UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Three-phase frequency conversion Berg, Gunnar Johannes 1960

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1960_A7 B3 T4.pdf [ 5.09MB ]
Metadata
JSON: 831-1.0105063.json
JSON-LD: 831-1.0105063-ld.json
RDF/XML (Pretty): 831-1.0105063-rdf.xml
RDF/JSON: 831-1.0105063-rdf.json
Turtle: 831-1.0105063-turtle.txt
N-Triples: 831-1.0105063-rdf-ntriples.txt
Original Record: 831-1.0105063-source.json
Full Text
831-1.0105063-fulltext.txt
Citation
831-1.0105063.ris

Full Text

T H R E E - P H A S E FREQUENCY  CONVERSION  by Gunnar Johannes Berg Siv.ing.  A THESIS THE  N.T.H. T r o n d h e i m ,  SUBMITTED IN PARTIAL REQUIREMENTS MASTER in  FOR  1957  FULFILMENT  THE DEGREE  OF A P P L I E D  OF  OF  SCIENCE  the Department of  Electrical  We  accept  standards degree  this  Engineering  thesis  required  as conforming  from  of Master  candidates  of Applied  to the f o r the  Science.  Members o f t h e D e p a r t m e n t of E l e c t r i c a l Engineering  THE UNIVERSITY  OF B R I T I S H  April,  1960  COLUMBIA  In the  presenting  this thesis  r e q u i r e m e n t s f o r an  of  3ritish  it  freely available  agree that for  Columbia,  Department  copying  gain  shall  or  not  his  shall  for reference  and  study-.  I  for extensive be  copying  granted  representatives.  the  It i s  of t h i s t h e s i s  a l l o w e d w i t h o u t my  The U n i v e r s i t y o f B r i t i s h Vancouver Canada.  by  Columbia  of  of  University  Library  publication be  the  the  p u r p o s e s may  o r by  that  advanced degree a t  fulfilment  I agree that  permission  scholarly  in partial  make  further this  Head o f  thesis my  understood  for financial  written  permission.  ABSTRACT This  thesis  regulation version, built  o f AC  and  source  and  be  phase  control  of  can the  tical  are  be  experimental  signals  from  from an  signals  fed  three-phase  a  conconverter  three-phase  auxiliary  570  cps.  sequence, 10  and  type.  operated  the  cps.  two  i n the  base-emitter  on-off  circuits  has  three-  output  The  I t c o n s i s t s of Each u n i t  cps.  frequency  fundamental  510  60  three-phase  When t h e  and  opposite  are  to  speed  frequency  70  i n each phase.  which  on  output  switching  one  continuous  whose f u n d a m e n t a l  v a r i e d between  transistors are  an  of  i s based  power  of  static  units,  which  accepts  generator  inputs  a principle  purposes.  v a r i e d between  quency is  test  converter  square-wave can  motors,  describes  f o r speed The  discusses  fre-  converter three  four  iden-  power  mode.  Control  through  isolating  transformers, A mines are  f r e e - r u n n i n g m u l t i v i b r a t o r i n the the  period  taken  sequence  from  content.  sequence  systems  of  A  principle  on  bistable  system  can  of  from  be  and  by  converter  harmonic  suppressed negative  speed  tests  which  the  on  unit  The  deter-  signals  triggered in  proper  gates.  the  this  signals.  circuits,  have  currents  sequences  type small  a  on  of  the  relatively  forming  entirely.  discussion i s given  when p o w e r e d  No-load  square-wave  Higher  positive  tolerated. motors  a  voltages  harmonic  monics  the  three  through  Output  of  control  har-  i n general  performance  of  be AC  converter.  i n d u c t i o n motors  experimental  zero-  Current  must  high  work has  been  confirm based.  the,  TABLE  .At  S "fe ~C 3f C *t  o  »  ©  o  o  o  o  o  »  Ld.S"b Of I 1 X US "t r 3/"fc X OHS  OP  o  o  o  o  CONTENTS  O  o  o  o  o  o  o  o  page  o  o  o  o  o  o  o  o  o  o  o  o  o  o  e  o  o  o  o  o  XI  o  v  o  Acknowledgeme 1. 2.  3.  4.  Intro An Approach t o Speed C o n t r o l o f Three-Phase I n d u c t i o n M a c h i n e s b a s e d on F r e q u e n c y C o n v e r s i o n 2.1  Rotating  2.2  MMF  General  power  3.2  Possibility  applied  o  » . . . .  voltages  „ „ .  relations  6  0  . „ . . . . . . . . . .  of application  4  . . . . . . . . . .  12 12 19  Frequency C o n v e r s i o n Networks C o n t a i n i n g Nonlinear Active Elements with Time-Varying ~  r  S  o  o  o  o  o  4.1  General  4.2  A  4.3  The t h r e e - p h a s e  o  o  o  o  «  23  . . . . . . . . . . . .  23  o  considerations  single-phase  «  switching  o  o  o  type  o  o  o  e  o  o  25  . . .  30  P e r f o r m a n c e o f AC M o t o r s w h e n P o w e r e d f r o m t h e S w i t c h i n g Type C o n v e r t e r . . . „ . . . . . . . . . . . '  37  switching  5.1  Three-phase induction  5.2  Synchronous motor  type  motor  load  converter  o  . . . .  converter  load  . . . . . . .  . . . . . . . . . . . .  E x p e r i m e n t a l F r e q u e n c y C o n v e r t e r and Square-Wave Generator . . . . . . . « . . . > - . o » . . . . . « < 6.1  Introduction  6.2  General  6.3  A-stable C i ~F C U l "t S  6.4 7.  f o r modified  3.1  >  6.  waves  machines  o  Frequency Conversion Networks C o n t a i n i n g Nonlinear Passive Elements . . . . . . . . . . . . . .  l?£H3/IHe"b 6  5.  mmf's i n i n d u c t i o n  o  . . . . . . .  description o  o  o  o  o  Gates and f l i p - f l o p s  No L o a d S p e e d T e s t s  o .  .  .  .  .  46  o  .  . . . . . . . . . . . . . .  multivibrator o  . . .  0  >  and p u l s e o  o  o  o  o  c  53 53 54  forming o  o  o  o  o  o  o  . . . . . . . . . . . . .  on T h r e e - P h a s e AC-Motors  in  37  . . . . .  57"  59 62  8.  Conclusions  Appendix  «  Bibliography  . „  LIST  OF  ILLUSTRATIONS  Figure  page  3.1  Symbolic  4.1  Single-phase  4.2  Mechanical  switching  arrangement . o o o o o o . .  4.3  Electrical  switching  arrangement . . . . o » . .  4.4  Single-phase  4o5  Output waveform  4.6  Transient switching  4.7  4.8  representation  of converter  circuit  switching  circuit  o f common-emitter . o o o o . . o o o o o » o o o o  » « . •> . « . ., »  O  of  Phaser  5.1  Circuit  5.2  Transformed  5.3  Typical  5.4  Maximum r e l a t i v e  0 !  » . . . . <> t o f o l l o w i 30  follow  6.3  6.4  30  XO£LCL o o o o o o o o o o o o e o o o o o o  representation of induction  of system voltages motor  induction  torque-slip  34  . . . . . . . . . . . . .  38  circuit  motor  curve  . „ . . . . J  0  o o o o . o o o . o o o o  as a f u n c t i o n  Block diagram of three square-wave g e n e r a t o r  of converter  42 44 48 50  to follow  54  . . . . . . . . . o o o t o follow  55  phase  V o l t a g e waveforms i n t h e square-wave generator . . . . o o o o o o o o o o o o Circuit  0  as p a r a m e t e r . . . . . . . . . . . . . .  Block diagram of experimental: converter o . . . o . . o . . . ' « » . o .  39,  0  o f k = •y— . s Phasor diagram f o r synchronous motor . . . o o o o o E x c i t a t i o n c u r r e n t as a f u n c t i o n of f r e q u e n c y torque  torque  31  . . . •» . «  f  6.2  30  three-phase  o o . . o . o o o o . . o o o o t o  4.10  with  28  system r e p r e s e n t a t i o n f o r  3?£tX£LSX"b "L C  6.1  »  28  diagram  4© 9  5.5 5.6  0  27  , . „ <> o „ <» <, . „ <> . . t o f o l l o w  converter  >  o  0  25  28  Simplified  analysis  . . . . . . . o . o <,  . . „ <. » . t o f o l l o w  converter  response circuit  Converter  „ „ . . . . <, „ ... 19  unit  v  to follow  55  „ . . . . . . . to follow  56  Figure 6»5  page  A-3table multivibrator forming  circuits  and  ,„ »  0  0  pulse 0  0  0  ,  0  0  0  to follow  57  , to follow  59  0  6.6  G a t e s G.