . F o r i n s t a n c e , w h e n t h e c u r r e n t t o u c h e s t h e u p p e r b a n d , 5 6 5 6 ! → . 7 1 7 5 8 1 7 6 61 ω . → ! 2 ! 2 2 .₂ 2 − − ! 5 6 5 6 !→ . . → ! 7 1 7 5 8 1 7 6 61 2 ! 2 2 .₂ 2 ! − − 9 : ; 4 9 : ; 4 c u r r e n t n e e d t o b e r e d u c e d a n d t h i s i s a c c o m p l i s h e d b y t u r n i n g o n t h e l o w e r s w i t c h o f t h a t p a r t i c u l a r l e g . T h i s i s i l l u s t r a t e d i n F i g u r e 5 . F i g u r e 5 H y s t e r e s i s – b a s e d c u r r e n t – c o n t r o l l e d H y s t e r e s i s b a s e d c o n t r o l l e d h a s l a r g e b a n d w i d t h . H o w e v e r , t h e s w i t c h i n g f r e q u e n c y v a r i e s w i t h o p e r a t i n g c o n d i t i o n s a n d c o n t r o l s i g n a l . T h u s t h e m a x i m u m s w i t c h i n g c a p a b i l i t y o f t h e s w i t c h i n g d e v i c e s m u s t b e b a s e d o n t h e w o r s t – c a s e c o n d i t i o n . I f t h e s i m u l a t i o n d o e s n o t r e q u i r e d e t a i l i n f o r m a t i o n r e g a r d i n g t h e r i p p l e , h y s t e r e s i s -b a s e d c o n t r o l c a n b e m o d e l e d b y a s i m p l e l a r g e D C g a i n d u e t o i t s l a r g e b a n d w i d t h . T h e n o n -l i n e a r b e h a v i o r o f t h e h y s t e r s i s -b a s e d c u r r e n t c o n t r o l c a n b e i n v e s t i g a t e d u s i n g l a r g e s i g n a l s i m u l a t i o n . A s w i t h t h e f i x e d f r e q u e n c y c o n t r o l , e a c h p h a s e c u r r e n t n o t o n l y d e p e n d o n t h e c o r r e s p o n d i n g p h a s e v o l t a g e , b u t a l s o o n o t h e r p h a s e v o l t a g e s . I n o t h e r w o r d s , t h e r e i s i n t e r f e r e n c e s b e t w e e n p h a s e s . T h e b e h a v i o r o f t h e h y s t e r e s i s c u r r e n t c o n t r o l c a n b e d e s c r i b e d u s i n g t h e c o m p l e x p l a n e s w i t c h i n g d i a g r a m , a s s h o w n i n F i g u r e 6 . T h e p h a s e c o m p o n e n t s o f t h e c u r r e n t e r r o r v e c t o r ∆ i ( w h i c h i s t h e d i f f e r e n c e b e t w e e n r e f e r e n c e c u r r e n t v e c t o r a n d t h e a c t u a l c u r r e n t v e c t o r ) c a n b e o b t a i n e d b y r e s o l v i n g i t t o t h e r e s p e c t i v e p h a s e a x i s . I f t h e c u r r e n t e r r o r o f a p h a s e t o u c h e s t h e h y s t e r e s i s b a n d o f t h a t p a r t i c u l a r p h a s e , i t s h o u l d b e s w i t c h e d t o t h e o t h e r d i r e c t i o n b y t o g g l i n g t h e s w i t c h o f t h a t p a r t i c u l a r p h a s e . T h e r e f o r e , i d e a l l y , t h e c u r r e n t e r r o r v e c t o r s h o u l d b e c o n f i n e d w i t h i n t h e h e x a g o n a l d e f i n e d b y t h e h y s t e r e s i s b a n d s . H o w e v e r , d u e t o t h e i n t e r a c t i o n s b e t w e e n p h a s e s , t h e c u r r e n t e r r o r m a y g o o u t s i d e t h e h y s t e r e s i s b a n d . A s a r e s u l t , c u r r e n t e r r o r m a y b e c o m e a s l a r g e a s t w i c e t h e h y s t e r e s i s b a n d ( F i g u r e 7 ) 4

