UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Measurement of induction motor speed from induced slip frequency signal Cai, Ji-Yuan 1985

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
[if-you-see-this-DO-NOT-CLICK]
UBC_1986_A7 C35.pdf [ 2.52MB ]
[if-you-see-this-DO-NOT-CLICK]
[if-you-see-this-DO-NOT-CLICK]
Metadata
JSON: 1.0065063.json
JSON-LD: 1.0065063+ld.json
RDF/XML (Pretty): 1.0065063.xml
RDF/JSON: 1.0065063+rdf.json
Turtle: 1.0065063+rdf-turtle.txt
N-Triples: 1.0065063+rdf-ntriples.txt
Original Record: 1.0065063 +original-record.json
Full Text
1.0065063.txt
Citation
1.0065063.ris

Full Text

MEASUREMENT OF INDUCTION MOTOR SPEED FROM INDUCED SLIP FREQUENCY SIGNAL  by  CAI  >  JI-YUAN  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  i n the Department o f Electrical  Engineering  We a c c e p t - t h i s t h e s i s as conforming t o the required Research  standard  Supervisor  Members o f t h e Committee,  P air Head o f t h e Department  Members o f t h e Department of E l e c t r i c a l  Engineering  THE UNIVERSITY OF BRITISH COLUMBIA December, 1985 © C a i Ji-Yuan,  1985  In presenting degree  at the  this thesis  in partial fulfilment  of the  advanced  and study. 1 further agree that permission for extensive  copying of this thesis for scholarly purposes or  for an  University of British Columbia, I agree that the Library shall make it  freely available for reference  department  requirements  by  his  or  her  representatives.  may be granted It  is  by the  understood  that  head of my copying or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  DE-6(3/81)  ABSTRACT  A method t o d e t e c t t h e s l i p o f a three-phase o r s i n g l e - p h a s e i n d u c t i o n motor and hence determine t h e motor speed i s proposed.  A  p i c k - u p c o i l i s p l a c e d i n c l o s e p r o x i m i t y t o the motor and by s u i t a b l e a m p l i f i c a t i o n and f i l t e r i n g  o f t h e weak s l i p f r e q u e n c y s i g n a l ,  the  s l i p f r e q u e n c y s i g n a l component i s i s o l a t e d f o r t h e speed measurement. No d e v i c e needs t o be a t t a c h e d t o t h e motor s h a f t . E x p e r i m e n t a l r e s u l t s from f o u r t e s t i n d u c t i o n motors proposed method o f speed measurement i s f u l l y normal l o a d s , i . e . , f o r s l i p  show t h a t the  f e a s i b l e and a p p l i e s t o  l e s s than 13% w i t h c o n s i d e r a b l e a c c u r a c y .  T h e r e f o r e , even t h e speed o f s e a l e d i n d u c t i o n motor such as t h o s e used i n r e f r i g e r a t o r s can be e a s i l y measured.  ii  TABLE OF CONTENTS Page ABSTRACT  i i  TABLE OF CONTENTS  i i i  LIST OF ILLUSTRATIONS  iv  LIST OF TABLES ACKNOWLEDGEMENT 1.  4.  5.  vi 1  1.1 1.2  1  General P r i n c i p l e o f O p e r a t i o n and Performance E q u a t i o n o f I n d u c t i o n Motors P r e v i o u s Method o f S l i p Frequency Measurement Proposed Method o f S l i p Frequency Measurement  FEASIBILITY 2.1 2.2 2.3 2.4  3.  •  INTRODUCTION  1.3 1.4 2.  v  TEST OF SPEED MEASUREMENT TECHNIQUE  D e s c r i p t i o n o f Measurement and C i r c u i t C a l c u l a t i o n o f S l i p Frequency f Measurement o f Supply Frequency Experimental Results  1 4 4 6 6 7 8 8  CONSTRUCTION OF SPEED MEASUREMENT CIRCUIT  14  3.1 3.2 3.3 3.4 3.5  14 18 18 22 23  The O v e r a l l S l i p Frequency Measurement C i r c u i t I n d u c t i v e Device Choke Second Order B u t t e r w o r t h Low-Pass F i l t e r Input B u f f e r and A t t e n u a t o r Phase S h i f t e r  EXPERIMENTAL RESULTS  25  4.1 4.2 4.3  36 39 39  Discussion o f Experimental Results Supply Frequency S i g n a l Component Measurement o f P o l e P a i r s  CONCLUSION  43  REFEFENCES  44  APPENDICES  46  Al. A2.  D e r i v a t i o n of t h e f r e q u e n c y Response E q u a t i o n f o r Second Order B u t t e r w o r t h Low-Pass F i l t e r C a l c u l a t i o n o f S l i p Frequency S i g n a l Component  iii  46 49  LIST OF  ILLUSTRATIONS  Figure 2.1  Page  S i g n a l Pickup and D e t e c t i o n w i t h Choke and Galvanometer  »....  6  2.2  Equivalent C i r c u i t  o f F i g u r e 2.1  7  2.3  S l i p Frequency Waveform o f Motor ml  10  2.3  S l i p Frequency Waveform o f Motor m2  10  2.5  Supply Frequency Waveform o f Motor ml  10  2.6  Supply Frequency Waveform o f Motor m2  10  2.7  Speed Comparison  12  3.1  O v e r a l l Block Diagram o f S l i p Frequency Measurement  15  3.2  Circuit  16  Components  (a) (b) (c) (d) (e) (f)  4.1 4.2  4.3  Phase S h i f t e r Choke Phasor Diagram o f Phase S h i f t e r Input B u f f e r G Attenuator G Second Order B u t t e r w o r t h Low-Pass F i l t e r (each o f them from F i l t e r 1 t o F i l t e r 6) E x p e r i m e n t a l Setup  16 16 16 16 16 Stage  Comparison o f Expected and D e t e c t e d S l i p Frequency f o r I n d u c t i o n Motor (a) Ml (b) M2 ( c ) M3 (d) M4 Amplitude R a t i o o f 60 Hz Supply Frequency to  16 26  •  38  Signal  S l i p Frequency S i g n a l  40  4.4  L o c a t i o n Diagram o f P o l e P a i r s Measurement  41  4.5  Choke V o l t a g e Waveform a t S e v e r a l L o c a t i o n s (a) Supply Frequency Reference Waveform  41 41  (b)  at  L  ±  (c) at  L  ?  (d) a t L  3  A l . l Second Order B u t t e r w o r t h Low-Pass F i l t e r A2.1 G a i n Curve o f Three I d e n t i c a l Second Order B u t t e r w o r t h F i l t e r Stages i n Cascade  iv  41 46 50  LIST OF TABLES Table  Page  2.1  S p e c i f i c a t i o n o f Motors and Instruments  2.2  Measured Values a t No Load  10  2.3  Measurement  11  3.1  S p e c i f i c a t i o n o f C i r c u i t s i n F i g u r e 3.2  17  4.1  Motor S p e c i f i c a t i o n s  25  4.2  E x p e r i m e n t a l R e s u l t s o f Motor Ml  27  4.3  E x p e r i m e n t a l R e s u l t s o f Motor M2  29  4.4  E x p e r i m e n t a l R e s u l t s o f Motor M3  31  4.5  E x p e r i m e n t a l R e s u l t s o f Motor M4  33  4.6  Waveform E v a l u a t i o n o f D e t e c t e d S l i p Frequency  37  Results  A2.1 C a l c u l a t e d G., (N<3) Nodb A2.2 C a l c u l a t e d G (N>3)  9  52 54  M  v  ACKNOWLEDGEMENT  I wish t o express my deepest g r a t i t u d e t o my s u p e r v i s o r , Dr. Malcome Wvong f o r h i s v a l u a b l e a s s i s t a n c e , c o n s t a n t  encouragement and  c o r r e c t guidance i n my graduate program and t h e p r e p a r a t i o n o f t h i s thesis. I s i n c e r e l y wish t o thank my w i f e , H u i Chun, f o r t h e c o o p e r a t i o n , understanding  and encouragement d u r i n g t h e e n t i r e p e r i o d o f t h e  study.  vi  1  1.  1.1  INTRODUCTION  General One  o f t h e most important t r e n d s i n v a r i a b l e speed d r i v e s i s t h e use  o f i n d u c t i o n motors.  T h i s t e c h n i q u e has been s t u d i e d i n r e c e n t y e a r s  [1-6],  because the i n d u c t i o n motor can be used i n a v a r i e t y o f poor working cond i t i o n s and a t l e s s expense than dc machines. d r i v e s w i t h c o n s t a n t speed r e q u i r e m e n t s .  I t i s w i d e l y used i n ac  Great p r o g r e s s i n t h y r i s t o r and  p o w e r - t r a n s i s t o r i n v e r t e r d e s i g n has made i t p o s s i b l e t o use motors  i n v a r i a b l e - s p e e d d r i v e systems.  motor d r i v e s can be o b t a i n e d by complex s l i p frequency c o n t r o l phase-locked loop  High performance o f i n d u c t i o n l o o p c o n t r o l systems, such as  [2-4], f l u x c o n t r o l  [pLL] c o n t r o l  induction  [2-4], and v e c t o r c o n t r o l o r  [5,6].  But the parameter o f r o t a t i n g speed must be a v a i l a b l e as a measurement f o r t h e s e c o n t r o l systems.  C o n v e n t i o n a l speed measurement r e q u i r e s  a tachometer, t r a n s d u c e r o r d e v i c e t o be a t t a c h e d t o t h e motor s h a f t . There would be l e s s i n s t a l l a t i o n and maintenance  c o s t s i f speed c o u l d be  measured w i t h o u t s h a f t attachment o f any such gadget.  1.2  The P r i n c i p l e o f O p e r a t i o n and Performance  E q u a t i o n o f Induction Motors  Some important a s p e c t s o f i n d u c t i o n motors w i l l be b r i e f l y reviewed here.  As we know, t h e r e a r e two k i n d s o f ac motors,namely,synchronous  and  asynchronous motors, the l a t t e r b e i n g more commonly known as i n d u c t i o n motors.  I n d u c t i o n motors a r e w i d e l y used i n ac d r i v e systems because o f  low c a p i t a l and maintenance  costs.  