^, G | a n d f l i p - f l o p  6.7  Trigger  6.8  sqaure-wave . . . . . . . . . . . . . . . t o Output voltages a t f l i p - f l o p t e r m i n a l s 1 and 2 . . . . . ' . . . o . o . t o  6.9  pulse  train  square-wave  signal  6'.. 10 O u t p u t l i n e - t o - l i n e converter o  A - l  output  Output l i n e - t o - n e u t r a l voltage and  7.1  and  F F ^ , „ ..„..,  at  follow  61  to. f o l l o w  61  at . t o follow  0  Experimental speed-frequency characteristics „ „ , o o , . Simplified circuit  61  converter  . . . . . . . . . . .  voltage  follow  o  for analysis  v i  V  ,  o  o  .  of v ,  o  t  o  follow  . . . . . . .  61  62 67  ACKNOWLEDGEMENT  The  work d e s c r i b e d  National  Research  The Dr.  author  this  expressed several trical  Engineering  Canada  Assistant  to express  undertaken.  f o rthe willing  author  by The  o f Canada. h i ssincere  Their  i s greatly appreciated.  occasions  The  t h e s i s i s supported  and D r . J . F. Szablya,  work h a s been  encouragement  of  Council  wishes  Prank Noakes  i n this  assistance  by other  members  gratitude to  u n d e r whose advice  and c o n t i n u a l  Appreciation given  guidance  i s also  to the author  of the Staff  on  i n the Elec-  Department.  i s indebted  f o rthe f i n a n c i a l i n 1958-59,  t o The N a t i o n a l support  he r e c e i v e d  and f o ra Studentship  vii  Research  Council  as a Research  i n 1959-60.  1.  A  frequency  bination voltage vers  converter  may  of  devices  that  and  current  components  i t at  phases  INTRODUCTION  a  may  different  or  may  not  be  defined  receives of  changed  a  electrical a  given  frequency. be  as  The during  device  energy  the  com-  with  frequency  voltage  or  and  and  deli-  number  conversion  of  pro-  cess. Several They  are  Rotating  types  of  frequency  generally classified frequency  generator  set.  changers  The  input  converters  as  either  consist  have been  static  or  in principle  electrical  energy  to  built.  rotating. of  the  a  motor-  motor  is  i converted again  to  mechanical  converts  converters which  i t to  are  means  that  the  inputj frequency.  can  be  type,  converters  involving  development rectifiers very  converting  of  high  has  the  generator  Most  of  frequency  has  a  fixed  a variable"output  high DC  rectifier voltage  AC,  be  and  output the  s u p p l i e d from  types  of such  static as  The  grid-controlled  r a t i n g s and the  AC-to-DC  components.  high  to  elements  the  power  load  Other  u s u a l l y of  design  equipment  must  are  made p o s s i b l e t h e  auxiliary  linear  ratio  frequency  However,  to  energy.  constant  output  fed  the  which rotating  changers, relation  to  frequency  obtained.  Static  with  and  electrical  so-called  the  AC)  energy  of  r e a c t i v e power sources  on  frequency  nonlinear  the  mercury-arc static  converters When  i s controlled r e q u i r e d by AC  by the  side.  converters  reactors  to  successful  efficiency.  frequency  ( o r DC  employ  non-  or c a p a c i t o r s ,  electron  tubes  and  Frequency power such  field  as  The  o l d as most  unsurpassed  much  of view t o be  control the  there are  be  the  low by  of  AC  itself.  AC  machine  maintenance other  speed  At  the  can  be  impedance,  i n which  case  the  rotor  and  slip  rings,  core  type  reactors  resistance lower  i n series  i n the  speed  results  control, The'speed closely  terminals. tional  60  sidered a  new  rotor  efficiency.  ductance  ways  have  On  with the circuit the  i n lower  therefore, of an  have  induction  Assuming  that  power speed  of changing that  f ^  the  impedance the  the  wound  Adding  increased  These  losses  and  input i n -  methods  of  disadvantages.  synchronous  motor  is a l -  frequency at the  is available control  a  from  problem  frequency,  becomes  rotor  saturable  additional  serious or  speed-torque  windings.  factor.  the  leaves  varying Of  motor.  from  input  be  such  operation  motor  connecting  stator  applied  sources, the  f r e q u e n c y f , so  by  leads to  power  to the  a problem  must  o t h e r hand,  related  cps.  machine  by  be  problem  smooth  the  either  or  a  But  induction  changing  done  where  induction  motor.  in  One  reliability in  present time,  a c h i e v e d by  This  and  the  fields  machines,  i s the  electric  control  large  equipment.  o f AC  desired.  motor.  reliable  machine  extensively  provided i t could  regulation  any  i s usually  quite  still  desirable,  widely used  ruggedness,  point  of  speed  devices. applied  i n e x p e n s i v e and  concerns  almost  is  But  c o n v e r s i o n would with  state  converters are  engineering.  achieved  Its  solid  say  input  convenmay  60  controllable  be  con-  cps.,  to  ratio.  The seek  a  purpose  solution  of to  result  i n the  should  c o n s i s t of  of  a l l be  the the  design  project reported frequency  of  a  static  solid-state  a p p l i c a b l e to  speed  in this  thesis is  conversion  problem  converter.  The  which  to can  converter  components  only  and  should  c o n t r p l of  conventional"  first  AC  motors. A ready The that  fair been  done  results the  frequency  energy A  amount or  of  so  DC  some s i m p l e  conducted  and  power  amplifiers,  12  3  approach  theoretical  these  success  use  '-'  work has  along  i s to  sources  different  development  f a r i n d i c a t e some  approach  oscillators  from  and  is presently  reported  general  research  and  lines.  and  transistorized  a l -  show-  variable  take  the  * has  been  results  adopted  derived  here,  based  i n Chapter  2.  on  4 2.  AN A P P R O A C H T O  S P E E D CONTROL OP T H R E E - P H A S E  MACHINES B A S E D  ON  FREQUENCY  CONVERSION 4  2.1  Rotating  mmf's  i n induction  INDUCTION  machines  5 '  When b a l a n c e d t h r e e - p h a s e v o l t a g e s a r e a p p l i e d symmetrical the  resulting  space a  three-phase  wave  simple  air-gap  stator  mmf  winding  c a s e , where t h e phase  sinusoidally  distributed  of an i n d u c t i o n  c a n be r e p r e s e n t e d by a  of constant amplitude.  and p l a c e d  machine,  travelling  To d e m o n s t r a t e  windings  to a  this for  a r e assumed t o be  120 e l e c t r i c a l  degrees  apart, l e t co = a n g u l a r f r e q u e n c y o f i n p u t 8 =  P  space  angle  i n elect,  phase  winding  = magnetomotive across  t  = time  p  = number  a,  force  the air-gap.  measured  radians per sec.)  from  the axis  of  (mmf) F =  o r magnetic  potential  difference  F(8,t).  i n seconds, of poles,  b, c, s u b s c r i p t s  F  Where  denoting  F  a  phases.  distributed  = F (wt)cos a  windings,  = F (u)t)cos(6  F  = F „ (cot)cos(6 - 2 4 0 )  c  b  - 120)  . . . 2.1  c  (cot),  F,(cot),  a t time  currents"so F  one c a n w r i t e :  6  F^  amplitudes  phase  degrees,  (elect,  a.  With s i n u s o i d a l l y  mmf  voltage  that  (wt) = F  t.  and F These  (wt) a r e t h e i n d i v i d u a l  amplitudes vary with the  one c a n w r i t e : c o s wt  phase  5 F, (cot) = F c o s (cot - 1 2 0 ) D m F (tot) = F c  The fore  currents  of equal  i n the three  amplitude.  