-0 .2 -0 .1 5 -0 .1 -0 .0 5 0 0.0 5 0.1 0.1 5 0.2 -0 .2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 R e f e r e n c e s : N . M o h a n , “ P o w e r E l e c t r o n i c s : C o n v e r t e r s , a p p l i c a t i o n s a n d d e s i g n ” J o h n W i l e y a n d S o n s , 1 9 9 5 . N . M o h a n , “ E l e c t r i c D r i v e s – a n i n t e g r a t i v e a p p r o a c h ” M N P E R E , 2 0 0 0 . W . L e o n h a r d , “ C o n t r o l o f e l e c t r i c a l d r i v e s ” , S p r i n g e r -V e r l a g , 1 9 8 4 . J . M . D . M u r p h y a n d F . G . T u r n b u l l , “ P o w e r e l e c t r o n i c c o n t r o l o f A C m o t o r ” , P e r g a m o n p r e s s , 1 9 8 8 . % & : 2 ∆ ( : 0 < 0 = * 7 = < . ;

http://encon.fke.utm.my/courses/mep1422/rl_2q_average.mdl Model { Name "rl_2q_average" Version 5.0 SaveDefaultBlockParams on SampleTimeColors off LibraryLinkDisplay "none" WideLines off ShowLineDimensions off ShowPortDataTypes off ShowLoopsOnError on IgnoreBidirectionalLines off ShowStorageClass off ExecutionOrder off RecordCoverage off CovPath "/" CovSaveName "covdata" CovMetricSettings "dw" CovNameIncrementing off CovHtmlReporting on covSaveCumulativeToWorkspaceVar on CovSaveSingleToWorkspaceVar on CovCumulativeVarName "covCumulativeData" CovCumulativeReport off DataTypeOverride "UseLocalSettings" MinMaxOverflowLogging "UseLocalSettings" MinMaxOverflowArchiveMode "Overwrite" BlockNameDataTip off BlockParametersDataTip off BlockDescriptionStringDataTip off ToolBar on StatusBar on BrowserShowLibraryLinks off BrowserLookUnderMasks off

Created "Thu Sep 11 20:51:10 2003" UpdateHistory "UpdateHistoryNever"

ModifiedByFormat "%<Auto>" LastModifiedBy "Nik Rumzi" ModifiedDateFormat "%<Auto>"

LastModifiedDate "Mon Jul 19 11:38:36 2004" ModelVersionFormat "1.%<AutoIncrement:14>" ConfigurationManager "None" SimParamPage "Solver" LinearizationMsg "none" Profile off ParamWorkspaceSource "MATLABWorkspace" AccelSystemTargetFile "accel.tlc" AccelTemplateMakefile "accel_default_tmf" AccelMakeCommand "make_rtw" TryForcingSFcnDF off ExtModeMexFile "ext_comm" ExtModeBatchMode off ExtModeTrigType "manual" ExtModeTrigMode "normal" ExtModeTrigPort "1" ExtModeTrigElement "any" ExtModeTrigDuration 1000 ExtModeTrigHoldOff 0 ExtModeTrigDelay 0 ExtModeTrigDirection "rising" ExtModeTrigLevel 0 ExtModeArchiveMode "off" ExtModeAutoIncOneShot off ExtModeIncDirWhenArm off ExtModeAddSuffixToVar off ExtModeWriteAllDataToWs off http://encon.fke.utm.my/courses/mep1422/rl_2q_average.mdl (1 von 10) [17.05.2005 17:11:15]