The p r i n c i p l e o f o p e r a t i o n o f an i n d u c t i o n motor i s d i f f e r e n t  from  t h a t o f a synchronous motor which has a speed always e q u a l t o the synchronous speed produced by a r o t a t i n g magnetic f i e l d .  When t h r e e phase  voltage  2  i s a p p l i e d t o the s t a t o r o f a three  phase i n d u c t i o n motor, a r o t a t i n g  magnetic f i e l d e x i s t s i n t h e space, a t a a n g u l a r speed c o r r e s p o n d i n g t o the s u p p l y f r e q u e n c y and c u t s t h e r o t o r w i n d i n g s . s h o r t - c i r c u i t e d r o t o r w i n d i n g s , and s e t s up a which i n t e r a c t s w i t h t h e s t a t o r f i e l d . r o t a t i o n o f t h e motor r o t o r .  Current flows i n the  rotating  magnetic  field  The torque produced r e s u l t s i n  With motion the c u r r e n t  i n the r o t o r i s a t  a f r e q u e n c y which r e f l e c t s t h e r e l a t i v e movement o f t h e r o t o r w i t h t o t h e r o t a t i n g magnetic f i e l d s e t up by t h e s t a t o r w i n d i n g s . s i g n a l corresponding t o the frequency o f the r o t o r current  respect  If a  can be p i c k e d  up i n t h e v i c i n i t y o f t h e motor t h e n t h e speed o f t h e motor can be r e a d i l y determined. Indeed, t h e speed o f t h e motor w i l l never r e a c h t h e speed o f t h e r o t a t i n g magnetic f i e l d s e t up by t h e s t a t o r , t h e s o - c a l l e d synchronous speed.  The speed o f t h e i n d u c t i o n motor depends upon t h e motor s i z e ,  mechanical l o a d , e t c . the  U s u a l l y , f o r r a t e d l o a d f o r most i n d u c t i o n  motors,  speed range i s from 99% t o 92% o f synchronous speed ( i . e . r o t o r  current  f r e q u e n c y i s 0.6 Hz t o 5 Hz f o r 60 Hz m o t o r s ) .  In o r d e r t o c a l c u l a t e t h e motor speed by measuring r o t o r  current  f r e q u e n c y , t h e r e l a t i o n s among i n d u c t i o n motor parameters need t o be reviewed. slip  s i s d e f i n e d as f o l l o w s : n-n^ s  =  (1.1)  ~n~~  where n i s synchronous  speed.  n^ i s r o t o r speed, i . e . asynchronous  speed.  The r o t o r r o t a t e s a t asynchronous speed such t h a t t h e s l i p g i v e n by  frequency i s  3  f  =  g  where f  sf  (1.2)  is slip  frequency  f i s supply frequency. The f r e q u e n c y o f r o t o r c u r r e n t i s a t s l i p f r e q u e n c y . if  s = 1 the motor s t a y s a t r e s t because  the r o t o r at synchronous slip  duced  the r o t a t i n g magnetic  that  f i e l d cuts  speed c o r r e s p o n d i n g t o t h e supply f r e q u e n c y .  can never be zero because  speed and t h i s  I t i s obvious  the r o t o r would need t o run a t  synchronous  would n o t be p o s s i b l e because no v o l t a g e would be i n -  i n the r o t o r . Substituting  (1.1) i n t o (1.2) we  get:  n-n, f  s  =n ^  since  60f  f  n = —  (1.*)  where p i s the number o f p o l e p a i r s . Eliminating f  s  f i n (1.3) and  = (n-n. ) 1  (1.4)  -2— 60  and e l i m i n a t i n g n i n (1.3) and  ,„  x  c  (1.5)  (1.4)  f-f rij  =  60  (1.6)  I f supply f r e q u e n c y f = 60 Hz then eqn. n. = - (3600 - 60 f ) 1 p s Let N = 60 f  (1.6) becomes  (1.7) (1.8)  s  or f = - ^ r s  1  9  C - )  bO  then n = - (3600 - N) ' 1 P  (1.10)  Here N i s t h e f r e q u e n c y o f the s i g n a l which w i l l be found i n the simple experiment  d i s c u s s e d i n c h a p t e r 2.  The  4  From (1.6) d e r i v e d above, a c t u a l motor speed  can be  determined  through measurement o f s l i p f r e q u e n c y , f , the g i v e n s u p p l y f r e q u e n c y , f and t h e number o f p o l e p a i r s ,  1.3  p.  P r e v i o u s Method o f S l i p Frequency I s h i d a , M. and  Iwata, K.  s l o t harmonics t o determine  (7) and  the s l i p  Measurement (8) have proposed  the use o f r o t o r  frequency o f i n d u c t i o n motors.  The a i r - g a p f l u x o f an i n d u c t i o n motor when f e d by a b a l a n c e d power s u p p l y c o n t a i n s the space harmonics, of  the r o t o r and  windings.  The  due t o v a r i a t i o n o f r e l u c t a n c e  s t a t o r s l o t s and t o s p a t i a l d i s t r i b u t i o n o f  stator  i n f l u e n c e o f s t a t o r s l o t s t o harmonics i n s t a t o r v o l t a g e  can be n e g l e c t e d i n comparison  with that of r o t o r s l o t s .  motor i s o p e r a t i n g each s t a t o r winding fundamental  sinusoidal  Thus, when the  c o n t a i n s b o t h components o f t h e  and s l o t harmonic v o l t a g e s .  The d e t e c t i o n o f the s l o t  har-  monics can be o b t a i n e d by the use o f t h r e e s i n g l e - p h a s e t r a n s f o r m e r s . primary o f each s i n g l e - p h a s e t r a n s f o r m e r i s connected s t a t o r winding and a l l the secondary windings for  output.  The s l i p frequency f  i n p a r a l l e l t o each  of the transformer i n s e r i e s  can be d e t e c t e d o n l y f o r motors w i t h  the number o f r o t o r s l o t s e q u a l t o 3n + 1 f o r n = 1, 2, 3, by the a u t h o r s .  reported  E x p e r i m e n t a l r e s u l t s show good l i n e a r i t y o f s l i p  measured by the proposed method w i t h r e s p e c t t o t h a t measured by t i o n a l means i n the range o f s l i p frequency up t o +30% frequency.  of the  frequency conven-  stator  However, t h i s method can o n l y be used where the number o f  the r o t o r s l o t s per p o l e p a i r i s known and i s equal t o 3n ± 1.  l.H  Proposed Method o f f l i p Frequency In  The  Measurement  t h i s t h e s i s a d i f f e r e n t method o f s l i p frequency measurement o f  an i n d u c t i o n motor and hence i t s speed w i l l be r e p o r t e d .  By the use o f  5  an i n d u c t i v e p i c k - u p c o i l  i n t h e v i c i n i t y o f t h e i n d u c t i o n motor i t i s  found t h a t o n l y two major s i g n a l components, namely, t h e supply (60 Hz) and t h e s l i p frequency  frequency  (f" )» e x i s t i n t h e i n d u c t i v e choke v o l t a g e . g  Even though t h e r e i s v e r y poor s i g n a l t o n o i s e r a t i o , f o r t h e s l i p quency s i g n a l w i t h r e s p e c t t o t h e supply f r e q u e n c y , t h e supply ("noise") s i g n a l can be f i l t e r e d out.  The unique  fre-  frequency  s l i p frequency, f , s  /  can then be d e t e c t e d t o determine r e q u i r e t h e attachment  the i n d u c t i o n motor f o r more  the s l i p  thesis  project  frequency  T h e r e f o r e , i t o f f e r s t h e measurement o f convenience.  i s aimed at  investigating this  component o f l e a k a g e f l u x  around an i n d u c t i o n motor t o d e t e r m i n e m i z a t i o n o f the of  the  The method does n o t  o f any d e v i c e t o t h e motor s h a f t and t h e number o f  r o t o r s l o t s t o be known, e t c .  The  t h e motor speed.  a v a i l a b l e i n the  the motor s p e e d .  inductive pick-up transducer,fi1ter  speed s i g n a l  i s not a t t e m p t e d  A simple experiment  in t h i s  field  However,the  s t a g e s and the  optidisplay  thesis.  t o check t h e f e a s i b i l i t y  i s d i s c u s s e d i n c h a p t e r 2.  concept o f u s i n g  o f t h e proposed  The r e s u l t o f t h a t experiment  method  guided t h e  d e s i g n o f t h e f i n a l speed measurement c i r c u i t , d e t a i l s o f which a r e g i v e n i n c h a p t e r 3.  F i n a l l y , t e s t r e s u l t s f o r f o u r i n d u c t i o n motors a r e g i v e n  and d i s c u s s e d i n c h a p t e r 4.  6  2.  FEASIBILITY TEST OF SPEED MEASUREMENT TECHNIQUE  To t e s t t h e f e a s i b i l i t y experiment  o f t h e speed measuring t e c h n i q u e , a simple  was done u s i n g an i n d u c t i v e p i c k - u p c o i l and a galvanometer.  The experiment  will  i n d i c a t e two important  (1) The s l i p frequency s i g n a l f  p o i n t s , namely;  e x i s t s i n t h e space around t h e  motor when i t i s r u n n i n g . (2) The s l i p frequency  s i g n a l t o g e t h e r with a s t r o n g s i g n a l a t  the supply f r e q u e n c y , 60 Hz, would be p i c k e d up by an i n d u c t i v e d e v i c e , such as a choke and c o u l d be d e t e c t e d a f t e r  suitable  filtering.  2.1  D e s c r i p t i o n o f Measurement and C i r c u i t The  Fig.  circuit  2.1.  diagram o f s i g n a l p i c k - u p and d e t e c t i o n i s shown i n  The choke i s p l a c e d b e s i d e t h e motor i n o r d e r t o p i c k up t h e  s i g n a l from t h e r o t o r c i r c u i t  o f t h e motor.  