a  found F  resultant  mmf  240)  phases  Whence (cat) = P max  F (u>t) = F max The  c o s (art -  m  . 2 . 2  are balanced  = F =  F  m  a  F  +  b  F  +  there-  also (art) = max  a t any p o i n t  F  m  8 and time  b y s u p e r p o s i t i o n o f t h e component  (8,t)  and  t c a n now  be  mmf's:  c  " c o s 8 c o s cot +  c o s (8 - 1 2 0 ) c o s (tot -  +  120)  c o s (8 + 2 4 0 ) c o s (tot + 2 4 0 ) ]  F = ~  [ (cos(8 +  + cot) +  cos(8  cos(8  - ait - 120 + 1 2 0 ) ) +  c o s (8 - tot + 1 2 0 F  (8,t)  = |  P  Equation direction  m  presents in  of rotation of this  wave  F(8s,t)  equal  radians  8 — art =  dt In  general  -  the travelling  + art '+ 2 4 0 )  i n mechanical  of poles speed  Thus  const.,  w  <  mmf  wave.  positive. radians  of t h e machine. equal  angular  per s e c . deItre-  The to  Its  The  i n t h e machine.  per sec. i s simply  a constant.  240)  120))]  i s by d e f i n i t i o n  the synchronous  electrical  (cos(8  + cot -  . . . 2.3  represents  o n OJ a n d t h e n u m b e r  pends  (cos(8  c o s (8 - tot)  2.3  space v e l o c i t y  - art)) +  dt  speed for  0)  = CO ,  ,  The  I—)  elo  mech.  synchronous  speed 6  0  h  2 120f - = ~  V * " 2 i ~ In  an  ,  synchronous  machine  the  mmf  purely  sinusoidal  because  the  windings  But  careful  keep  design,  saturation  sign.  Therfore, the  2.3  can  i n g e n e r a l be  2.2  MMF In  fied  waves this  and  low  assumption  are  applied  line-to-neutral  s i n (co t -  a^)  =. E  s i n (co t _ s  -  240)  s i n (co t c  a  g  c  the  co  angular  frequency  of  an  time  phase  Equations  =  e  =  i  , al  e  are  modi-  ...  2.4  ) ,  c  power  source  auxiliary  three-phase  =  angle 0  or  control  system  2it  let E 2.4  i n the  ..»  =  and  1.0  per  unit  (p.u.).  - co t s  -  write:  a  a  ) -  c o s ( c o t - a c a  +  2.5  Then  „ a2  [ c o s (co t c  c. I  Equation  source  convenience,  expand  de-  )  of  For  a  a  frequency  aa + a,b + a c  e  a  angular  control  and  -  co = s  =  of  To  by  120)  a  slots.  lot.  a matter  voltages  -  =  not  voltages.>  s i n (co t  where  kept  made i n d e r i v i n g  E  c  be  largely  e  =  confined to  can  s i n co„t s i n (co t s c  fe  is  justified.  the  represented  distribution  content  i s also  for modified  section  space  e„ as E a  e„  can  harmonic  effects  sy  r.p.m.  actual  by  = to  is  el  )  = CO  p'  co t ) l s J  one  e  b  ~  b l  e  =  e  b2  = i|^cos(a) t  -  c  s , cl  e  al  **  e  a2  =  e  bl  e  b2  e e  cl  c2  I I  C  2  =  °  c  G  o  S  (  c o s  =1  c  w t  +  s  o  In  s  (  (  w  c  e  240)  -  cos(w t c  -  a,  +  (a t g  s  w  H  ~  a  )  a  a  b  a  b  a  s  a  ,  a  +  1  2  2  e  ^  0  )  -  2  and  a  then  240)  4  0  )  can  be  chosen  there are  signal;.  = a,  a  p  the  a^lid^;;^hat = a  three  1  one  b2  f t l  findsj  at w i l l  in  possibili-  input  e^^  a  n  d  e  is  .= . 0.  c  v o l t a g e s are  instantaneous value  »  0  +  c  c  a,  1  ~  a) )t -  "s?* "  }  and  c2°  ©j_> c  A  s  s  ^  balanced.  of the and o  w  n  let  mmf  wave  F^Pjt)  above,  one  P (p,t) = P (P,t) + P (p,t) + P ( P , t ) , and  g  interests  case  a 2  a  E q u a t i o n 2.5,  &  to  ~  +  the v o l t a g e s e ,,  related  ~  angles  sequence  (8,t) denote  "s^* "  e  with  this  +  c +  a  by  -  c  0 t  ((a) -  of p r a c t i c a l Zero  w  ( ( w  cos  c  (  s  I  = |  accordance  F  - a  K ~  C0S (  Assume-the  (i)  + wt ~  r e  ties  - cosCc^t -  120)  g  120)  „ c2  = Ifcos^t where  - wt +  b l  o l  Let produced be  then  similarly hass  240)]  8  F, ( B , t ) CF c o s x  cos  = £ That  a  =  120)  +  cos  ((to - to )t c  The  l max  P  =  2  F  2  c  o  way m  a  s  (  less  than  F,  them  are of equal  C(to  cos  + u  c  with  applied  t o an  rotate  and F,,(8,t)  at a  three-phase  speed  Since  not  a  120)  cos(8+120)  resembles  Positive =  In  this  0,  e  al  +  e  e  a2  +  e  b l b2  =  e  6  F  but  with is  0  that  produce  impedance i n voitages are  machine,  (ca  than  F  2.7  mmf  directions  components  to  ...  amplitude  induction  the  machine  - to ) i n n e g a t i v e 0  „„, the  machine  ^ max i n other  of the single-phase  r e s p e c t s i t s beinduction  machine,  s i g n a l s •'applied,.-.'that Is  120°, a  =  -120°  case +  e  +  e  c l =1 c2  =  C  0  S  K  ~  w  s  H  Thus:  >  °"  input voltages are of equal  balanced.  n)  r e p r e s e n t two  o  torque,  sequence a,  8 -  I f these  i s larger  that  2  the input  proportional  F,  starting  a  The  while  1 max  The  -  )  i n opposite  amplitudes.  amplitude  ideal  P  ) t -  increased frequency.  direction.  have  s  because the v o l t a g e  creases  (ii)  - to H s•  120)  finds  i n the air-gap, t r a v e l l i n g  haviour  ((co_ - t o _ ) t +  8)  ~ " a * * .+  functions F^(B.t)  speeds and  will  cos  ((oo ©  +  g  K  one  x  different  will  cos 8 +  D  (8 -  cos  similar  F (B,t)  wares  - to )t  i s  *ll*>V In  (to  maximum a m p l i t u d e s ,  but  9 F-i ( P s t ) ^ = c o s ( o ) x  cos(6  x  cos  =  cos(»  - co_)t c o s P + - 120) +  (p +  cos((a)----- to ) t -> 1 2 0 +  cos((oi  - &i ) t +  120 +  240)  120)  ^ a)  )t[cos p +  cos(p  - 120)  +  „ o o 2.8  cos(P + 120)] = 0 F (b,t) 0  j p c o s ^  cos(p  x  120)  + to  ) t cos p + COS((OK  - 12G-) +-eo«-((b)  c  = -o c o s ( ( o )  + as ) t +  ) t-  + w  1  + w  )t.+"120)  cos(p  120)  +  120)  s,  o o . 2.9  p)  Hence  F,(p t) s  = F  From t h i s single  wave,  m  a  x  cos((o)  i t i s seen  rotating  stator  windings  sequence  s  neutral.  neutral  means  i n negative  of angular  t h e sum  at  t h e n e u t r a l becomes ^ ( e + o a (presuming to  (iii)  Negative a For  X  2.10  a t an  grounded 0)„ - 0)  still e, + D  t h e phase v o l t a g e s  = 0,  line  holds, e  c  X  will  ) = ^  &>  this  case,  a  i t c a n be  shown  that  zero-  flow i n  currents i s zero the voltage  c o s (to  ©  are symmetrical  = + 120°.  a  Isolated  b u t now  signats''.kppli'eS^that  a, = - 1 2 0 ° ,  I f the  neutral,  ground). sequence  2.10  angular  i s undesirable.  of a l l three  any time.  spect  with  frequency  at  p.u.  Equation  direction  radians p e r second.  This, of course, that  ...  t h a t t h e a i r - g a p mmf*s a d d u p t o a  are X-connected  current  p)  + 0) )t +  c  o f (OJ + 0) ) e l e c t r i c a l c s•  speed  the  2  i s  - . c os_ ) t with r e -  10 F-^B.t) = ? Fg (8, t ) Hence, tion If  —  x  m  a  ( ( u - to )t c  (to  - 8 )  ooo  o o  the resulting  the neutral  g  0  a t an angular  frequency  co«  x  mmf  speed  of  (co - c a ) , e l . c  i s grounded, + to ) f l o w s ,  -wave t r a v e l s  i n positive radians  a zero-sequence  2.11  o  2 o 12  direc-  per.sec.  current  and i f i t i s i s o l a t e d ,  of angular  i t s voltage  -i  becomes  c o s (to  s e e what  nected. the e  ab  h a p p e n s when t h e s t a t o r  The l i n e  line-to-line ~~ a e  =  e  I t i s also  easy  3  C  Ca  to  + to ) t p . u . t o g r o u n d .  currents  must  windings  are delta-con-  be b a l a n c e d  and  likewise  voltagess  b  i J cos  - to )t  -  g  c o s (to + ' t o ) t c  -  g  V*-  c o s C (w ;  cos e, be  =  e, b  ffi  •• e  cos  ca  } )t  +  g  ((co  — co ) t — 1 2 0 +  =  e  c  = i  [ c o s ( (OJ  cos .