http://encon.fke.utm.my/courses/mep1422/rl_2q_average.mdl ExtModeArmWhenConnect on ExtModeSkipDownloadWhenConnect off ExtModeLogAll on ExtModeAutoUpdateStatusClock on BufferReuse on RTWExpressionDepthLimit 5 SimulationMode "normal" Solver "ode5" SolverMode "Auto" StartTime "0.0" StopTime "100e-3" MaxOrder 5 MaxStep "auto" MinStep "auto" MaxNumMinSteps "-1" InitialStep "auto" FixedStep "1e-6" RelTol "1e-3" AbsTol "auto" OutputOption "RefineOutputTimes" OutputTimes "[]" Refine "1" LoadExternalInput off ExternalInput "[t, u]" LoadInitialState off InitialState "xInitial" SaveTime on TimeSaveName "t" SaveState off StateSaveName "xout" SaveOutput on OutputSaveName "yout" SaveFinalState off FinalStateName "xFinal" SaveFormat "Array" Decimation "1" LimitDataPoints off MaxDataPoints "1000" SignalLoggingName "sigsOut" ConsistencyChecking "none" ArrayBoundsChecking "none" AlgebraicLoopMsg "warning" BlockPriorityViolationMsg "warning" MinStepSizeMsg "warning" InheritedTsInSrcMsg "warning" DiscreteInheritContinuousMsg "warning" MultiTaskRateTransMsg "error" SingleTaskRateTransMsg "none" CheckForMatrixSingularity "none" IntegerOverflowMsg "warning" Int32ToFloatConvMsg "warning" ParameterDowncastMsg "error" ParameterOverflowMsg "error" ParameterPrecisionLossMsg "warning" UnderSpecifiedDataTypeMsg "none" UnnecessaryDatatypeConvMsg "none" VectorMatrixConversionMsg "none" InvalidFcnCallConnMsg "error" SignalLabelMismatchMsg "none" UnconnectedInputMsg "warning" UnconnectedOutputMsg "warning" UnconnectedLineMsg "warning" SfunCompatibilityCheckMsg "none" RTWInlineParameters off BlockReductionOpt on http://encon.fke.utm.my/courses/mep1422/rl_2q_average.mdl (2 von 10) [17.05.2005 17:11:15]

http://encon.fke.utm.my/courses/mep1422/rl_2q_average.mdl BooleanDataType on ConditionallyExecuteInputs on ParameterPooling on OptimizeBlockIOStorage on ZeroCross on AssertionControl "UseLocalSettings" ProdHWDeviceType "Microprocessor" ProdHWWordLengths "8,16,32,32" RTWSystemTargetFile "grt.tlc" RTWTemplateMakefile "grt_default_tmf" RTWMakeCommand "make_rtw" RTWGenerateCodeOnly off RTWRetainRTWFile off TLCProfiler off TLCDebug off TLCCoverage off TLCAssertion off BlockDefaults { Orientation "right" ForegroundColor "black" BackgroundColor "white" DropShadow off NamePlacement "normal" FontName "Helvetica" FontSize 10 FontWeight "normal" FontAngle "normal" ShowName on } BlockParameterDefaults { Block { BlockType Constant Value "1" VectorParams1D on ShowAdditionalParam off

OutDataTypeMode "Inherit from 'Constant value'" OutDataType "sfix(16)"

ConRadixGroup "Use specified scaling" OutScaling "2^0" } Block { BlockType Gain Gain "1" Multiplication "Element-wise(K.*u)" ShowAdditionalParam off

ParameterDataTypeMode "Same as input" ParameterDataType "sfix(16)"

ParameterScalingMode "Best Precision: Matrix-wise" ParameterScaling "2^0"

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OutScaling "2^0"

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OutScaling "2^0"

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Position [610, 431, 640, 464]

SourceBlock "simulink/Math\nOperations/Dot Product" SourceType "Dot Product"

}

Block {

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Amplitude "2" Frequency "1000" SampleTime "0" } Block { BlockType SubSystem Name "Subsystem" Ports [0, 1] Position [110, 130, 150, 190] TreatAsAtomicUnit off

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OutDataTypeMode "Inherit via internal rule" } Block { BlockType Sum Name "Sum1" Ports [2, 1] Position [350, 325, 370, 345] ShowName off IconShape "round" Inputs "|++" InputSameDT off

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http://encon.fke.utm.my/courses/mep1422/rl_2q_average.mdl