p r i a t e volume i n t h e choke would be b e s t .  High  inductance and appro-  The i n d u c t i v e v o l t a g e i n t h e  choke ( c a l l e d choke v o l t a g e i n t h i s t h e s i s ) i s i n d i c a t e d i n t h e c i r c u i t by t h e galvanometer, which i s h i g h l y s e n s i t i v e t o dc o r low frequency  OSCILLOSCOPE  Fig  2.1  S i g n a l Pickup and D e t e c t i o n w i t h Choke and Galvanometer  7  current. spot  The  current value  on the s c a l e d g l a s s .  i s determined by the The  d e v i a t i o n o f the  light  galvanometer a l s o p l a y s the r o l e o f a  m e c h a n i c a l low-pass f i l t e r because i t s m e c h a n i c a l i n e r t i a w i l l o n l y it  t o f o l l o w the dc o r low  f r e q u e n c y s i g n a l and  s i g n a l s , such as t h a t o f the supply ponent o f supply  frequency may  scope by c o n n e c t i n g  cuts o f f higher  frequency 60 Hz.  be measured and  The  frequency  s i g n a l com-  d i s p l a y e d on the  the l e a d s of t h e choke d i r e c t l y to the  oscillo-  input of  oscilloscope.  2.2  C a l c u l a t i o n o f S l i p Frequency f  r  i  O S C I L L O S C O P E  F i g 2.2  The  equivalent  Equivalent  C i r c u i t of F i g  2.1  c i r c u i t t o measure the s l i p frequency f  amplitude produced by r o t o r c u r r e n t  i s shown i n F i g . 2.2  r. x  :  choke r e s i s t a n c e  r  :  galvanometer r e s i s t a n c e  :  external variable resistance  :  s l i p frequency v o l t a g e  g r V  e g  I t s amplitude V  g  can be  d e r i v e d as  component i n choke  follows  allow  and i t s  where  the  8  We have  f = s  — 60  (1.9)  A l s o V = Ig( r , + r s i  +r g  6  e  The  c u r r e n t Ig i n eqn. Ig  where A  and  (2.1)  (2.1) i s g i v e n by  = A - sg  (2.2)  g  i s t h e amplitude  g  )  of the l i g h t  spot d e f e c t i o n o f t h e galvanometer  s g i s t h e galvanometer s e n s i t i v i t y . -6  sg  = 0.0035 x 10  Sustituting V s  a.  (2.2) i n t o (2.1)  = A • sg ( r , + r s i e to  +r g  )  (2.3)  In  t h e experiment  of  o s c i l l a t i o n s N o f t h e galvanometer l i g h t  2.3  the s l i p frequency  Measurement o f supply The  f  i s o b t a i n e d by c o u n t i n g t h e number spot f o r one minute.  frequency  s t r o n g r o t a t i n g magnetic f i e l d e x i s t s i n t h e space around t h e  motor when i t i s r u n n i n g so t h a t t h e s u p p l y frequency i n t h e choke.  60 Hz i s a l s o  induced  I f t h e s w i t c h K i s opened so t h a t t h e choke v o l t a g e i s  a p p l i e d d i r e c t l y t o t h e i n p u t o f t h e o s c i l l o s c o p e , as i n d i c a t e d i n F i g . 2.2, t h e wave form on t h e o s c i l l o s c o p e i n d i c a t e s 60 Hz frequency and i t s a m p l i tude which can be r e a d from t h e s c a l e . contains  Even though t h e choke v o l t a g e  b o t h s i g n a l components o f s l i p f r e q u e n c y and supply  60 Hz, they a r e f a r enough a p a r t and t h e s u p p l y f r e q u e n c y  frequancy  s i g n a l i s much  s t r o n g e r than t h e s l i p f r e q u e n c y s i g n a l , t h a t t h e wave form o f supply frequency 2.4  i s hardly d i s t o r t e d at a l l .  Experimental  Results  Measurements were c a r r i e d out on two s i n g l e - p h a s e i n d u c t i o n motors at  no l o a d .  The s p e c i f i c a t i o n o f motors and i n s t r u m e n t s employed i s shown  i n T a b l e 2.1.  The measurement, wave form and r e s u l t s a r e g i v e n i n t h e  9  T a b l e s and F i g u r e s as f o l l o w s : (1) Table 2.2 shows t h e measured s i g n a l components o f s l i p and supply  frequencies.  (2) F i g u r e 2.3 and F i g u r e 2.4 show t h e s l i p frequency for  wave forms  t h e two motors.  (3) F i g u r e 2.5 and F i g u r e 2.6 show t h e supply frequency  wave form  seen.on the o s c i l l o s c o p e . (4) T a b l e  2.3 shows t h e f i n a l r e s u l t s f o r f , n , V , V  Table 2.1  S p e c i f i c a t i o n o f Motors and Instruments  choke  Motor Pole Pairs  and  Phase  Power  r  i  L (H)  (Hp)  P  galvanometer Sens. Sg (a/mm)  r " (0)  c ml  (Motorl)  2  1  1/3  92  18.9  3.5*l(f  m2  (Motor2)  1  1  1/40  92  18.9  3.5X10  9  25  -9  25  10  Table 2.2  Measured Values a t no Load  Actual  S l i p Frequency  Speed  Component  Component r  Stroboscope Motor  Supply Frequency  Circuit  i  r  r  g  e  Frequency Amplitude  Frequency Amplitude  (n) (rpm)  (N/min)  A  s  (mm)  f  60  (  H  z  )  V  6 0 ( v . pk-pk)  Oscilloscope  Galvanometer  ml  1792  16  48  92  330  25  60.5  1.2  m2  3495  105  10  92  3300  25  59.7  1.83  3490  110  10  92  3300  25  59.7  1.83  3486  114  10  92  3300  25  59.7  1.83  Fig  2.4  S l i p f r e q u e n c y Waveform o f Motor m2  Fig  2.6  Supply Frequency Waveform o f Motor m2  11  T a b l e 2.3  Actual  Measurement R e s u l t s  Measured  Slip  Speed Motor  Slip S  Stroboscope  Motor Speed  Frequency  n'  f  (rpm)  (Hz)  s  Components  Ratio  Slip Frequency Voltage V (v)  Supply Frequency Voltage  s  V  60  (  V  v,  60  /v  s  )  (rpm) Oscilloscope  Galvanometer  1.2  16 10  6  1.83  15.3 10  3  6  1.83  15.3*10  3  1.83  15.3 10  6  ml  1792  0.44%  0.26  1792  75*10~  m2  3495  2.91%  1.75  3495  119 10"  x  3490  3.05%  1.83  3490.2  119 10~  x  3486  3.16%  1.90  3486  119X10  - 6  X  X  The f o l l o w i n g i n d i c a t e s sample c a l c u l a t i o n f o r motor m^ u s i n g t h e measurements and v a l u e s o f T a b l e s 2.1 and 2.2. and T a b l e  From eqn. ( 1 . 9 ) , (1.10)  2.2 f  =  N 60  =  - (3600 - 60 f ) = - i (3600 - 60 x 0.26) = 1792 rpm. p s 2  s n, 1  s  .16 60  0  >  2  6  H  z  with p = 2  From eqn. (2.3) and T a b l e s 2.1 and 2.2, t h e amplitude  of s l i p  frequency  voltage i s  V  s  =  V  S  S  r  < i  +  r  e  +  r  g>  _9 Sg = 3.5 x 10 r. i and A So  = 92  r  = 48 mm  s V  fi  s  a/mm = 330 Q  e  r  g  = 25 n  p = 2 r  = 48 x 3.5 x 1 0 "  9  (92 + 330 + 25) = 75 x 1 0  From T a b l e 2.3 t h e a c t u a l speed n ^  - 6  v.  measured by a s t r o b o s c o p e as compared  w i t h the speed n^ measured by t h e arrangement o f choke and galvanometer are drawn i n F i g . 2.7.  any p o i n t on the s t r a i g h t  3  X  3  12  Fig  2.7  Speed  Comparison  l i n e means t h a t t h e measured speed i s e q u a l t o a c t u a l speed.  The  measured v a l u e s o f T a b l e 2.3 agree e x a c t l y . From t h e above r e s u l t s some important p o i n t s can be made: (1) The speed as c a l c u l a t e d u s i n g t h e o s c i l l a t i o n galvanometer l i g h t  p e r minute o f t h e  s p o t a r e i d e n t i c a l w i t h those measured w i t h  a s t r o b o s c o p e , which i s g e n e r a l l y a c c e p t a b l e as a c c u r a t e speed measurement. the  So t h e proposed method w i l l c o r r e c t l y  determine  i n d u c t i o n motor speed a t no l o a d .  (2) There e x i s t s m u l t i p l e frequency s i g n a l components i n t h e f i e l d around t h e motor when i t i s o p e r a t i n g .  O b v i o u s l y , t h e choke  v o l t a g e w i l l c o n t a i n a t l e a s t the s i g n a l components o f s u p p l y f r e q u e n c y 60 Hz and s l i p f r e q u e n c y f motor speed. range o f f  t h a t a r e r e l a t e d t o the  From T a b l e 2.3 i t c a n be e s t i m a t e d t h a t t h e f o r t h e i n d u c t i o n motors v a r i e s from 0.2 5 Hz a t no  l o a d up t o 5 Hz, i . e . t h a t t h e speed range from 1792 rpm down to  1650 rpm f o r a f o u r - p o l e  motor.  13  (3) The  amplitude  r a t i o o f Vgg  to V  reaches  up t o 16 x 10  approximately  f o r motor m^.  component V g  i s much g r e a t e r than the amplitude  frequency  Q  component.  amplification frequency signal (4) The  g  of the  from the much s t r o n g e r supply  slip  with  frequency  Vg . Q  but the synchronous speed has t o be known i . e . the  (1.6) e x h i b i t s  p  f o r the motor has this  t o be known.  t o be g i v e n f o r the speed  may  be determined  experimentally  T h i s parameter p w i l l be  or  assumed  determination.  i s t o be noted t h a t the e r r o r  o f t h e s l i p frequency  i n c r e a s e w i t h frequency because o f t h e mechanical  amplitude  will  i n e r t i a o f galvanometer.  So the galvanometer can be c o n s i d e r e d as a m e c h a n i c a l frequency.  equa-  s  number o f p o l e p a i r s  corner  The  relation  from nameplate s p e c i f i c a t i o n .  