*» ^  g  c o s ((co + w  ) t - 120 - 2 4 0 ) S  90)  c  g  c o s ((co ©  120 +  240) -  S  (co - t o ) t  Whence  -  - co ) t C  Cs  S-  a  L  Ct  ©  ©  V | c o s ((co - to )t  = e  240) +  ((oa + to ) t -  S  =  c  30)  ((co • - to_) - 1 2 0 ) - c o s S C  e  £ l  c  I  Ct  7  ( ( ^ „  \ cos  = i  - w^Jt - 1 2 0 ) + c o s ( ( w + co ) t - 120 + 120)  +  cos  (u> + co ©  c o s (co -  - co ) t +  c  )t =  S  (i) )t g  o.o  150)  S  the line-to-line  voltages  constitute  a  positive  2.13  11 sequence per  system  sec.  these  When a p p l i e d  i n accordance  Now  suppose  available,  co  three-phase  achieved also  fore, and  local  i s made  angular  instead  range  speed  t o be most  then  above  subject  speed  control  of three-phase  how  to generate  described quired  problem  or produce  by Equations  by t h e l o a d  2.4  i s taken  control  The above  This  1.4 a r e  approach  associated three-phase  c a n be analysis  signal seo f t h e machine  as below t h e case,  there-  with  investigations„  to the problem  adopted  i n this  this  that  entirely  of  thesis.  approach i s  v o l t a g e components  o r 2.13, s u c h  by  the three considered,  f o r further  machines  almost  speed  as w e l l  g i v e n b y co .  i t represents the p a r t i c u l a r  technical  speed  the three-phase  Also,  major  that  frequency produced  p r o m i s i n g among  the principal  windings,  by Equations  t h e synchronous  a wide  stator  i n the air-gap  o f co »  by c h o o s i n g  (iii),  over  The  wave  - co ) e l . r a d i a n s  Equation 2 o i l .  with  co  synchronous  appears  mmf  oscillator,  that  as i n case  "normal"  a space  the voltages described  by v a r y i n g  be v a r i e d  (co  to the idealized  i s the constant  indicates  quence can  angular v e l o c i t y  v o l t a g e s produce  behaves  a  with  as  t h e power r e -  from  t h e co  g  source. Chapters considerations  3 and 4 p r e s e n t on t h i s  some t h e o r e t i c a l  problem.  and  practical  12 3.  FREQUENCY CONVERSION NETWORKS C O N T A I N I N G PASSIVE  General The  can  be  ment  Two  for  our  iron  source  general  frequency The  have  can  of  The  be  will  be  f  and  will  be  the  be  judged. f o r the  and  leads  i s given  to  single-valued, by  case  follows the  characteristic  It  In  of  but  a  nonlinear  The  energy  the  s i g n a l frequency  as  which  a  the  nonlinear  current,  i , as  power-  elements. con-  capacitor. to  that  inductor  i t s shape  f^.  power r e l a t i o n s  similar  results.  by  is  show t h e  nonlinear  same  otherwise  6,  ele-  supplied  nonlinear  reference of  is  a power-frequency  pattern  s p e c i f y i n g the  A  inverter  whose n o n - l i n e a r  derived  modulator  type  examined„  considered,  of  analysis  be  is a  is  The  in  re-  assumed  arbitrary. function  of  l i n k a g e f ': /  i  the  s i n g l e modulator  inductor.  expressions  derived  to  flux  will  frequency  then  subsequent 6  purpose  a  relationships associated with  been  fernce  whether  core  feasibility  verter  of  modulator  i s an  AC  power r e l a t i o n s  question  used  reactor  In  ELEMENTS  6  3,1;  an  NONLINEAR  general,  inductor  an  infinite  can  m = —oo  be  =  f(n  number  described  of  by e  n  =  -co  m,n  3.1 frequencies  a  j(mx  double +  ny)  can  Fourier  be  present,  series: 3.2  where x =  2-ref t ,  The  voltage  taking  v  y =  across  the t o t a l d £ at  =  f  =  2rcf t  .„„  the nonlinear  derivative v  £  <—>  {-•  inductor  o f *p w i t h  m,n  e  J  (  m  X  respect  i s obtained to  by  times  ^  +  3,3  ...  3,4  where V  „ = m.n  The  where the  j2n(mf  series, =  The  i , a l s o may  Y  I  = F(x„y)  = - ^ /  n 9  d y /  ^ f 4it J o  Multiplying •oo t o +oo  in x  and y  ..  3,6  Thus  Q  over  m  x  ...  +  3.7  by  I  F(x y) 9  e ^  (  m  X  +  n  ^ d x  ...3.8  o -m,-n  2TC -  0  2TC  =  has:  (  are given  o  I* m,n  e J  m.n  coefficients  ^  ^  double  since  2TI  one  by a  3o5  c a n be w r i t t e n a s  V  Since  be r e p r e s e n t e d  i s s i n g l e - v a l u e d and p e r i o d i c  j—  1 ^  oo„  y  8  current =  ) if „ ^m^n  f ( ^ ) = f [^(x y)]  F(x,y)  i  + nf s  c  v  current,  Fourier i  d  =  the complex conjugate  of I  m.n  ,  2TC  J  F(x,y)  e J  (  m  x  ^  +  dx  ...3.9  u  Equation m a n d n,  3,9 one  by  jra^F  obtains  m  n  and  summing  from  14 oo i=-oon=-oo  '  '  4 it  J  on  changing  the right  t h a order  hand  side,  F(x,y)  Equation  »  9  4n  i->l  J  co  0  m=-con=-oo  Therefore Equation  .  x  3.11  L.  0  Equation  3.10 b e c o m e s 2n  3.11  -"-nun 3.2 w i t h  respect  to x  yields:  3.12  m,n 9  t h e double  summation  equals  _ m.n ~ 2 i t j Combining  under  Equation  the integral i n  3.5 g i v e s f o r  m,n (mf + mf ) c s Equations  o  3.12 a n d 3.13 w i t h 2 TI  3.11  o  The suitable  c  s  next  step  form.  accordance  with  -*  Equation  211  3.14  P(x,y)  Equation  the last  integral 1  M2nif)  V(o,y) 3 . 1 4takes  O  c o n s t a n t , Q~- d x = d ^ „  dx  o and  J  3.6, t h e r e f o r e ,  <  3.13 U o  2u  i s to transform  With y held  O  yields  ) Vp- d x m=-oon=icx)  3 o l 0  a n d t h e summation  oo  J-> ^J-^  Ox  d  J  m=4oo n = — o o Differentiating  ^  +  of integration  2% m=-oon=-oo  »  2n  dy J o  o By  m=-oon=-oo  2 it  !  oo  on t h e form  O O  O  t o a more In  20 fc  00  mv  I*  ~oon=-oo Starting  it  ,  c  s  J  again with  by jiff"  r'  r  oo  I  2,1  3ol5  one c a n n e x t  procedure.  multiply  Thus:  271  d y  1  9  3.9,  t h e above  -hiJ mE =-o©nE = - o o l » \ n S' . " 4TC  '  V(o,y)  Q  Equation  and repeat  ^ •'  o  Jj ' F(x  y)fa  o  ? jj£ E E n=-e>on=-co OO  OO  ...3.16  > a  s  m=-oon=-oo  00  3 o IT  00  E E  ,  m=—00 n = ~ 00 nV  T r  nx  I*  m,n  m,n  Hence, _  =  ? *  m  n  I* '  m  ;j2Tt(mf  3.16  Equation  „ y  n  +  nf  . 3 o 18  )  becomes: 2n 2n  E E >? 5'" - h\ **f  00  I*  00  m=-oon=-oo The able  c  second  s  J  integral  to letting  constant.  Hence  co  co  nV 11 v  one  00  I*  m  m=  'fif)  the  right  from  -  2  r  J  o  *  that  the vari-  the variation  of  ^  0 t o 2rc wjb.ile x i s k e p t  .nx,2v.)  rf  «*fJim <^>  ^ \ 4  .  .  3.20  r(x,o)  i s single-valued  hand s i d e  by changing  finds: -,  m  Since  indicate  y vary  E - -s = k E E- t <lrS'" n=-oon=-co c 00  J  i s transformed  t o V^, a n d t h e l i m i t s  corresponds  00 3.19 u o U < 7  of Equations  and p e r i o d i c 3.15  a n d 3.20  i n x and y, must  equal  zeroobecause '/ (2TC,y) /  j  V(o,y)  d ^  =  r  ^(x,27t>  .  f ( f ) d>  V(x,o)  =  0  3.21  16 It  follows  oo  co  h  L  oo  co  roV f  mss^oon=-c>o m  m  m=—oon=~oo  voltage  appear  with  u  <  "  > 0  I* ?  n  c  n  s  = 0  equations  ...3.23  both  as a r e s u l t  a particular or phasors  positive  and negative  fre-  technique.  The  of theFourier of a certain  frequency  of half  amplitude,  f >0, i s s p l i t  amplitude  asso-  i n t o two  a n d f r e q u e n c i e s +.f  - f respectively. Let  powers  S  , P and Q denote v e c t o r , r e a l m,n' m,n *ra,n '  respectively,  frequencies S  m,n  S* m,n  P  m,n m  Q Ta,n  n  m  = 2 V m,n  I* m,'n  ...  3.24  = P  = 2 V* m,n  I mj,n  ...  3.25  d  m,n  = .V  - jQ *m,n J>  3.24 a n d 3 . 2 5 i t f o l l o w s  that:  „ I* + V* I m,n m,n m,n m,n  ...  n  = j(V I * ' - V* I ) *• m,n m,n m,n m,n' I = V m,n -m,  -n  I* -m,  ...  , Equations -n' ^  3.26  3.27  3.26 a n d  be w r i t t e n  = V m,n  I* + V m,n —m,  —n  Q = j (V I* - V ^m^n ^ m,n m,n -m, v  Thus,  t h e n o n l i n e a r element a t  = P + jQ m,n *m,n  can also  m,n  and r e a c t i v e  Then:  e  Since V* m,n  P  flowing into  ± | ^ + *J°  From E q u a t i o n s  3.27  -  s  o r c u r r e n t component  components and  + nf  f<*  t h e above  quencies  ciated  c  I*  nV  E E  In  that  combining  pairs  I* —m,  -n  .  —n  I* ) -m, - n '  o f terms  . . . d.zo ...  (VI*)corresponding  O  Q  3.29  to the  17 positive rewrite elated the  and  Equations with  real  3.28,  negative  the  P  Equations  oo  )  ffi  m  oo  oo  nP  £ L  mf  m=-co n=0 These provide a  are  two  have  been  assumption of  Equations flowing  linear  3.30  into  presence  c  well  the  of  =  +  nf  final  of  only  Equation  , . o 3 , 30  J  0  0  results.  Equations  at  various  3.30  0  0  3  and  powers  flowing  frequencies.  case  of  a  nonlinear  inductor,  to  the  case  of  a  nonlinear  capacitor.  3.31  be  hysteresis  at  level,  one  can  double-valued, oscillator  assume  and  output  the  that  of  loss  a l l the is  powers  zero.  but  compared  to  the  actual loop f  ,  is  The non-  fairly  taken  frequency  The  charac-  be  hysteresis  of  B-H  can  the  into  Consequently  c e r t a i n simple  i s low  B-H  into  power function  formed  According  by to  s  g the  literature  Equations  3.15  on and  this 3.20  subject, i s not  the  analysis  a l t e r e d by  1  but  d i s s i p a t i o n i n the  hysteresis  level  sum  the  frequencies  power  Under  this  the  various  implies  element.  that  single-valued.  show t h a t  element  conditions,  r e s u l t s was  3  They  the  these  o  3.31  for  oscillator  power  use  that  asso-  become:  element  signal  the  Making  can  powers  i s seen  r e l a t i o n s among t h e  I f the  only  =  s  account.  is  It  the  0  s i  inductor and  reactive  realistic  nf  behind  the  introduce  one  n  derived  major  teristic  m  +  reactive  equally  3.23  frequency,  n  the  apply  each  and  appear.  and  independent  nonlinear  3.23  of  frequencies.  n *' „ n  „ m=Un ~n = — o o mf c ™  and  will  n  3.22 mP  )  3.22  various  powers  oo  components  the  leading new  up  to  condition  18 for  i = f(V^).  the  right  But the values  hand  Consider  s i d e ©f t h e s e t h e double  flux  y  D  W  loop  a minor  at  t h e same p o i n t  one of  minor  Next  loop  equals  consider  J  entirely  represents the  nonlinear  frequency.  branches.  zercr,  power  loss  Therefore,  and t h e above  loop  to  the area  integral  equals  0,  integral  ^(x,2i0  (  f<V)  d^  T(x o) Jib  _  9  and y v a r y i n g from  of the loop,  reactor during Let this  and ends  hysteresis  must be c o n f i n e d  the hysteresis loop.  the area  starts  entirely  dx  x = constant around  loop  loop  2n  With  This  represented  f ( ^ ) i s only  the double  h i  means t h a t t h e  to the condition that  of the h y s t e r e s i s loop t h e minor  integral  on one o f t h e p r i n c i p a l  this  re-evaluated.  0, t o 2 i t , c a n b e  i n t h e B-H p l a n e .  But according  double-valued,  be  &  i n the second  when x v a r i e s f r o m  by  branches.  *^')  >(o,y)"  constant  variation,  must  on  y(2u,y)  o Keeping  equations  integrals  integral  2n  k  of the double  0 to^ H j ^ w i l l  This  integral  or the energy  travel  therefore  dissipated i n  one c y c l e o f t h e power  oscillator  q u a n t i t y be denoted h and t h e average  due t o h y s t e r e s i s be d e n o t e d  H„  T h e n H = f h,,  H or can  h =, be  6 Under these s written?  oo  oo  • E * <Fms=0  conditions, Equations  3.30 a n d 3.31  mP w mf '+ n f -ns-Sb c m  • » s  • • •  3  -  3  2  19  oo  C O  Y  Y  °  m==oon=0 Equation from  m  +  f c  3»35  n  H  n  ~  f s  3  f  indicates  that  the hysteresis  bined  com-  s  2  Possibility As  a  33  i s taken  power  P  3.2  loss  H source» T h e t e r m -s— c a n t h u s b e 0 1 w i t h t h e t e r m — y — on t h e l e f t hand side„ s  the l o c a l  o  s  of  application  concluded i n Chapter  circuit  which  a c c e p t s power  different  frequencies  frequency  (to  to represent  reactor  modulator  c a n be  illustrated  P  Fig.  The  f  - to ) . '  assumed  P  c s  2,  of of J  3.1  The  signal  two  i s t o be u s e d  power  associated  Now  for this  suppose purpose.  at  at angular  w i t h to a  i s  nonlinear The  system  i n F i g . 3„X„  Nonlinear  c  establish  sources, operating  power  power„  as shown  i s to  , and d e l i v e r s  input  -P  O  y  reactor •  of Q  modulator  s  Symbolic  representation  P  C  =  input  signal  P  G  =  input  power  P  q  =  o u t p u t power  power  from  and f  g  the object  gain,  G  —  f =  converter  f  from t h e l o c a l at frequency f  , i s defined P  Cp  power,  of  source, f = f  s  .  as.  Q  p~  =  f ( f  c  , f  g  )  o o o 3o34  20 In assume are  analyzing this that  method use  the of  to flow  being  suppressed  of Equation  only  Now,  power,  1  by  L  L n=0  mP  important  frequency 3.32  3.34  f = . o  that  f  c a n be  +  carries  f a  s  a l lother Adopting  frequency  c  frequencies frethis  s o l v e d by t h e has  been  and takes  significant  this  as  amount  yields * 1  L  nf  filters.  the output  chooses  „  mf„ +  ideal  Equation  once  i f one  Equation  1  m=0  3.32  output  i t i s customary t o  i n t h e n o n l i n e a r element,  or s i m p l i f i c a t i o n ,  chosen.  o f problem,,  c u r r e n t s a t o n l y t h e most  allowed  quencies  type  mP  r  I  mf  m=0  mf  c  =.^* +  +  c  f  £  '' • '  mP^  ^ c  1  c  f  J  s  ° Q s  f  ...  3.35  where P  I,o  ~ e> P  P  l„l  ° V  =  Hence O  G Equation 1  Y  Y  ^ m*0  ^ n=0  n  3.33  ••  o  oe  gives?  p  p m n  p o J  9  mf  o  = "it" s  nf e  s  ,  -1.1  ±~ + * e  =  H 7"  ...  „ „«, 3.37  where P  —  P  Hence P  - H  s f  The powers P , g  P  ~  S P  c  o f •o  and  3o o o o fcj o  are a l l positive  and t h e  38  system  21 will  normally  less  than  tive  according  the  power  verted to  f  o  , e.g. f  g  the signal  cases,  s  =  Q  f  g  to Equation  t o frequency  = nf  I f t h e output, f r e q u e n c y - f  , the signal  c  3.32.  d e l i v e r e d by t h e main  potentially f  be s t a b l e .  f  c  source  source.  Under  sucb  T h e same  means t h a t  element  therefore, will  nega-  some o f f  and  i s con-  returned  c o n d i t i o n s t h e system i s  conclusion holds f o r  where m and n a r e l a r g e r t h a n °  c  i s made  Q  power becomes  at frequency  i n the nonlinear  unstable.  - mf  This  f  be e x c l u d e d  from  zero.  These  t h e subsequent  dis-  cussion. Equation can  3.36  be a c h i e v e d  rated.  indicates that  a s u b s t a n t i a l power  i f the frequencies  Earlier  i t was  assumed  f  and f  g  that  P  gain  are widely  was  t o be  sepa-  supplied  s directly istic ^ may,  from  a 60 c p s s o u r c e .  because  a low power  gain  This  appears  results.  t o be  unreal-  The f r e q u e n c y  f s  however,  represent  a harmonic  Equation  3.36, a s u b s t a n t i a l power  provided  f  has  a comparatively  o f 60 c p s . gain  According  can then  be  to  obtained  low v a l u e .  C  As the  mentioned  assumption  selected  above,  that  frequencies  suppressed  by i d e a l  but  i t i s n o t always  the  unwanted  not  likely  t h e power only,  powers  i n t h e above  tailed  circuit  a l l other This  justifiable. may  analysis.  