Block {

BlockType ToWorkspace Name "To Workspace4" Position [925, 160, 985, 190] VariableName "iau" MaxDataPoints "inf" SampleTime "-1" SaveFormat "Array" } Block { BlockType ToWorkspace Name "To Workspace5" Position [665, 320, 725, 350] VariableName "iave" MaxDataPoints "inf" SampleTime "-1" SaveFormat "Array" } Block { BlockType ToWorkspace Name "To Workspace8" Position [715, 435, 775, 465] VariableName "iD" MaxDataPoints "inf" SampleTime "-1" SaveFormat "Array" } Block { BlockType TransferFcn Name "Transfer Fcn" Position [785, 157, 845, 193] Denominator "[0.01 10]" } Block { BlockType TransferFcn Name "Transfer Fcn1" Position [500, 317, 560, 353] Denominator "[0.01 10]" } Line {

SrcBlock "Sine Wave" SrcPort 1 Points [25, 0] Branch { Points [5, 0] Branch { Points [0, -35]

DstBlock "To Workspace2" DstPort 1 } Branch { Points [35, 0] DstBlock "Sum" DstPort 1 } } Branch { Points [0, 265] DstBlock "Gain3" DstPort 1 } } Line { SrcBlock "Subsystem" SrcPort 1 http://encon.fke.utm.my/courses/mep1422/rl_2q_average.mdl (8 von 10) [17.05.2005 17:11:15]

http://encon.fke.utm.my/courses/mep1422/rl_2q_average.mdl Points [60, 0] DstBlock "Sum" DstPort 2 } Line { SrcBlock "Sum" SrcPort 1 DstBlock "Relay" DstPort 1 } Line { SrcBlock "Relay" SrcPort 1 Points [65, 0] Branch { DstBlock "Gain1" DstPort 1 } Branch { Points [0, 345] DstBlock "Dot Product" DstPort 2 } } Line { SrcBlock "Gain1" SrcPort 1 Points [20, 0; 0, 45; 80, 0; 0, 20; 30, 0] Branch { DstBlock "Transfer Fcn" DstPort 1 } Branch {

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Branch {

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DstBlock "To Workspace8" DstPort 1 } Line { SrcBlock "Gain3" SrcPort 1 http://encon.fke.utm.my/courses/mep1422/rl_2q_average.mdl (9 von 10) [17.05.2005 17:11:15]

http://encon.fke.utm.my/courses/mep1422/rl_2q_average.mdl DstBlock "Sum1" DstPort 1 } Line { SrcBlock "Sum1" SrcPort 1 Points [20, 0] Branch { DstBlock "Transfer Fcn1" DstPort 1 } Branch { Points [0, -80]

DstBlock "To Workspace1" DstPort 1 } } Line { SrcBlock "Constant" SrcPort 1 Points [10, 0] DstBlock "Sum1" DstPort 2 } Annotation {

Name "2-quadrant with iD" Position [736, 44] FontName "Arial" FontSize 20 FontWeight "bold" } Annotation { Name "qA" Position [399, 99] } } } http://encon.fke.utm.my/courses/mep1422/rl_2q_average.mdl (10 von 10) [17.05.2005 17:11:15]

! " ! # $ ! % &! ! ' ( ( ' ) * % + # "

,

-. ' " ' ' ' " ' '

$" $ . ' ' − = *+ ' ' , ' */ 0 $" $

(

/

)

. ' − = *1 2 $" $ , ( 3 0 , / "+ * 2 $" $ , ( 43 % / , 2

(

)

'_5. 3 ( 3 ( . / ' _/ = − = *6 "# " ' "' # " $" $ . ' ' − = *3 ' ' , */ "+ ' *7 0 $" $

,

-' ' ' ' '

/ . ' / − = *4 # 2 , ( 3 ( 3 2 $" $ % , 2

(

)

. / ' 3 ( 3 ( . ' / _/ = − = *5 8 *6 *5 2

SPACE VECTOR MODULATION

In contrast to Sinusoidal Pulse Width Modulation (SPWM), which treats the 3-phase quantities

separately, in SVM, the 3-phase quantities are treated using single equation known as space vector.