w i t h low  frequency  d e t e r m i n a t i o n o f t h e motor speed not o n l y depends upon t h e  1 The  o f supply  Therefore, a multi-stage f i l t e r  component V  number o f p o l e p a i r s tion  The amplitude  has t o be designed t o i s o l a t e the weaker s l i p  component  measured f  It  g  low-pass  filter  14  3.  CONSTRUCTION OF SPEED MEASUREMENT CIRCUIT  The simple experiment feasibility  d e s c r i b e d i n c h a p t e r 2 demonstrated t h e  o f t h e t e c h n i q u e t o measure t h e speed o f an i n d u c t i o n motor  by use o f t h e s l i p frequency s i g n a l p r e s e n t the magnetic f i e l d around the motor.  But t h e simple measurement s e t up u s i n g a galvanometer  in i t s ability  t o measure t h e whole range  d i f f e r e n t k i n d s o f i n d u c t i o n motors. filter  circuit  i s limited  o f speeds with v a r y i n g l o a d f o r  T h e r e f o r e , a more s o p h i s t i c a t e d  needs t o be d e s i g n e d t o r e p l a c e t h e mechanical  filter  r e p r e s e n t e d by t h e galvanometer. Major f a c t o r s t o be c o n s i d e r e d i n t h e c i r c u i t (1) The measurement c i r c u i t  design are:  i s t o use an a c t i v e f i l t e r ,  with  o p e r a t i o n a l a m p l i f i e r s as t h e a c t i v e elements, and w i t h multi-stage f i l t e r i n g .  Each f i l t e r  stage s h o u l d a t t e n u a t e  the s u p p l y frequency s i g n a l and a m p l i f y t h e weak s l i p  frequency  component. (2) Each f i l t e r  stage s h o u l d work w i t h i n t h e l i n e a r range o f t h e  o p e r a t i o n a l a m p l i f i e r i ^ . w i t h i n ± 13 v o l t s (3) F o r t h e d e t e c t i o n o f t h e s l i p frequency f o r v a r i o u s motors, the multi-stage f i l t e r  circuit  s h o u l d p r o v i d e output a t each  because t h e number o f f i l t e r  stages needed v a r i e s w i t h t h e  type o f i n d u c t i o n motors and t h e i r (4) The amplitude o f the d e t e c t e d s l i p  speed. frequency can a l s o be  c a l c u l a t e d through t h e m u l t i - s t a g e f i l t e r .  3.1  The O v e r a l l S l i p Frequency The o v e r a l l c i r c u i t  following:  stage  Measurement C i r c u i t  i s shown i n F i g u r e 3.1 and comprises t h e  15  (1) S i g n a l p i c k u p i . e . i n d u c t i v e d e v i c e , choke (2) Second o r d e r Butterworth low pass  filter  (3) Input b u f f e r and attenuato_r (4) A l t e r n a t i v e s u p p l y frequency phase The  shifter  d e t a i l e d c i r c u i t s f o r each o f t h e above a r e shown i n F i g u r e 3.2.  PHASE SHIFTER  Fl L T E R 1  INPUT BUFFER CHOKE  G-  o u t Put 1  Fl L T E R 3  Fl LTER 2  ATTENUATOR G  a  O output 2  FILTER4  G  output 4  3.1  F I L T ER 6  Fl LT ER 5  °f  Fig  output 3  f  G  f  output 5  O v e r a l l Block Diagram o f S l i p Frequency  Measurement  ou t pu 16  16  Input Buffer  (b)  (d)  (e)  F i g 3.2 (a)  Phase  Shifter  (c)  Phasor Diagram o f Phase Shifter  (e)  Attenuator G  (f)  C i r c u i t Components (b)  Choke  (d)  Input B u f f e r  (f)  Second Order B u t t e r w o r t h Low Pass F i l t e r Stage (each o f them from F i l t e r 1 t o F i l t e r 6)  G  i  17  T a b l e 3.1  Specification  Choke  150  T Phase Shifter  p  in Fig  120/6.3 0-20  C  5 yF  R_.  Circuits  3.2  Henry  R . yi 3  of  R  0-500kQ  v2  kfi  100 kn  FX  Input  Buffer  R  f l  155 k?2 22 kfi.  R P R . al  1. Mfi  G  R  150 kfi  G .  al  =1 R  a  + R , — — =1/7.4 R . + R _+R , al a2 ai  a  Attenuator  R  a3  5  6  '  G  ^  a3= R  9  a  1  +  al  R ' R = a2 ai  1  +  /  2  °  6  22 kfi  R P R_ F  Filterl  to  50 kn  R  10  Rj^  6.8  R2  56 kn  C  0.32,  kS2  Filter6  Note:  kn  0.47,  Selection  of R ,  R ,  R ,  described  in section  3.3  p  f  1  1.0,  2.47,  4.4  R  C for  Filterl  2 >  yF  to  Filter6  is  18  3.2  I n d u c t i v e Device Choke T h i s d e v i c e p i c k s up, by magnetic  components o f i n t e r e s t .  i n d u c t i o n , the s i g n a l  frequency  The s t r e n g t h o f the s i g n a l p i c k u p depends upon  the core s i z e , i n d u c t a n c e and the l o c a t i o n o f the choke r e l a t i v e t o the motor. The 3.3  A choke w i t h an i n d u c t a n c e o f 150  design o f b e t t e r t r a n s d u c e r was  Second Order B u t t e r w o r t h Low-Pass T h i s i s the most impotant  of  not  selected.  attempted. Filter  part i n the o v e r a l l c i r c u i t  s i x second o r d e r B u t t e r w o r t h f i l t e r  Butterworth f i l t e r has f l a t  Henry i s a r b i t r a r i l y  stages.  and  consists  The second o r d e r low-pass  c h a r a c t e r i s t i c s i n the pass band and r a t h e r  sharp c u t o f f i n the s t o p band. makes the c a l c u l a t i o n o f f i l t e r  No r i p p l e i n the frequency response gain easier.  curve  The p r o p e r s i z e o f a m p l i f i e d  s l i p f r e q u e n c y s i g n a l component a f t e r each stage o f f i l t e r i n g s h o u l d be obtained.  The d e s i g n o f a second o r d e r B u t t e r w o r t h low-pass f i l t e r  be d i s c u s s e d i n the (1) Frequency The  following. Response E q u a t i o n  f r e q u e n c y response  e q u a t i o n o f each f i l t e r  stage can be  as f o l l o w s G(OJ)  G  =  1 _ (JL_)  2 +  2 5  c where G  1 + ——  will  (j ± - ) c  i s the dc g a i n o f t h e f i l t e r  1  i s the c u t o f f frequency  (3.3) (3.4)  c  + i s the damping  coefficient  See t h e d e r i v a t i o n i n d e t a i l i n Appendix A l .  (3.5)  expressed  19  When  _ _1_  the f i l t e r  e q u a t i o n i s then  J2  |G(co)| - • , * 1 + (—V  and  the E;  =  (3.6)  s u b s t i t u t i n g i n t o eqn.  (3.5)  /2  (3.7) R  The f i l t e r  stage has an a t t e n u a t i o n of -40  C  2 2 dB/decade and  the  total  a t t e n u a t i o n c h a r a c t e r i s t i c i s p r o p o r t i o n a l t o the number o f f i l t e r The  v a l u e s o f the c i r c u i t elements can be determined  equations  Parameter Value  Determination  For s i m p l i c i t y o f c i r c u i t capacitors  f  from the above three  (3.3), (3.4), (3.7).  (2)  for  stages.  "  and  - i -  c  2TTC  where f  / AJ  c a l c u l a t i o n i n d e n t i c a l v a l u e s were chosen  i n F i g u r e 3.2  -  i  ( f ) . Now  eqn.  ( 3 . 4 ) , (3.7) become  -  R^R  (3.8)  i s the corner  frequency  (3.9)  From eqn.  (3.8) the c o r n e r frequency  i f R^ and from eqn.  are kept c o n s t a n t .  f  o n l y depends upon t h e c a p a c i t o r C  Furthermore, R^ and  R^  ( 3 . 3 ) , ( 3 . 8 ) , (3.9) once the parameters G, f  The v a l u e s o f f , G and c' a.  f  =18  can be and  calculated  C are given.  C a r e s e l e c t e d as f o l l o w s :  Hz  c The  c o r n e r frequency  f  i s s e l e c t e d t o be 18 Hz because i t i s b a s i c a l l y  s u f f i c i e n t f o r measurement o f the whole range o f motor s i n c e the  overall  20  cut  o f f frequency w i l l decrease w i t h an i n c r e a s i n g number o f f i l t e r  stages.  Most i n d u c t i o n motors o p e r a t e a t speeds c o r r e s p o n d i n g t o ^ s l i p s o f l e s s 8% i . e . f  than  l e s s than 5 Hz.  s  R  b.  F i.e. G = 1 t — — f  G = 6  = 6  R  and R  = 50 K ft, R  p  = 10 K ft a r e s e l e c t e d .  f  C o n s i d e r i n g t h e minimum s l i p f r e q u e n c y component V  i n choke v o l t a g e , which  g  —6 happens t o be a t t h e lowest s l i p f r e q u e n c y , a v a l u e o f 75 x 10  was  o b t a i n e d (see T a b l e 2.3) i n t h e simple experiment o f c h a p t e r 2.  So t h i s  value o f V six  . s h o u l d be a m p l i f i e d t o t h e proper v a l u e V"*" . a f t e r smin . smm  filter  total  stages.  For t h e s e l e c t i o n G = 6, t h e a m p l i f i e d output V" . s h o u l d be smin c  V  . = smin  G  5  ' V  . smin  =  6  5  x 75 x 1 0 ~  6  =  T h e r e f o r e , t h e output o f 3.5 v a t t h e s i x t h f i l t e r  3.5 v o l t s  stage i s a r e a s o n a b l e  v a l u e t o be measured and t h e s e l e c t i o n G = 6 i s r e a s o n a b l e . c.  C = 0.47 uF  By use o f t h e above s e l e c t i n g parameter v a l u e s , (G=6, f =18 Hz c  —6 C=0.47 x 10  F ) , R^ and R  can be worked out by s o l v i n g t h e eqn. ( 3 . 3 ) ,  2  ( 3 . 8 ) , (3.9) as f o l l o w s : F G = 1 +_£. = 6 R. R  s i n c e _p c R, R, 1  so  2  R R 1  2nC  ^  1  2  1  (27if C ) c 2  R R  2  = 0.0354 x 1 0  6  ( 2 T T x 18 x 0.47 x 10 ) N  1 0  (3.10)  2  21  (6-1)  =  1.414  (3.11)  R„  Let  s u b s t i t u t i n g x i n t o eqn. (3.11). _5 X  +  X  1.414  x  - 4 = 0  X  X = 2.83  or  -1.39  The c o r r e c t s o l u t i o n i s x = 2.83 R  Then  C  V R  2  /  R_  R  1 2  =  =  R  2.83  =  2  Q.O^  8 R,  =  0.0354 x 10  10  6.65 x 10 ft  =  53.5 x 10 ft  T h e r e f o r e , R^ = 6.8 K ft and  = 56 K  Now v e r i f y t h e E, and f R  ±  = 6.8 K ft, R  2  using available r e s i s t o r values.  