i s based  place  frequencies procedure Although  be s m a l l ,  entirely. equation  f o r G^  t r a n s f e r takes  filters.  frequencies  to vanish  the expression  a t a few  being  i s very  useful,  t h e powers o f  i n p r a c t i c e they are  Of c o u r s e ,  can only  on  the individual  be d e t e r m i n e d  For the purpose  by a de-  of variable  22 frequency  conversion,  parameters. '  by  the  f  -  S  the  The  fact  2f  C  output  practically  If  the  power  r  as  version  a  of  filters •  close  is  •  to  one  nonlinear  frequency  regulation,  f. ... reg  s p e c i f i e d i s 25, ' r  reactor  dealt  with  the  here.  f  s  sideband  regulation  variable  complicated -  +  another.  f  c  ,  Even i f  frequency  appears  conclusion,  based will  to  be  on  the  is  rather - f , i s d e f i n e d as o s „ ^ s f i s o n l y 4.16$. reg, max  modulator,  s o l u t i o n to  have  further  ' '  suitable filters  i s that  upper  Another  must  frequencies  design  of  an  used  at  gain °  problem  be  sideband  come q u i t e  impossible.  possible  such  to  available  Frequency  The out  may  the  analysis,  poor.  filters  successive  power  substantial,  above  design  that  , etc.  the  therefore,  be f  can  be  p a r t i c u l a r frequency  ruled con-  23 4..  FREQUENCY CONVERSION NETWORKS CONTAINING- NONLINEAR A C T I V E E L E M E N T S WITH T I M E - V A R Y I N G  4.1  General The  to  use  considerations  common m e t h o d  variable  amplifiers. -  DC  of providing  frequency  The  power  converters.  arrangements  The  i s taken  quite  output  i s required.  easily  amount  t o more t h a n  native  method  i s to use  necessarily the  amount  the  transistor, design.  a  the  The  o f power  have  with  output.  This but  state  has  been  taken  in static  near  been p u b l i s h e d  some  does  reduce itself.  d e v i c e s as  i t i s expected  which  But  describe  so  been  that  very  solid-  f a r , only  f r e q u e n c y power  These type  and  consist  oscillator have  been  converters are  and  c o n v e r t e r s f o r AC of  principally  an  ommitted  output i n the  the of  switching  a  machines.  ' '  square-wave  a three-phase  square-wave  of  i n each  few  7  or  amplifier case  will  transistorized 12  variable  the  inverter  power h a n d l i n g c a p a b i l i t y future.  form  not  inverter  a r t has  may  alter-  i t does  solid  d e v i c e s o f much g r e a t e r  papers  inverter  such  state  i n the  An  efficiency,  years,, and  i f sinu-  conversion loss  power.  solid-state  AC  such  especially  case  power  from  of  of  i n the  available  capability  i n the  rapid  become  or  dissipated  progress i n the few  wave  frequency i s  i f necessary,  batteries  input  at the  overall  interest  past  of  square  development  new  50$  a t any  and  limited,  In t h i s  filtering  improve  relative With  or  from  power-handling  i s , however,  waveshaping  power  oscillators  soidal  of  PARAMETERS  phase.  three-phase  Filters  induction  motor  24 drive.  I t i s claimed  three-phase the wave  system,  with  isolated high  of  provided  power  available  The for  rectifier pool  rent  ly  but flow  The  is  on  once  a  is  the  of  turned  low  forward For by  on  control  on  and  conceptual a  switch  by  the  controlled  the  until  the  cur-  other  means. While  firm-  when o f f a v e r y the  opening  and  mercury-  will.  purpose,  whose  as  presently  a c t i o n of  is lost  and  transfer  rectifiers  of  the  off at  impedance  kind.  components  i s made z e r o  the  future  controlled by  in  the  controlled  grid  turned  element  be  a  converter  essentially  characteristic  grid  can  this  behave  sine-  high  transistor  and  closing  zero. attempts  phase v o l t a g e s acceptable  switching output  be  i t i s on,  impedance.  Several  only  that  type  i n which  state  are  of  a  by  T-connected  assume t h a t  that  solid  electrical  often represented  time  of  are  from  requirements  i s one  purposes  to  to  tests,  when p o w e r e d  switching  e v e n t u a l l y be  devices  I t can  through  i t has  by  well  windings  the  converter  types  similar  transistor  forward is  two  rectifier.  grid,  The  type  switching  transistors.  when p o w e r e d  do  i t i s reasonable  i s performed  switches.  they  dissipation  will  laboratory  as  statbr  low  to  perform  Since  or  converters  switching  as  the  neutral.  device,  high-power A  converter  efficiency  switching  according  i n d u c t i o n motors  square-wave  has  that  type  wave  as  were  d e s c r i b e d by  approach devices  forms  made t o  will  produce  Equation  appeared  to  be  time-varying  2.4  or  2.13.  conversion  such  as  the  transistor.  only  be  approximations  The  based  This to  three-  those  way,  on the  given  25 by  Equation  the  2 . 4 , b u t a t t h e same t i m e  converters w i l l  cuit  developed  4.2  A  single-phase  Consider and E  @2  will  the  now  discussed.  be  The s i m p l e  switching type  two  r^  Fig.  The  small.  the single-phase  represent  s i n u) t , s  become  4.1  converter  i n F i g . 4.1.  voltages  cir-  such  that  L e t e^ e^ =  6g  =  resistance.  Single-phase  time-varying  losses i n  converter  circuit  sinusoidal  i s a load  the internal  switching  resistors  circuit  a n d Rg v a r y  with  time i n  f o l l o w i n g way: R, = r , 1  R  Rj^  = R,  R  H  S  = R  for  2nx < t <  2  = r  for  (2n +  ^  R »  e  »  •  r  t h e above 6  o  L  l ) x < t <  4o1  >  r  4.2  circuit:  1  =  2  =  1  e  (2n + 2)x  o »  H "h  (  r  B  1 :  +  r  ) L  -  =  6  2  =  X  2  + *2  L  -  6  (2n + 1 ) T oe  0 9 1. 9 2 y  A s sume For  9  6  l .  r  L ( R  2  +  r  L  }  .  4.3  26 The  current i  ^ becomes  L  , 2 ~ l .r (B + B )  hence  the voltage v:  i  and  load  =  e  B  B  L  1  +.11^2  2  r  In a  4.1  4 . 4 . , g i s a f u n c t i o n , o f R^,  Equation  function  of time.  and 4 . 2 , For  L  one  Using  the c o n d i t i o n s g i v e n by  =  l)x<t  7  L  < ( 2 n + 2)x  ...  r  square-wave  to Equations  4.5,  g ( t ) c a n be r e p r e s e n t e d by a  amplitude  and periode  fundamental, angular  frequency  of this  expansion  4 g(to t) = c  v(t)  o f g(to t ) c  ( s i n to t c  +  Hence  the output  . = - E  s i nto t s  When t h e l o a d  4.5  L  of unit  Fourier  Equations  & 1  1 + r + r R  According  and r ^ , hence  (2n + 1 ) T , 1 - r  <  g(t)  (2n +  2  finds:  2nT<t  For  R  2 T.  w a v e b y to  Denote t h e 2TC = Q— .  yields:  s i n 3to t g — — +  s i n 5to t g — — +  ... 4.6  ...)  v o l t a g e v ( t ) becomes  ( s i n to t c  +  s i n 3to t g — — +  i s resistive,  s i n 5to t g — — +  i h a s t h e same  form  ...) , as v .  4.7  But  t h e above  partly  not hold  f o r i i f the load i s  reactive.  The voltage  voltage  v(t) given  components  components is,  a n a l y s i s does  as E q u a t i o n  of smaller  therefore,  by E q u a t i o n 2.4  amplitude  4.7  (phase  t h e same  a) and i n a d d i t i o n  and higher  an approximation  contains  frequency.  to the desired  It  voltage-time  function.  R  o—s/VWS -o  r  o—vAAAA-f R  Pig.  The can  resistors  be r e p r e s e n t e d  4.2.  S.^ a n d  times the  4.2  equal  same  on-off  cated  zero.  switches.  