Therefore in terms of microprocessor or digital implementation, SVM gives less computational

burden. The space vector of a 3-phase voltage is defined as:

2 4 j j 3 3 s a b c

2

v

v (t) v (t)e

v (t)e

3

π π

=

+

+

,

where v

a

, v

b

and v

c

are the phase voltages.

In 3-phase VSI, there are 8 possible switch configurations, hence there are eight possible voltage

vectors that can be generated or obtained from the VSI. SVM utilized these 8 voltage vectors to

synthesize the reference voltage.

Given a location of the reference voltage in any of the sectors, the actual voltage can be synthesized,

within a sampling period, by selecting the two adjacent voltage vectors and zero voltage vectors. For

example, if the reference voltage is located in sector 1, voltage vectors v

1

, v

2

, v

0

and v

7

should be

selected. This is illustrated in Figure 2

v

d

*

v

q

*

Space vector

modulator

AC

Motor

+

V

d

−

Figure 1 Space vector modulator applied to AC motor drive

(2/3)V

d

Sector 1

Sector 3

Sector 4

Sector 5

Sector 2

Sector 6

(1/√3)V

d

[100]

[110]

[010]

[011]

[001]

_[101]

* s

v

0 0.005 0.01 0.015 0.02 0.025 0.03 -100 -50 0 50 100

a

b

c

sector 6 sector 1 sector 2 sector 3 sector 4 sector 5

Figure 3 Sinusoidal reference voltage Figure 4 Example of modulated waveform in sector 2 000 010 110 111 110 010 000 Phase a Phase b Phase c T T

d

q

Figure 2 Voltage vectors of a 3-phase VSI T0 T1 T2 T7

The interval for each voltage vector, as shown in Figure 4, is determined by equating volt-second

integral of v

s

with the sum of all voltage vectors within a cycle. Thus, for example in sector 1,

7 7 2 2 1 1 o o s

T

v

T

v

T

v

T

v

T

v

⋅

=

⋅

+

⋅

+

⋅

+

⋅

Note that v

1

and v

2

equal

V

d

3

2

_{. Thus in terms of d-q components this can be written as:}

0

T

)

60

sin

j

60

(cos

T

V

3

2

T

V

3

2

0

T

T

v

s

⋅

=

o

⋅

+

d

⋅

1

+

d

⋅

2 o

+

o

+

7

⋅

Also, we need to satisfy the time constraint: T= T

0

+ T

1

+ T

2

+ T

7

If we let T

0

= T

7,

we can calculate all the required time intervals. If the angle between the reference

voltage and the adjacent vector (to the right of the reference voltage) equals

α, it can be shown that

for any sector, the time intervals T

1

and T

2

are given by:

1 s

3

1

T

T v

cos

sin

2

3

=

⋅ ⋅

α −

α

2 s

T

=

3 T v sin

⋅ ⋅

α

In the above equation, v

s

is the normalized reference vector. The interval for the zero voltage vector is

given by: T

0

+ T

7

= T – (T

1

+T

2

). The ratio between T

0

and T

7

essentially control the amount of

triplen harmonic components in the fundamental phase voltage.

Further readings:

PG Handley and JT Boys, “Practical real-time PWM modulators: an assessment” IEE Proceedings-B,

Vol 139, No. 2 March 1992

1 DC DRIVES

Princ iple of ope ration and c onstruc tion – a re vie w

DC m a ch in e con s is ts of

s ta tor – s ta tion a ry – wh ere th e field flu x is produ ced rotor – rota tin g – wh ere th e a rm a tu re win din g is pla ced.

Field flu x is obta in ed eith er from perm a n en t m a gn et or from field win din g excita tion . Field flu x in tera cts with cu rren t ca rryin g con du ctors in a rm a tu re to produ ce torqu e. Com m u ta tor in a rm a tu re circu it will en s u re th a t th e torqu e produ ction is a lwa ys m a xim u m , rega rdles s of rotor pos ition .