by s u b s t i t u t i n g t h e s e l e c t e d a c t u a l  values  = 56 K ft, C = 0.47 uF i n t o eqn. ( 3 . 5 ) , ( 3 . 8 ) .  so  { hri J +  R  =  =  0.5 {  2  I 6.8  8  (G-l)  \hr  R  +  t  -  _  (6  .  x  1}  2  }  6.8 56  0.74  2irc  =  56 6-  R  17.3 Hz  ./R^  6.28 x 0.47 x 10  6.8 x 56 x 10  22  The a c t u a l £  =0.74 and f  = 17.3  Hz are c l o s e t o the s e l e c t e d  c values of f  = 18 Hz and  £ = 0.707.  The  c o r n e r frequency  by u s i n g d i f f e r e n t v a l u e s o f c a p a c i t o r s w h i l e keeping values  f  i s varied  a l l o t h e r element  fixed. C = 0.32,  0.47,  1.0,  2.47,  4.4  yF  The a t t e n u a t i o n f o r each f i l t e r stage i s -40 a t t e n u a t i o n from i n p u t t o output employed.  i s p r o p o r t i o n a l t o the number o f  S u c c e s s f u l d e t e c t i o n of s l i p frequency  speed h e a v i l y r e l y upon the r e a s o n a b l e and  t o t a l attenuation.  3.4  Input B u f f e r and The  dB/decade and the  total  stages  f o r measuring each  s e l e c t i o n o f both c o r n e r  frequency  Attenuator  i n p u t stage o f the measuring c i r c u i t  c o n s i s t s of a n o n i n v e r t i n g  o p e r a t i o n a l a m p l i f i e r w i t h h i g h i n p u t impedance, as i n d i c a t e d i n F i g u r e 3.1,  F i g u r e 3.2(d).  I f the s i g n a l from the choke, which has h i g h  t a n c e , i s d i r e c t l y connected  t o the f i r s t  c a p a c i t a n c e feedback c i r c u i t  the c i r c u i t  o s c i l l a t i o n s w i l l take p l a c e .  induc-  f i l t e r i n g stage t h a t i n c l u d e s the w i l l become u n s t a b l e  and  T h e r e f o r e , the i n p u t o p e r a t i o n a l a m p l i f i e r  a c t s as a b u f f e r stage between the choke s i g n a l source and the i n p u t o f the f i r s t  filter  stage.  As mentioned p r e v i o u s l y , the d e s i g n of each f i l t e r t h a t the weak s l i p frequency supply frequency two ing.  s i g n a l s Vg^  s i g n a l component V*  s  w i l l be a m p l i f i e d and  s i g n a l component w i l l be a t t e n u a t e d . to V  a t any f i l t e r  stage must be  The r a t i o o f  stage i n d i c a t e s the degree o f  such the  these  filter-  Where more than f o u r stages o f f i l t e r i n g i s r e q u i r e d , the a t t e n u a t o r  circuit  i s r e q u i r e d t o m a i n t a i n o p e r a t i o n i n the l i n e a r r e g i o n o f the  operational amplifier.  A l t e r n a t i v e l y , a t t e n u a t i o n c o u l d have been  p r o v i d e d i n each f i l t e r  stage.  23  3.5  Phase One  frequency  Shifter  of the o b j e c t i v e s t o use a f i l t e r  i s t o a t t e n u a t e the supply-  s i g n a l component 60 Hz i n the mixed source s i g n a l so t h a t the  s l i p frequency component can be i d e n t i f i e d , thus the r a t i o o f V g can be minimized  a f t e r each f i l t e r  stage.  to  Q  The phase s h i f t e r h a v i n g  V  g  supply  frequency can a l s o be used i n the same r o l e as the f i l t e r by p r o v i d i n g a phase-shafted 3.1,  3.2.  supply frequency  The phase s h i f t e r  signal.  How  ( F i g u r e 3.1)  i t works i s shown i n F i g u r e s  can vary i t s output phase  r e l a t i v e t o the phase o f the supply frequency component o f the choke voltage.  L e t the phase s h i f t e r output superpose magnetude o f  and a d j u s t the phase and  on the choke v o l t a g e  phase s h i f t e r  output.  Then the  put o f b u f f e r w i l l have a g r e a t l y a t t e n u a t e d supply frequency Hence the number o f f i l t e r  stages r e q u i r e d would be l e s s .  in-  component.  The major  components i n the phase s h i f t e r shown i n F i g u r e 3.2(a) i n c l u d e the c e n t r e tapped  transformer T , variable r e s i s t o r  f u n c t i o n o f T^ i s t o reduce  the supply  and c a p a c i t o r C^.  v o l t a g e to  a suitable  The size.  The p r i n c i p l e o f phase s h i f t e r can be r e a d i l y found i n the diagram F i g u r e 3 . 2 ( a ) , ( c ) .  The  s u p p l i e s t h e s e r i e s RC c i r c u i t v o l t a g e V^g R 2_»  across  f i x e d supply v o l t a g e V  w i t h c e n t r e tap  i n the c l o s e d - l o o p c i r c u i t  always l a g s by 90°  ACBD.  a c r o s s the v a r i a b l e  The resistor  The p o t e n t i a l D r e l a t i v e t o ground C w i l l move on the c i r c l e  i n F i g u r e 3.2(c) i f the r e s i s t o r R ^ v a r i e s . v a r i a b l e v o l t a g e output R  v 2  i s used.  The  example, once the  switch  i s a r e v e r s i n g switch  r e v e r s e s , the p o t e n t i a l D w i l l f a l l  D' i n the next h a l f c i r c l e this circuit  (Figure 3.2(c)).  shown  In o r d e r t o o b t a i n a  which a l l o w s the phase s h i f t e r t o work i n another h a l f c i r c l e .  for  circuit  into  For symmetric  So the range o f phase a n g l e  can change t h e o r e t i c a l l y from 0° t o 360°.  However, i t  24  i s impossible  t o use an i n f i n i t e v a l u e o f c a p a c i t o r , and t h e a c t u a l  range o f phase s h i f t e r mainly depends upon the s i z e o f c a p a c i t o r  used.  25  4.  EXPERIMENTAL RESULTS  The proposed method o f speed measurement was i n d u c t i o n motors.  The s p e c i f i c a t i o n s f o r these motors a r e g i v e n i n Table  T a b l e 4.1  Full Motor  t e s t e d on f o u r d i f f e r e n t  Motor S p e c i f i c a t i o n s  Load  Horse  Speed (rpm)  Type of Motor  Rated  Rated  Power P (Hp)  Voltage V (v) .  Current I (a)  Ml  1725  3-phase wound r o t o r  1/3  1.7  220  M2  1725  3-phase s q u i r r e l  cage  1/4  1.5  220  M3  1725  1-phase s q u i r r e l  cage  1/4  5.5  120  M4  1690  3-phase wound r o t o r  8  208  2.5  The e x p e r i m e n t a l setup i s shown i n F i g u r e 4.1. the v i c i n i t y o f the t e s t motor and Its  4.1.  output i s taken t o a f i l t e r ,  is left  A choke i s p l a c e d i n  i n p o s i t i o n throughout  as d i s c u s s e d i n c h a p t e r 3.  The  the  test.  filter  output i s then f e d t o a U n i v e r s a l Waveform A n a l y z e r which i s a 68000 m i c r o processor-based instrument.  The dc g e n e r a t o r i s m e c h a n i c a l l y - c o u p l e d t o  the i n d u c t i o n motor, and with i t s r e s i s t a n c e l o a d p r o v i d e s a v a r i a b l e motor l o a d . comparison filter for  w i t h t h a t determined  stages, value of f i l t e r  from t h e s l i p f r e q u e n c y .  The number o f  c a p a c i t o r and a t t e n u a t o r g a i n were a d j u s t e d  best s l i p frequency determination.  cy s i g n a l and  The a c t u a l motor speed i s measured by a stroboscope f o r  component can be determined  The  amplitude o f the s l i p  by the use o f the f i l t e r  c h a r a c t e r i s t i c curves as d e s c r i b e d in appendix  A2.  frequen-  g a i n fomulae  26  Resistanse Load  Choke Universal  Filtering  I-  I  Waveform  Circuit  Analyzer Induction Motor  z F i g 4.1  Experimental  Setup  E x p e r i m e n t a l r e s u l t s f o r t h e f o u r i n d u c t i o n motors a r e g i v e n i n Table 4.2 t o 4.5. I  The f o l l o w i n g symbols a r e used i n t h e T a b l e s .  Motor c u r r e n t  n  A c t u a l speed measured by t h e s t r o b o s c o p e , 3.  (* means speed  at o r c l o s e s t  N  Number o f t h e f i l t e r  t o rated speed )  s t a g e s a t which t h e output i s measured  C  Value o f c a p a c i t o r used i n the f i l t e r  G a  Gain o f t h e a t t e n u a t o r  f  Slip  v  Amplitude  of f  Frequency  of "noise" signal  Amplitude  of f  stages  frequency at the f i l t e r  output  n at f i l t e r  circuit  output  n Motor speed c a l c u l a t e d from t h e measured f C a l c u l a t e d i n p u t amplitude o f t h e f  v  sin v n  ^  n  eqn.  v Q g  component by  (A2.14)  C a l c u l a t e d i n p u t amplitude o f t h e f eqn.  through (1.7)  component by  (A2.14)  Induced  s u p p l y frequency (60 Hz) amplitude  i n t h e choke  T a b l e 4.2 M oto r  Filtering  Experimental R e s u l t s of Motor Ml  Circuit  Output  of F i l t e r i n g  Circuit  Input' V  1  fa)  n  a (rpm)  N  3 1.14  1.2  1.3  1.45  1.65  C (MF)  .  