fc  i  2  by t h e s w i t c h i n g  An  switches  electrical  n  To p e r f o r m  where  arrangement  "the c i r c u i t arrangement whose  the switching  i g . 4.1  shown i n P i g .  with  using  square-wave  F  opening  arrangement  i s shown i n F i g . 4.3,  are triggered with  i n F i g . 4.3,  switching  R-jJt) a n d R (' )  are ideal  features  transistors  Mechanical  and  closing  essentially  transistors  operations, s i g n a l s as  as  these  indi-  28  4  o o o  Pig.  The as  i n P i g . 4.4.  successive  In  Electrical  single-phase  shown  For  4.3  time  order  Pig. pairs  When  each  T g a n d T^,  T^ be  If  the load  that  turned  and c u r r e n t assuming  i t c a n n o t be  attempt  t o do  switch,  results  be  Tg and T^ Tg  block.  block.  illustrated  i n  transistor  on and o f f s i m u l t a n e o u s l y . for resistive  to > to = 2TC60 r a d i a n s , p e r c s  switched  arranged  conduct i n  of the  waveform  has an i n d u c t i v e  through  may  a n d T^,  operations  T^,  4.5,  while  the switching required  i n Pig.  and Tg  alternately  i t i s also  8  arrangement  converter  o f l e n g t h x,  4.2,  voltage  type  e > o, T^  conduct  to simulate  Output shown  switching  intervals  e < o,-Tg and T^  switching  o  component,  i n a r c i n g and a h i g h  with  voltage  second.  the current  o f f instantaneously.  so i n a s i n g l e - l o o p c i r c u i t  load i s  a  An  mechanical  spike  across  the  Fig.  4.5  Output  waveform  29 switch  gap.  switching  transistor  down a l m o s t Pig.  4.4  sient such  The c o r r e s p o n d i n g  be s e e n  v o l t a g e peaks w i l l that  the load  Consider ducts.  t h e case  to  approximately  T^  case will  not occur.  These  be t u r n e d  exceeding So  f a r i t has been  be  zero  a n d T^  con-  Now l e t  from  going  because  drops  below -e i n transistor  as Tg and a c t s as a  when T g i s t u r n e d  +e b y t h e a c t i o n  c o n d i t i o n s must  than  component,  o n a t t h e same t i m e  Similarly,  tran-  The v o l t a g e v t h e n  I t i s prevented load  i n  conditions,  e approximately.  simultaneously.  -e.  break  opened.  e i s larger equals  would  of the circuit  certain  i s never  operation of a  the transistor  under  when  o f an i n d u c t i v e  clamp.  turned  that  Then t h e v o l t a g e v  open and Tg c l o s e  mode"  By i n s p e c t i o n  circuit  T^  the  i s not possible,  immediately.  i twill  "arcing  off, v  i s prevented  from  of Tg.  assumed  that  on and o f f i n s t a n t a n e o u s l y .  the transistors I n most  c o u l d be  practical  appli-  cations,  s w i t c h i n g times t , and t . . a r e of the order o f ° on o i l microseconds. Generally, t i s l e s s t h a n t ««. With i n " * on o i l creased base d r i v e , t decreases while t tends to increase. on A typical  transient  off  characteristic  f o r common  emitter con-  3 8 figuration interval pair  when  performs  break in  i s shown i n F i g . 4.6.  cumstances  Even though with  the short  to that  the actual  of a  make-before-  transition  time  constant, under  circuit  will  n o t be opened  up.  time  transistor  the load  the load  can b u i l d  During  place, the active  an o p e r a t i o n s i m i l a r  switch.  comparison  spikes  switching takes  '  time normal  so t h a t  i s short cirvoltage  30  t  t  Pig.  tion  4.6  Transient  The  expression  4.7,  former. is  not  4.3  To given type  ideal  Transfer  of  by  Equation  described  principal  an  voltage  voltage  2.4,  derived  source  and  switching  above,  circuit  Equa-  split-phase transsource  and  load  case.  switching  type  approximation  i n the  circuit  common e m i t t e r  r e a c t i v e power between  i n that  three-phase produce  of  f o r output  presumes  restricted  The  response  three  to  converter the  system  single-phase  previous  i n Pig.  voltages  converters  s e c t i o n can  is illustrated  of  be  4.7  used. (control  of  the  The circuits  omitted). Now  consider  the  output  voltages  from  the  converter.  (1*2)  three-phase load  switching  Fig.  4.7  S i m p l i f i e d diagram  of three-phase  '^witching  Fig.  4.8  Converter  system  units  converter  units  representation  for  analysis  The  i n p u t rms  and  the  ther fed  in  winding  i t will from  these  line-to-line  an  be  ratio  F i g . 4.8,  i s chosen  assumed t h a t  ideal  simplified  voltages will  control  load is  4.8.  examined  sa  ^  =  signal  s  i n the  toJ f l o w  satisfy  this  sistive  three-phase  is  indicated  as  1.0  p.u.  f o r ^convenience.  transformers  cps. voltage circuit  can  are  Fur-  ideal  source.  be  and  Under  represented  as  ^s*  n  s i n (co t S s i n (co t + s applies  on  previous  the  condition,  one  load  the may  i n the  120)  4.7  to  phase  for a  finite  i s only  true  switches  when  circuit  n  load  4.9  v  by ,  resistive i f current  closed.  i n F i g . 4.8.  4.9.  Parasitic  i s given  introduce a parasitic  load  4.9  a  v o l t a g e , here  three—phase  Fig.  ...  120)  output  section  Equation  through  in Fig.  i  which  Its effect  (Equation 4.7).  allowed  taken  where  o  was  60  conditions the  e , = \[2 SD e „ = \J2 sc  Equation  1.2  these  three-phase  e  The  as  be  To reThis  32 In trol  accordance  voltages  trical  degrees  phasors the by  60  whose cps  form  .  = — f - sin  =  sin  7t  Output  CO  s  t  sin  (co t S  i b -  4  V  (co„t +  2  S  120)  D  trigonometric  2^2  be a  symmetrical  The  input  sequence  must  form  angular  a  s i n 3co  sin-5to't + i  c  [sin(u t c  5(co  sin  +  of  120)  t  -  120)  + ~ o  120)  7(co  sin  these  can  CO ) t  be  to_)t] + s j  i [ cos o L  (5co„ - c o ) t c s  c  +  120)+  120)  -  120)+  .  written:  cos(3co  c  (7co  120),  C  c o s (3 co„ + c  cos  t  +  +  ~  c  /  -  are:  s i n 3(co t  + ~  C  )t ;  the  control  s i n 7(co t e  +  (7 M  given  s i n 3(co t C  cos(co.  [cos  of  ...1 J  +  co ) -  Y  that  sequence  the  £ c o s ( CO,  an  of  hence  voltages  + JI  + |  transformation  elec-  t  120)  -  con-  set  voltages  negative  s i n 7co t c  l  s i n 5(co t c  c  to  system,  frequency  line-to-neutral  co t + c o  the  d i s p l a c e d 120  i s opposite  120)[sin(co„t + l_ C  + i By  form  rotation  a positive  + io cn  and  must  phasors.  fundamental  +  'bn  of  voltage  i s to . c  4V§~  i n time  direction  ( i i i ) s e c t i o n 2.2,  phases  control voltages  The  voltages  condition  three  apart  4.9  Equation  system.  the  input  fundamental  an  to  with  co ) t I + s J  COS(5OJ  ...  -  CO ) t  +  c  co ) t s  33 [cos((w_  Tt  bn  + !|pos  ((3w  120) - cos(u)  - c o _ ) t •=  - to )t  c  + 120)  g  - cos  +  to_)t|§;  ((3w  + £o )t -  e  120)p  g  + ||>os (5u - u ) t - cos ((5o) + » )t + 1 2 0 ) | c  8  c  _  a  4  a  Q  + i k o s ((7ai - t o ) t = 120) - cos (7co + w_)tf|+ . e  —  cn  - cos ( l a + w_)t ,  |cos((co„ - 0» )t + 120)  it  o  'til + i  fc-os  ((3w  + ^ t|cos  (5OJ  + ~  ((7w  |2os  These be  be  +  e  = ^rT [  C  O  as  S  K  —  2v2 T1  = -—-  H L 3  f o r t h e output  of frequency  =  Ti I  "  c a n how  components.  interest  I ~ y  c  o  s  (  3  a  >  c  "s**  +  co.s(9a> + to_)t + e s  here  1  c o s (9«o  c o s (to^  C  + to ) t +  C  S  i i  = to ) t +  and  The  will  cos  O  c o s ( 7 to ©  7  c  o  . „.  s  (  7  w  c -  V* ...  