Mode ling of DC m otor

Th e torqu e is produ ced a s a res u lt of in tera ction of field flu x with cu rren t in a rm a tu re con du ctors a n d is given by

Te = kt Φ ia (1)

wh ere kt is a con s ta n t depen din g on m otor win din gs a n d geom etry

Φ is th e flu x per pole du e to th e field win din g

For th e m otor with wou n d field, th e flu x ca n be va ried to con trol th e s peed, bu t for perm a n en t m a gn et m otor, th e flu x is fixed a n d th u s ca n be written a s :

Te = Ktia

wh ere Kt depen ds on th e perm a n en t m a gn et m a teria l

Th e direction of th e torqu e produ ced depen ds on th e direction of th e a rm a tu re cu rren t

Wh en th e a rm a tu re rota tes , th e flu x lin kin g th e a rm a tu re win din g will va ry with tim e a n d th erefore a ccordin g to Fa ra da y’s la w, a n em f will be in du ced a cros s th e win din g. Th is gen era ted em f, kn own a s th e ba ck em f, depen ds on s peed of rota tion a s well a s on th e flu x produ ced by th e field a n d is given by:

ea = kt Φ ω (2)

2

ea = Kt ω

Th e pola rity of th e ba ck em f depen ds on th e direction of th e m otor rota tion For s epa ra tely excited DC m otor, th e a rm a tu re circu it is s h own :

Ra – lu m ped a rm a tu re win din g res is ta n ce

La – s elf in du cta n ce of th e a rm a tu re win din g

ea – a s defin ed before, is th e ba ck em f of th e m otor

Us in g KVL,

(3) In s tea dy s ta te con dition ,

(4) In term s of torqu e a n d s peed th e s tea dy s ta te equ a tion ca n be written a s :

(5) wh ich gives :

(6)

Th u s th ree m eth ods ca n be u s ed to con trol th e s peed: Vt , Φ a n d Ra

Speed con trol u s in g a rm a tu re res is ta n ce by a ddin g extern a l res is tor Rext is s eldom u s ed,

es pecia lly for la rge m otor du e to th e los s es a s s ocia ted with Ia 2Rext. Vt is n orm a lly con trol for

s peed u p to ra ted s peed. Beyon d ra ted s peed, for s epa ra tely excited DC m otor, th e s peed con trol is a ch ieved by flu x con trol, Φ. Wh en speed con trol by flu x con trol is u sed, th e m a xim u m torqu e ca pa bility of th e m otor is redu ced s in ce for a given m a xim u m a rm a tu re cu rren t, th e flu x is les s th a n th e ra ted va lu e a n d th u s th e m a xim u m torqu e produ ced is les s th a n th e m a xu m u m torqu e. Als o it s h ou ld be n oted th a t , with perm a n en t m a gn et excita tion , s peed con trol u s in g flu x wea ken in g is n ot pos s ible – th u s m a xim u m s peed of perm a n en t m a gn et m otor is lim ited. Wh en des ign in g con trollers for DC m otor drives u s ed in s ervo or h igh perform a n ce a pplica tion s , a s m a ll s ign a l m odel of th e m otor is requ ired. A s epa ra tely excited DC m otor with fixed field excita tion , or a perm a n en t m a gn et DC m otor, is des cribed by equ a tion s (3), (1) a n d (2). If a s m a ll pertu rba tion a rou n d a DC opera tin g poin t is in trodu ced, th es e equ a tion s ca n be written a s (7)-(9). Th e ‘~’ in dica tes a s m a ll pertu rba tion , wh ich is a dd to th e DC com pon en ts of vt, ia, ea, Te, TL

a n d ω : + ea − a a a a a t i R L di_dt e v = + + + vt − Ra La ω Φ + Φ = _{a t} t t _kT R k V a a a t I R E V = +

( )

₂ a t t t _R k T k V Φ − Φ = ω

3

(7)

(8) (9) Equ a tion des cribin g th e dyn a m ic of th e m ech a n ica l s ys tem is given by:

(10) wh ere Tl = TL + Bω

Tl is th e loa d torqu e com pos ed of workin g torqu e of th e loa d, TL a n d torqu e du e to friction , Bω.