G  «n  Waveform (H«)  (v)  0.98 0.24  1.0  (H»)  V„ (v)  2.93 .032  60Ain  Calculated (»v) »in nin  n , (rpm)  v  1770  0.73 0.1  v  (*10 ) 3  (V)  1.4 1.02  1770 6  0.32  3  1.0  0.18  1/206  1.56 0.44  30  0.14  4.3  .034  0.52 8.3  1753  1.41 0.12  0.7C 1.07  1755 6  0.32  3  1.0  0.32  3  1.0  1/206  1710* 6  0.32  3  1.0  1/206  1680 6  0.32  1/206  /\/\s A A A A A A A A AAAA  A A A A  9.5  0.16  1742  1.94 0.74  1740 6  30  1/206  2.25  0.49 1.1  0.65  30  1.87 10.7  0.18  1711  2.96 0.12  3.8  0.3 1.18  1.13  3.9  30  3.3  0.23  1.86  1683  13.6  0.20  6.5 1.29  1.67  30  0.3  4.96 17.8  T a b l e 4.2 Motor 1  "a (rpm)  Filtering Circuit N  .  •  1.0  1650 6  0.32  1/206  . 3. 1.0 2.06  1620 6  0.32  t  Output  of F i l t e r i n g  Circuit  Input'  G  C (MF)  3 1.86  (continued)  1/206  :  Calculated Waveform  AAAAA AAAAA AAAAAA AAAAAA  In t h e s e two cases each one has about i t s c e n t r e speed  (Hi)  (v)  5.08  2.16  (Hi)  (v)  n, (rpm)  1647  v  (mv) tin nin v  60 (v)  V  J  (X10 )  0.16  8.49 1.33  2.23 30  0.31  5.86 2.03  6.7  1624  18.4  0.14  9.89 1.37  2.75 30  a low r a t e of o s c i l l a t i o n  1770 & 1755 as observed  8.4  0.33  19.6  2.93 Hz & 4.3 Hz  by a s t r o b o s c o p e  T a b l e 4.3 M  oto r  Filtering  Experimental R e s u l t s of Motor M2 Output  Circuit  of  Filtering  Circuit  Input' V  1 fa)  "a (rpm)  N  3 1.1  c  6  a  (Hi)  1.15  (v)  n, (rpm)  1778  3  (*10 )  (mv) V » i n ^n i n  (V)  0.186  2.04 0.38  0.32  30  1/7.4  1.1  40.8  19.1  1767  .067  0.27  1.15 0.362  1767 6  0.32  3  4.4  30  1/7.4  37.4  17.5  1754  1.52 .055  0.34  1.01 0.345  1755 6  0.32  3  4.4  h  1740 6  0.32  5  1.0  1/7.4  1725 6  0.32  30  1/7.4  * 1.15  (Hi)  0.73 .052  4.4  3 . 4.4  1.15  (v)  V„  1778 6  1.1  'n  Waveform  (MF)  6oAin  Calculated  1/7.4  AA\A AAA  32.5  15.2  1741  1.96 .034  0.46  0.75 0.346 <•  30  2.5  28.9  13.5  1725  .74  0.72  0.50 0.362  30  9.51  20.4 i  T a b l e 4.3 (continued) M  oto r  Filtering  Circuit  Output  of  Filtering  Circuit  1 nput' V  1 (a)  a (rpm) n  N  1.27  1.37  1.0  (Hi)  1/7.4  1710 6  0.32  1/7.4  5.  1.0  1/7.4  1695 6  0.32  1/7.4  5  1.0  1/7.4  1680 6  0.32  'n  Waveform  (MF)  5 1.2  C  1/7.4  A A A /  (v)  (Hi)  n, (rpm)  1712  v  (mv) sin nin v  V  60  3  (*10 )  (V)  0.63  0.552 0.346  8.14  3.46 .865  17.4  1696  0.52  0.646 0.337  30  AAAAA  (v)  2.92 .792  30  A A A A  V„  6o/Kin  Calculated  4.1  7.14  .933  15.2  1677  0.46  0.728 0.334  30  6.28  13.4  T a b l e 4.4 E x p e r i m e n t a l Motor  Filtering  Circuit  R e s u l t s o f Motor M3 Output  of  Filtering  Circuit  Input' V  1  (a)  "a (rPm)  N  5.1  5.1  5.16  5.4  5.7  C  Waveform  (MF)  5  1.0  60Ain  Calculated. (Hz)  1/7.4  (v)  (Hi)  (v)  0.61 0.31  n (rpm) 1  1782  ( xio )  (mv) in  v  3  n in  (V)  0.21  18.5  1882  3.88 6  0.32  1/7.4  5.  1.0  1/7.4  30  4.5  1.03 .547  9.6  1769  0.37  9.3  1770  3.44 6  0.32  1/7.4  5  1.0  1/7.4  1755 6  0.32  1/7.4  5  1.0  1/7.4  1740 6  0.32  1/7.4  5  1.0  1/7.4  1725* 6  0.32  1/7.4  30  3.7  1.48 .801  /\y\ / w AAA AAA AAA  7.9  1756  0.56  5.5 3.11  30  3.2  6.8  1738  2.07 .946  0.68  4.4 2.96  30  2.5  5.3  1724  2.53 1.0  0.73  3.7 2.7  30  2.1  4.5  T a b l e 4.4 Motor  Filtering  (continued)  Circuit  Output  of  Filtering  Circuit  Input'  c 1  <a)  "a (rpm)  N  ( J* F )  5 5.95  C  1.0  .  1/7.4  1710 6  0.32  1/7.4  Calculated Waveform  AAA/ AAA/  (Hi)  (v)  2.94  1.09  (Hi)  (v)  n  t  (rpm)  1712  v  (mv) »in nin v  (*10 ) 3  (v)  0.814  3.0 2.48  30  1.8  3.9  T a b l e 4.5 Motor  Filtering  Experimental  Circuit  R e s u l t s o f Motor M4 Output  of  Filtering  Circuit  Input'  ^W^in 1 fa)  1.3  1.72  2.2  2.7  "a (rpm)  1770  1740  N  C  Calculated 6  a  (MF)  Waveform (Hi)  (v)  (Hi)  (v)  n  t  (rpm)  v  (mv) »in nii»  3  4.4  1.05  .029  1769  0.11  3  1.0  1.04  .034  1769  0.128  6  0.32  3  4.4  3  1.0  6  0.32  3  1.0  1/7.4  / \ / \  / X X X  0.32  3  1.0  0.32  0.23  2.05  .079  1739  0.24  1/7.4  3  0.45  3.5  0.48  2.0  29.5  1708  1.45  0.35 0.50  30  A A A A  13.8  0.11  1/7.4  1680* 6  1739  3.08  ( X10 ) (V)  15.9  .017  30  AAA  5.55  2.03  1/7.4  1710 6  30  v  3.9  33.6  15.7  0.17  1683  0.87  0.58  15.6  1.74  0.28  24.4  6.9  4.9  0.50  Table 4.5 (continued 1) Motor  Filtering  C ircui t  Output  of  Filtering  Circuit  Input' V  1 (*)  "a (rpm)  N  3 3.23  4.57  4.9  a  (Hi)  A A A A A  1.0  0.32  6  0.32  VUAVWUAi  3  1.0  1620 1/7.4  AAAAAAA  1590 0.32  3  1.0  0.32  3  1.0  6  0.32  1  (rpm)  1647  v  (mv) sin nin v  ^ V V V A A / V \ A A  3  (V)  0.63 0.27  22.5  4.6  10.8 1617  (X10 )  0.79  12.7 2 . 0  6.04 0 . 2 3  0.50  1.1  0.45 0.50  21  12.5  4.04  33.2  4.4  14.2  7.01 0.24  1586  1.55  0.33 0.51  8.0  0.23  1/206  1/206  (v)  n  19.5 0 . 6 6  AAAAM  1530  (Hz)  V„  0.2  1/206  1560 6  5.1  (v)  1/7.4  AAAAAA/  6  'n  Waveform  1.0  3.  4.22  G  1650 6  3.7  C  (<"F)  6oAin  Calculated  9.1  0.16  4.8  1560  2.02  0.25  12.7 0 . 3 7  1.35  17.6 0 . 6 5  3.5  6.84 16.6  0.12 .015  1527  2  '  0 3  0.74 .075  0.51  0.25 0.51  T a b l e 4.5 (continued 2) Muto r  1 <a>  Filtering  a (rpm) n  N  5.7  G  a  (MF)  3 5.3  C  Circuit  Output  of  Filtering  Circuit  Waveform (Hi)  V\AMMA  1.0  Input'  (v)  0.32  3  1.0  V„  (Hi)  (v)  10.0 0.13  Calculated n, (rpm)  1500 4.9 14.7  1500 6  'n  V  (mv) V* i n  ^n i n  60 (V)  2.4  0.25 0.12 2.66  .031 .024  3  (X10 )  0.52  1/206  12.1  1440  .068  0.16  3.3  1440  0.52 6  0.32  1/206  0.91  13.7 5.9  1380  6  0.32  1/206  6.4  1320  6  0.32  1/206  6.8  1260  6  0.32  1/206  7.5  1200  6  0.32  1/206  0.67  1369 8.8  u  1.76 1257  0.92 1200  1.95  0.17  0.12  0.55  0.10  0.56  .078  0.58  .076  7.16  0.32  1.0  0.53  5.3  0.57  1.24 2.73  20.0  2.8 1323  5.86  4.5  0.87  15.9 1.16  18.1  3.02  1.06  7.7 0.48  36  4.1  Discussion of Experimental Results The b a s i c i d e a o f u s i n g t h e s l i p frequency t o determine  t h e speed o f  i n d u c t i o n motors r e l i e s on c o u n t i n g t h e zero a x i s c r o s s i n g s o f t h e output s i g n a l waveform from t h e f i l t e r .  T h e r e f o r e , the shape o f t h e wave-  form i s i m p o r t a n t .  Any d i s t o r t i o n i n t h e s i g n a l waveform w i l l  speed measurement.  The s i g n a l waveform can be c l a s s i f i e d as  (1)  There  frequency f .  The waveform c o n t a i n s some n o i s e but the frequency o f t h e  waveform i s s t i l l (3)  follows;  i s no d i s t o r t i o n and t h e s i g n a l f r e q u e n c y i s e x a c t l y  equal t o the s l i p (2)  a f f e c t the  There  e q u a l t o the s l i p frequency f .  i s c o n s i d e r a b l e d i s t o r t i o n i n t h e waveform so t h a t the  frequency o f the waveform i s no l o n g e r e q u a l t o the s l i p The r e s u l t s f o r a l l measured f i l t e r waveform c l a s s i f i c a t i o n a r e l i s t e d  frequency f .  outputs a c c o r d i n g t o t h e above  i n T a b l e 4.6.  The expected and  measured s l i p frequency f o r each motor speed a r e shown i n F i g u r e 4.2. From F i g u r e 4.2 and T a b l e 4.6 some important p o i n t s can be deduced as follows: (1)  The speed o f an i n d u c t i o n motor w i t h i n the r a t e d l o a d , i . e . f o r  s<13%, o r f (2)  K  & Hz, can be a c c u r a t e l y measured by the proposed  The measured s l i p f r e q u e n c y would be f u l l y  method.  e q u a l t o t h e expected  v a l u e i f t h e r e a r e no u n s t a b l e f a c t o r s , such as f l u c t u a t i o n s i n the motor speed. (3)  Proper f i l t e r  parameter v a l u e s , such as t h e c o r n e r frequency f ,  must be chosen t o e l i m i n a t e any n o i s e i n t h e output s i g n a l . of  C=1.0 y F , i . e . , f =4.5  the f o u r t e s t motors used.  Hz f o r t h r e e stages appears  The s e l e c t i o n  t o be s u i t a b l e f o r  T a b l e 4.6 Motor  C  (rpm)  1770  1755  1740  1725  1710  1695  3.5  1620  4  5  6  2  1.0  o  o  o  o  o  o  o  0.32  A  A  A  A  A  A  A  o  o  o  s  (Hz)  4 .4 or 1.0  3  1650  1680  1.5  f  2.5  S l i p Frequency  1  (yF)  Ml  n  Waveform E v a l u a t i o n of Detected  o  o  o  o  X  X  M2 0.32  X  X  X  X  X  1.0  o  o  o  o  o  0.32  A  A  A  A  A  4.4  o  o  1.0  A  A  o  o  0.32  X  X  X  X  M3  M4  o  o  1590  1560  1530  1500  1440  1380  1320  1260  1200  7  8  9  10  12  14  16  18  20  1.0  o  o  X  A  A  0.32  X  X  X  X  A  C  n  (rpm)  Motor (yF)  f  g  (Hz)  M4  Case (1) : o  Case (2) : A  X  A  Case (3) : x  o  A  38  *s (Hz)  M4 20 f  JO  s(Hz)  M3 10  1800  1700  (rpm)  0  0.05 5  s  1800 0  1600  1400  1200  (rpm)  0.11  0.2 2  0.33  s  (c)  F i g 4.2  (d)  Comparision o f Expected and Detected S l i p f o r I n d u c t i o n Motor x  expected v a l u e case (1)  (a) Ml o  (b) M2  (c) M3  detected value case  (2)  Frequency (d)  M4  39  4.2  Supply Frequency S i g n a l Component The amplitude  o f the s u p p l y frequency f  (60 Hz) component  that i s  induced i n the choke, o f c o u r s e , dejpends upon the l o c a t i o n o f t h e choke w i t h r e s p e c t t o the motor. to  U s u a l l y the choke i s p l a c e d as c l o s e as p o s s i b l e  the motor so as t o p i c k up a s t r o n g s i g n a l .  