Th e friction a l torqu e depen ds on th e rota tion a l s peed, wh ile TL depen ds on th e n a tu re of th e

loa d bein g driven . Sim ila rly, if a s m a ll pertu rba tion is in trodu ced in Te a n d TL a n d ω, equ a tion

(10) ca n be written a s :

(11) Sepa ra tin g th e DC a n d s m a ll pertu rba tion or AC com pon en ts in (7)–(9) a n d (11), th e s tea dy s ta te a n d s m a ll s ign a l equ a tion s des cribin g th e DC m otor ca n be obta in ed:

Th e tra n s fer fu n ction of th e DC m otor is obta in ed by ta kin g th e La pla ce tra n s form of th e s m a ll s ign a l equ a tion s .

Vt(s ) = Ia(s )Ra + Las Ia + Ea(s ) (12) Te(s ) = kEIa(s ) (13) Ea(s ) = kEω(s) (14) Te(s ) = TL(s ) + Bω(s) + sJ ω(s) (15)

(

)

_{(E e~ )} dt i ~ I d L R ) i ~ I ( v~ Vt t a a a a a a + a+ a + + + = + ) i I ( k T~ Te+ e = E a + a ) ~ ( k e~ Ee + e = E ω+ω dt d J T T m l e ω + = dt ) ~ ( d J ) ~ ( B T~ T T~ Te e L L ω + ω + ω + ω + + = + a a a a a t _dt e~ i ~ d L R i ~ v~ = + + ) i ~ ( k T ~ a E e = ) ~ ( k e~e = E ω a a a t I R E V = + a E e k I T = ω = E e k E dt ) ~ ( d J ~ B T ~ T~e = L + ω+ ω Te =TL +B(ω)

4

Th u s th e block dia gra m repres en tin g th e DC m otor is s h own :

Powe r e le c tronic c onve rte rs in DC drive s

Th e power electron ic con verters a re u s ed to obta in a n a dju s ta ble DC volta ge a pplied to th e a rm a tu re of a DC m otor. Th ere a re ba s ica lly two types of con verter n orm a lly em ployed in DC drives : (i) con trolled rectifier (ii) s witch –m ode con verter.

(i) Con trolled rectifier

Con trolled rectifier ca n be opera ted from a s in gle ph a s e or th ree ph a s e in pu t

Ou tpu t volta ge con ta in low frequ en cy ripple wh ich m a y requ ire a la rge in du ctor in s erted in a rm a tu re circu it, in order to redu ce th e a rm a tu re cu rren t ripple. A la rge a rm a tu re cu rren t ripple is u n des ira ble s in ce it m a y be reflected in s peed res pon s e if th e in ertia of th e m otor–loa d is n ot la rge en ou gh . Con trolled rectifier h a s low ba n dwidth . Th e a vera ge ou tpu t volta ge res pon s e to a con trol s ign a l, wh ich is th e dela y a n gle, is rela tively s low. Th erefore con trolled rectifier is n ot s u ita ble for drives requ irin g fa s t res pon s e, e.g. in s ervo a pplica tion s .

In term s of qu a dra n t of opera tion s , a s in gle ph a s e or a th ree ph a s e rectifier is on ly ca pa ble of opera tin g in firs t a n d fou rth qu a dra n ts – wh ich is n ot s u ita ble for drives requ irin g forwa rd brea kin g m ode. To be a ble to opera te in a ll fou r qu a dra n ts , con figu ra tion s u s in g ba ck to ba ck rectifiers or con ta ctors s h own below m u s t be em ployed.

T

k

a a

s L

R

1

+

)

s

(

T

_l

)

s

(

T

_e

s J

B

1

+

E

k

)

s

(

I

_a

_ω

₍

_s

₎

)

s

(

V

_a + - - + 3-phase supply 3-phase supply + Va - Converter A ω T Converter B Converter B Converter A