most important slip  s i g n a l t o be f i l t e r e d out i n the p r o c e s s o f d e t e c t i n g the  frequency.  The number o f f i l t e r  upon the amplitude  stages r e q u i r e d not o n l y depends  o f t h e 60 Hz s i g n a l but more i m p o r t a n t l y upon t h e  amplitude r a t i o o f the 60 Hz s i g n a l t o the f  of  The 60 Hz s i g n a l i s the  signal. s °  The v.^/v,. r a t i o s 60 f  f o u r motors a r e measured and i n d i c a t e d i n F i g u r e 4.3. It  i s found t h a t t h e amplitude  of the s l i p frequency s i g n a l increases  g r e a t l y w i t h i n c r e a s i n g s l i p whereas t h e amplitude o f the supply  frequency  s i g n a l v a r i e s very s l i g h t l y .  frequency  amplitude  v s  ^  n  From e x p e r i m e n t a l r e s u l t s the s l i p  can be c o n s i d e r e d t o be a p p r o x i m a t e l y p r o p o r t i o n a l t o the  p r o d u c t o f the motor c u r r e n t and the s l i p v .  = I«f  sin  (4.1) s  The r a t i o o f the amplitude of  frequency.  o f the s u p p l y frequency s i g n a l t o the amplitude  the s l i p frequency s i g n a l governs how  many f i l t e r  stages are r e q u i r e d .  However, the c l o s e r the s l i p f r e q u e n c y i s t o the s u p p l y f r e q u e n c y , the more d i f f i c u l t 4.3  i s the f i l t e r i n g .  Measurement  of Pole P a i r s  The motor speed n^ not o n l y depends upon the measured s l i p f  frequency  but i s a l s o r e l a t e d t o t h e p o l e p a i r s , p, as g i v e n i n e q u a t i o n (1.6)  and r e w r i t t e n below:  n. 1  =  f-f • S- 60 P  (1.6)  V  60/Vsin  (x10 ) 3  41  v a  Fig  4.5  Choke V o l t a g e Waveform a t Several  Locations  (a) Supply Frequency Reference Waveform (b) a t L  The f o l l o w i n g experiment i l l u s t r a t e s value.  one way  1  (c) at L  2  (d) L  3  t o measure the p o l e - p a i r  The experiment uses t h e i n d u c t i o n motor M3,  which i s s i n g l e phase.  With t h e motor r u n n i n g , t h e p o s i t i o n o f the choke i s v a r i e d around the  42  motor s u r f a c e , as i n d i c a t e d i n F i g u r e 4.4.  The choke v o l t a g e i s connected  o f a two-channel o s c i l l o s c o p e w h i l e t h e 60 Hz supply wave-  t o one channel  form i s connected t o the o t h e r channel and i s used as t i m i n g r e f e r e n c e (shown i n F i g u r e 4 . 5 ( a ) ) . location L , L  and L  The waveform  o f t h e choke v o l t a g e f o r d i f f e r e n t  (here a=90°, B= 180°) a r e shown i n F i g u r e  (c),(d), respectively.  With the choke a t l o c a t i o n  4.5(b),  t h e l e f t m o s t peak  v a l u e o f the choke v o l t a g e i s noted t o be on t h e assumed v e r t i c a l a a ' i n d i c a t e s t h a t t h e m e c h a n i c a l angle 180°.  line  a i s e l e c t r i c a l l y equivalent to  S i m i l a r l y , moving the choke t o l o c a t i o n L  back a t l i n e a a ' so t h a t the m e c h a n i c a l angle  w i l l p l a c e t h e peak  3 is electrically  value  360°.  T h e r e f o r e , the p o l e - p a i r v a l u e o f t h e motor can be o b t a i n e d as f o l l o w s :  P  =  360° —  (4.2)  ^  (4.3)  In t h i s example  a=90°  g=180°  so p=2. I t i s n o t v e r y hard t o e x p l a i n .  The choke v o l t a g e waveform a t each  l o c a t i o n w i t h r e s p e c t t o t h e motor u s u a l l y a r e t h e same but i t s phase a n g l e w i t h r e s p e c t t o the s u p p l y v o l t a g e v a r i e s .  Therefore, the r e l a t i o n  between e l e c t r i c a l and m e c h a n i c a l angle w i l l y i e l d t h e p o l e - p a i r v a l u e as g i v e n by e q u a t i o n s  (4.2) and ( 4 . 3 ) .  43  5.  CONCLUSION  A method o f measuring t h e motor speed o f an i n d u c t i o n motor by d e t e c t i o n o f t h e s l i p frequency s i g n a l has been proposed i n t h i s The measurement shaft.  thesis.  can be made without attachment o f any d e v i c e s t o t h e motor  In f a c t , t h e motor may be c o m p l e t e l y e n c l o s e d so t h a t  speed measurement  conventional  by t h e use o f tachometers o r s t r o b o s c o p e s would not be  possible. E x p e r i m e n t a l r e s u l t s i n d i c a t e good agreement between a c t u a l motor speed and t h a t measured by t h e proposed s l i p f r e q u e n c y method i n t h e range below a s l i p frequency o f about 8 Hz t h a t i s w i t h i n t h e range o f most 60 Hz i n d u c t i o n motors.  The method i s s u i t a b l e f o r b o t h s i n g l e - and  three-phase i n d u c t i o n motors and w i l l be a p p l i c a b l e t o s e a l e d motors, such as motors used i n r e f r i g e r a t o r s , s i n c e a c c e s s t o t h e motor s h a f t i s n o t needed. The proposed method does r e q u i r e knowledge o f t h e number o f p o l e s i n the  motor.  A simple method t o determine t h i s e x p e r i m e n t a l l y was d i s -  cussed i n c h a p t e r 4. The speed measurement c i r c u i t improved 0)  C2)  i n the  following  i n c h a p t e r 3 needs t o  t r a n s d u c e r s h o u l d be o p t i m i z e d  quency  pickup.  signal  Instead o f d e t e c t i o n o r the  available  for  be  respects:  The i n d u c t i v e  signal  (3)  described  f o r best  by the waveform a n a l y z e r  corresponding display or  for  motor use  the  speed s i g n a l  slip  slip-fre-  frequency  s h o u l d be made  i n any speed c o n t r o l  loop.  The f i l t e r i n g and a m p l i f y i n g s t a g e s c o u l d be o p t i m i z e d  for  per c u t  desired  slip  - o f f and  frequency  improved s i g n a l - t o - n o i s e  signal.  ratio  o f the  shar-  44  REFERENCES  1.  Sawaki, N. and Sato, N., " S t e a d y - S t a t e and S t a b i l i t y A n a l y s i s o f I n d u c t i o n Motor D r i v e n by C u r r e n t Source I n v e r t e r " , IEEE T r a n s , on IA.  2.  V o l . IA-13, pp.244-253.  P l u n k e t t , A.B.,  May/Jun., 1977.  " D i r e c t F l u x and Torque R e g u l a t i o n i n a PWM  Invertor-  I n d u c t i o n Motor D r i v e " , IEEE T r a n s , on IA, V o l . IA-13, pp.139-146, Mar./Apr., 3.  1977.  P l u n k e t t , A.B.,  D'Atre, J.D. and L i p o , T.A.,  "Synchronous C o n t r o l o f  a S t a t i c AC I n d u c t i o n Motor D r i v e " , IEEE T r a n s , on IA, V o l . IA-15, pp.430-437, J u l . / A u g . , 1979. 4.  Nabace, A., Otsuka, K., Uchino, H. and Kurosawa, R.,  "An Approach t o  F l u x C o n t r o l o f I n d u c t i o n Motors Operated w i t h V a r i a b l e - F r e q u e n c y Power Supply", IEEE T r a n s , on IA, V o l . IA-16, pp.342-350, May/Jun., 1980. 5.  B o u l e r , P. and McLarren, S.G.,  "Slip-Frequency Limited  Phase-Locked  Loop I n d u c t i o n - M o t o r D r i v e " , IEE P r o c , V o l . 127, p t . d , pp.51-54, Mar. 6.  1980.  Moore, A.W.,  "Phase-Locked Loop f o r Motor Speed C o n t r o l " , IEEE  Spectrum, 1973, A p r i l 7.  ( 1 0 ) , pp.61-67.  I s h i d a , M. and Iwata, K.,  "A New  S l i p F r e q u e n c y ' D e t e c t o r o f an  I n d u c t i o n Motor U t i l i z i n g Rotor S l o t Harmonics", IEEE/IAS, ISPCC Conf. R e c , 8.  1982  pp.408-415.  I s h i d a , M. and Iwata, K.,  "A S l i p Frequency D e t e c t i o n Method o f  I n d u c t i o n Motor U t i l i z i n g Rotor S l o t Harmonics", Conf. Rec. o f 1980 T o k a i Region Annual Meeting o f IEE o f Japan, pp.153, Nov.  1980.  45  9.  M a i s e l , J . E . and K l i n g s h i r n , E.A., "Low-Frequency S p e c t r a l  Analysis  Using a Dynamometer-Type Wattmeter", IEEE Trans, on E d u c a t i o n , V o l . E-25, No. 2, May 1982. 10.  Davold, W.,  C i r c u i t Design f o r E l e c t r o n i c I n s t r u m e n t a t i o n .  APPENDICES  Al.  D e r i v a t i o n of the Frequency Response Equation Butterworth  U  Low-Pass F i l t e r  in  Fig A l . l  The  f o r Second Order  Second Order Butterworth  Low-Pass F i l t e r  n o n i n v e r t i n g second order low pass f i l t e r  used f o r the d e r i v a t i o n . A n a l y s i s " and  The  shown i n F i g u r e A l . l i s  d e r i v a t i o n i s based upon the " V i r t u a l Ground  assumes t h a t  (1)  The  open-loop g a i n o f the o p e r a t i o n a l a m p l i f i e r i s i n f i n i t e ,  (2)  i.e. , G = ' o Input impedance i s i n f i n i t e  00  I (3) Equations  = °°.  +  = 0  f o r F i g u r e A l . l by normal c i r c u i t  -  input current  = o.  n Input v o l t a g e u - u  U  Therefore,  = R7T1C O  <AI.D  U  + = 1 + juR C 2  2  a  (A1.2)  a n a l y s i s are as f o l l o w s :  47  T = ^-(u. '1 R m  - u ) a  (A1.3)  I . = j c.(u - u ) cl 1 a o  (A1.4)  I. 2  (A1.5)  u  J  = ^ ( u - u ) R a + 2  S o l v i n g t h i s group o f e q u a t i o n s the g a i n o f f i l t e r G(u) Is expressed as  U  G(ju)  U  G(u))  ° in  ( j ^ ) c  =  2  2 (j^- )  +  5  +  1  —2  (A1.7)  2  i - <A •  f->  c  where 5 i s t h e damping  c  coefficient  C J i s the inherent angular corner frequency c  T h e r e f o r e , the amplitude response G.(w)  and phase response <f>(w) i s  r e a d i l y o b t a i n e d by e q u a t i o n (A1.7).  w  G( ) = —  (A1.8) 2 \2  1 w  c  2? -1 cf>( ) = - t a n " w  w  .  i -  c -  (A1.9)  <A c  2  48  W  h  e  r  e  G = l  +  ^  (ALIO)  1  u) C  J  g  R  R  C  1 2 1 2  pfg.  1 { 2  ' (Al.ll)  C  y  R  l C l  (,.,)•/¥! A  /R C 2  V  1  R  C 2  }  (A1.12)  2  when 5 /2~ e q u a t i o n (A1.8) becomes  l+(^-) c W  T h i s i s the second  order Butterworth f i l t e r  Substituting £ = — —  i n t o eqn. (A1.12)  ST we t h e r e f o r e r e q u i r e  Rc 9  /R.c  /^c  response.  49  A 2 .  Calculation  A2.1  o f S l i p Frequency Signal  Component  Gain Formulae and Curves of Three F i l t e r The  gain of three i d e n t i c a l f i l t e r  Stages i n Cascade  stages  i n c a s c a d e , as i n d i c a t e d  i n F i g u r e 3.1 i s g i v e n by:  ^12H1 in where  =  G.-G*  ^3out  ^  e  o  u  t  (A2.1)  P  u  t  °^  t  n  e  third f i l t e r  stage  i s the i n p u t t o the i n p u t b u f f e r G^  i s the gain of input b u f f e r  G^  i s the g a i n o f a s i n g l e f i l t e r  U s u a l l y the g a i n i s expressed  i n d e c i b e l s , which i s d e f i n e d as f o l l o w s :  m where  ^3oout  ^  S  e  ^  t  u  " P "  t  °^  / t  n  e  om third f i l t e r  stage a t f = 0  V . om  i s t h e i n p u t V. m  G^^  i s the t o t a l g a i n i n dB, t h e i n p u t b u f f e r and the t h r e e  r  filter The  ou  stage  g a i n curves  three f i l t e r  stages  stages  at f = 0  i n cascade  i n F i g u r e A2.1 a r e p l o t t e d from measurements made f o r i n cascade f o r v a r i o u s v a l u e s o f c a p a c i t o r s C.  curves w i l l be used as c a l c u l a t i o n o f measured s l i p frequency for  A2.2  These  amplitude  v a r i o u s i n d u c t i o n motor speeds.  C a l c u l a t i o n of Slip-Frequency o f F i l t e r Stages The  number o f f i l t e r  ments d i f f e r  stages  S i g n a l Amplitude Through Any Number  a c t u a l l y used i n s l i p frequency  f o r v a r i o u s i n d u c t i o n motors.  s i g n a l amplitude w i l l be d i s c u s s e d h e r e .  measure-  The c a l c u l a t i o n o f s l i p - f r e q u e n c y In F i g u r e 3.1 t h e a t t e n u a t o r , G , cL  50  51  is  i n s e r t e d between t h e t h i r d and t h e f o u r t h f i l t e r  s t a g e , so t h a t the  d e r i v a t i o n o f t h e g a i n e q u a t i o n w i l l be d i v i d e d i n t o two cases as f o l l o w s ; (1) When N ^ 3 Where N i s t h e number o f f i l t e r stages  stages.  The g a i n f o r N f i l t e r  i s g i v e n by  S  =  ^  (A2.3) in  Where  s  ^Nout ^  From F i g u r e  G  ^  e  output  NG  i  o f "the Nth f i l t e r  stage.  3.1  =  N  (  Assume  G  f  )  (A2.4)  V. Noout — T  _ • _ No-  G  , (  A  A O 2  c  '  . 5  )  o m  Where  G. No V  So  i s t h e g a i n G„ a t f = 0 N  T  6  i  h  e  O  G„ = (G_ ) No fo  N  G.  Noout  Where  G,. fo  and  G. = G.  G  t  U  t  P  U  t  V  Nout  =  at f = 0  (G ) fo  10  N  G. l  (A2.6)  i s t h e g a i n G_. a t f = 0 f &  10  1  Therefore,  s  t h e g a i n i n dB i s  Ndb  =  2  0  l o  8lor-  (A2.7)  No (A2  G  Ndb  =  2  0  l o  «10  G  N ~  2  0  1O  h0  So '  8)  S u b s t i t u t i n g ( A 2 . 4 ) , (A2.6) i n t o (A2.7) vN  = 20  logoff j  , ^  TO  Assume  „  3  (A2.9)  G_ fo  3  52  S  G  °  Substituting  I  =  Ndb  G  (  3db  (A2.3), (A2.10) i n t o  20 l o g  N _ 3  Nout 10 V.  A  2  .  1  0  )  (A2.8)  G  3db  + 20 l o g G °10 No  (  1  i n  (A2.ll)  in A  s  S  U  m  e  G  =  Nodb  Substituting  G  2  0  1 O g  G  1 0 No  (A2.6) i n t o  Nodb  Substituting  =  2  '  0  l o g  A  2  '  2  )  (A2.12)  10  (A2.12) i n t o  (  G  f o  }  G  (A2.13)  i  (A2.ll)  Nout  an  -UE 2CT3  G  +  3db  G  Nodb  (A2.14)  )  10  Nodb  inequation  (A2.13) i s independent o f c i r c u i t  c a p a c i t a n c e C.  I t can  be found i n t h e curves i n F i g u r e A2.1 t h a t G , a t f = 0 f o r a l l c u r v e s 3db OJ  s t a r t a t ^^^=0.  I t can be a l s o e x p l a i n e d as t h e c i r c u i t  t h a t when t h e c i r c u i t f o r both C a r i s e s .  Figure  i s working i n dc no e f f e c t o f by-pass and feedback  The G ^  Q  i s measured by experiment and l i s t e d i n  T a b l e A2.1 T a b l e A2.1  C a l c u l a t e d G.  (Ns=3)  T  N o d b  N G  3  N  Nodb  Note:  2  1  -(G. ) - G . No fo l  G_.  =20' l o g . _G. 1 0 No (dB) 6  3.2(f)  T  9.60  57.2  341  19.64  35.14  50.6  (1) C a l c u l a t e d : G =1.64  Experimental: G ^ l . 6 1 G . =5.96 fo  G. =6 fo (2) E x p e r i m e n t a l G ^ , G ^  q  a r e used  53  Equation  (  A  2  .  1  4  i s used t o c a l c u l a t e t h e component a t t h e measured  )  frequency. From Table A 2 . 1  G  when  N  =  30db  5  0  \  '  6  G ^ Q ^  = 3 substituting  V. in  V  _ - -  i oTo 20  Example: V g f  ( G  =  50.6 i n t o e q u a t i o n  (  A  2  .  '  1  1  4  )  3out  3 d 3d bb +  5  0  6  '  )  (  A  2  5  )  = 0.109 v. : output measured a t t h i r d f i l t e r  o u t  = • 3 Hz  :  measured output  stage  frequency  C = 1.0 uF : s e l e c t e d c a p a c i o r i n F i g u r e 3.2(f) f o r a l l t h r e e filter  stages  We f i n d G , = -0.64 dB from C = 1.0 uF curve a t f = 3 Hz odb OJ  in  F i g u r e A2.1..  so t h a t  _ =  V.  0.109 =  1 1 1  0.346 mv  (-0.64 + 50.6) 10  (2) When N > 3 In  t h i s case the a t t e n u a t o r , G , e x i s t s i n the middle o f f i l t e r , as 'a' '  i n d i c a t e d i n F i g u r e 3.1.  The way o f d e r i v a t i o n i s s i m i l a r t o t h a t f o r  N .$ 3 except t h a t t h e g a i n o f a t t e n u a t o r has t o be added. So where  G.  T  =  =  No  Equations  (A2.16)  N f I a G i s the gain o f the attenuator G._  where  N  ( G _ ) * G. • G  (G fo  ) ^ • G. io  • G ao  = (G_ ) fo  N  • G. • G I  (A2.17)  a  G i s t h e g a i n G a t f = 0 and G = G ao ° a ao a (A2.3),  (A2.5) and (A2.7 - A 2 . l l ) a r e s t i l l  applicable.  54  S u b s t i t u t i n g e q u a t i o n (A2.17) i n t o G  20 l o g (G. ) °10 fo  =  M  ' G. • G 1 a  N  i n  Nodb so t h a t e q u a t i o n  (A2.12) (A2.18)  (A2.14) becomes  V V. m  _ "  Nout :  —  20  1 Q  (— r  K  3  + r  ^3db  (A2.19)  )  Nodb'  which i s s i m i l a r t o e q u a t i o n (A2.14) except To c a l c u l a t e G J ^ ^ J G ,  G  a l  a 2  and G  respectively.  A 3  equation  i n the c a l c u l a t i o n o f G„ „ . Nodb  (A2.18),  G^ may be t h r e e  corresponding to switch K set i n p o s i t i o n a ^ , G  The g a i n s o f G  actual r e s i s t o r values.  c a l c u l a t i o n of equation  T a b l e A2.2  Therefore, G  N  O  (A2.18) and l i s t e d  C a l c u l a t e d G._  N  O  G  N  =  G  i a fo  G  Nodb  2  °-  a2  1/7.4  1  G  G  3.1  A2.2.  (N>3)  a3  1/206  3 <  through  N=6  N=5  G  and a  2  i n Table  ^ i s obtainable i n Table  N=4  al  a  and G „ simply c o n s i s t o f the  e m i t t e r f o l l o w e r o f g a i n 1 and r e s i s t a n c e d i v i d e r , l i s t e d for  selections  G  al 1  a2  1/7.4  G  a3  al  1/206  1  G  a2  a3  1/7.4  1/206  2031  274  9.81  12106  1636  58.5  72157  9751  350  66.15  48.75  19.8  81.7  64.2  35.4  97.16  79.8  50.9  G  l o  =  G  SlO No  Note :  Experimental  G ^ l . 6 1 , G -=5.96 a r e used f  The procedue t o c a l c u l a t e t h e (A2.14) i s s i m i l a r t o t h a t used  w i t h N>3 through f o r N$3  equation  

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
India 205 16
United States 93 3
Pakistan 17 0
France 7 0
United Kingdom 6 1
Germany 6 0
Unknown 5 2
Canada 4 0
South Africa 4 2
Iraq 4 0
Philippines 3 0
Brazil 3 0
China 3 4
City Views Downloads
Unknown 199 18
Mountain View 33 0
San Mateo 29 1
Islamabad 16 0
Mumbai 14 1
Bangalore 13 1
Kolkata 10 0
Howrah 5 2
Chennai 5 1
New Delhi 4 0
San Jose 4 0
Delhi 4 0
Pasadena 3 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0065063/manifest

Comment

Related Items