4.11  tojt s  1  „.. 4.32  J  S  1 •=  +  ... 1 j  cos(3w„ - to ) t CQS(5OJ c s o c -  This  + u> ) t ' | + .,  voltages  or sequence  120)1  follows:  «7  O  a  c  +  120)'^  + w )t -  phase  are of p a r t i c u l a r  + i 2v2 r—J—  c  - c o s (7OJ  g  components  -  e  - o ) ) t + 120)  c  + OJ H  - c o s ((3&i  - c o s ((5to  g  i n terms  presented  ) t - 120)  - co )t  c  expressions  arranged  sequence  - M  (5w  - to ) t  C  S  - to ) t +  . . . 1  S  i s not t h e c o n v e n t i o n a l form  ...  J  i n which  symmetrical  4.13  34 components several  are presented.  frequencies  The  positive  frequency  f  amplitudes. frequency sequence  nents  Pig,  not appear;  +  f  This  i s done  system,  i t appear  sequence  fundamental  the fundamental In the zero  voltage  w i t h t h e same  The system v o l t a g e s  can best  contains  amplitude  a  as t h e  be r e p r e s e n t e d b y  i n P i g . 4.10, f o r t h e compo-  i n E q u a t i o n s 4.11 t o 4 . 1 3 , u s i n g  fort =  4,10  n o r does  But t h e zero  of frequency f  diagrams.  ference  system.  has t h e d e s i r e d  I n t h e n e g a t i v e sequence  does  given  system  sequence  because  and i n a d d i t i o n components o f h i g h e r f r e , as 3f + f , 7 f - f , 9f + f , w i t h reduced c s ' c s c s'  fundamental. phasor  sequence  here  s  voltage.  component  a r e p r e s e n t i n each  adopted  - f  c such  quencies ^  I t has been  phase  a as r e -  o.  Phasor r e p r e s e n t a t i o n o f system v o l t a g e s i n d i c a t i n g amplitudes r e l a t i v e t o the fundamental, d i r e c t i o n of r o t a t i o n and a n g u l a r v e l o c i t y . A) p o s i t i v e sequence systems, B) N e g a t i v e s e q u e n c e systems, C) Z e r o s e q u e n c e s y s t e m s ( p u l s a t i n g v o l t a g e s ) .  Now  suppose  neutralo  the  A l l sequence  the  load.  the  zero-sequence  should  be  But  of  the  which  equals  Hence  the  case  currents This  zero  resultant  L  + l_ O  + 2\/i 1t  cos(9o3  c  c os (3co c  K cos(9w  +  +  -  w ) t s  - i  g  l\l2 'cL  +  i  torque  and  the  star-  at the  neutral  ( E q u a t i o n 4.13).  ,  load  becomes  + w ) t + i cos(7to s  - w )t  c  g  COS((3M  ((9w  -  c o s (5w„ + m  o  ©  )t  s 4.14  to J t  -  toJt  ~ cos((9u  c  o  +  s  ..  120)  +  -  J  cos-((5tu  a» )t  +  s  120)  to )t g  +  +  co )t  +  o  S  i  cos((3M  120) 120)  .  '  '  -  '  4.15  120)  -  -  as )t+12Q)  + O  O  +  120)  + ta_)t  ©  ^ •  S  c  ((3w  7? c o s  s -  cos((7to  -  + &* ) t - 120)  C  cos  .  120)  c  f c o s ((to  load,  S  to ) t s  c  O  I  c  through  - w ) t  y  M  cos(3w  co )t  a c o s ((9o)i •f i  occurs  voltage e  +  C  [cos((w  machine  isolating  voltages f o r the  i  current  no^degfr^l®  voltage then  sequence  AC  grounded  s  u  =  drive  three-phase  produce  A  phase  c g  then  i s a c h i e v e d by  windings. the  of  ,2\J2~ ['COs(<i) - u ) t  'aL  bL  i s star-connected with  v o l t a g e s can  i n the  avoided.  point  e  load  +  +  u>Jt  +  120)  36 + [ i c o s ((3co  + i The presented and  voltage  systems).  c  - w )t  -  120)  - ^  - w )t  -  120)  -  g  g  components  i n F i g . 4,10  a negative  pretation  cos((9co  c  sequence  i s used:  a  system.  B),  and  4.14  and  t o 4.16 a  common  negative  -  g  ooo  constitute  (Or i f t h e  set of p o s i t i v e  + co )t  c  „oo |  i n Equations  A), a n d  c o s ( (5C«J  120)  4»16  are r e positive  inter-  sequence  37 5.  PERFORMANCE  OF AC MOTORS WHEN POWERED.FROM T H E SWITCHING TYPE  This  chapter  performance described  load,  spective  an  section  these  driving  that  reactive  power  the  motor.  synchronous  All  components  produce  of lowest  amplitude, general, tional  motor  the positive  opposite +  sequence  and T  Similarly, which  c  - f  g  which  i s  require  of three-  Since the  has the l a r g e s t frequency.  s e t up a  i s termed  In  unidirec-  positive  since  i n t h e a i r gap as t h e d i r e c t i o n  the negative negative  to the positive  o r may  torque.  by t h i s  associated with  i s termed  be a s -  assumption  types  ) also  voltages  torque  o f t h e mmf-waves  torque  ( f  i s determined  electro-magnetic  voltages.  T  speed  h a s t h e same d i r e c t i o n  travel  a  frequency  i twill  positive  sequence  Denote  these  of  sequence  voltages  since i t acts  torque.  form  t h e i n d u c t i o n motor and  load,  component  it  Two  i n d u c t i o n motor will  i s of the  can l a g t h e i r r e -  n o t be j u s t i f i e d ,  be c o n s i d e r e d ,  motor  of current flow i n  the f i r s t  Three-phase voltage  with  currents  source.  o f AC  phase  To s t a r t  While  may  will  t o each  the load  voltages.  p h a s e , AC m o t o r s  examination  a r e independent  the latter  auxiliary  5.1  4.3.  voltages  and a l s o  reasonable,  a brief  applied voltage  above,  sumed t h a t the  when  presents  CONVERTER  i n a  s e t up direction  two t o r q u e s  respectively.  Then: =  T  f - f c s  +  T  3f  c  +f  + s  T  7f  -f  •s  +  T  9f  -*-f  s  +  oo  o  as  38  T  =  ~  T  3f  - f c  The  s +  T  5f  subscripts  frequencies.  s  +f c  The  +  T  on  -t c  9f  the  ~ To  find  components, cuit  will  approximate the  be  s  right  resultant  5o 2  °  +  hand  torque,  side T,  4 5 0  is 5.3  + ~ expressions  f o r the  conventional induction  used  denot e a s s o c i a t e d  This  circuit  motor  individual  torque  equivalent  cir-  i s shown i n F i g .  5.1,  Xg +  -e-  L  X  m  Fig.  rms-value  of  = stator  x„  = rotor leakage frequency f.  X  =  r  = rotor = slip  s The the  magnetizing  = stator  0  leakage  of induction  reactance  reactance  of  at  reactance  frequency  frequency  referred  at  motor  to  frequency  f.  f. at  stator,  f.  resistance. resistance  i n p.u.  referred  of  principal  electromagnetic torque  following  2  ET,  input voltage  x,  m r,  by  Circuit  5.1  _)  2  at  to  stator.  synchronous input  speed.  frequency  f is  given  equation: T  =  4rtf  E  l  I  2  C ( 9 S  ^2  0 0 0  5  o  <  39  2 where  cosdg  It one  s  =  p  m number  of poles  q  = number  of  phases  i s c o n v e n i e n t t o t r a n s f o r m t h e above  s h o w n i n F i g , 5.2.  By T h e v e n i n ' s  R, E  5.2  Transformed  induction  E  — m j ( x  lo  = V1  r J  The  torque  x  X  relation r„  +  m ^  r  1 J 1 « +  1  1  +  l  r  +  J(  c a n now  1  +  ( B  As termined  1  stated  + i^) + 2  above,  ^  x  V  to this  +  circuit  5o 5  l ^ l  5.6  )  be w r i t t e n  as  (Nm) < 1 X  +  t  Q  x  - f  frequency.  of frequency.f  motor  m  X  2  )  i s applied  5.7  2  the reference  by t h e f r e q u e n c y  related  s  2  — — x )  S  slip  theorems  x  E  age  to the  10  Fig.  be  circuit  =  synchronous  speed  f , and t h e s l i p  When a p o s i t i v e to the stater,  i s des  will  sequence  volt-  the apparent  becomes s.  f ^ - f (1 - s ) = _ _ _ _ _ _  o  5o8  40 For the  an  applied negative  apparent  slip  s  now  be  cies,  of  the  slip  s  is  t h a t no  of  the  find  f  s)  components  In  of  4.11  w h i c h make x  are and  the m  to  given  In  r» „ I n  5.7  approximated  E  10  =  i n the  frequency  will  on  be  x  m  and  5.2  =  V, 4TXf  Let a  new  x^  :  —  Per  values  simplify  JL  basis  adding torques  unit  for e  5>> x, , a n d  and  +  the to  x  m  e  Xg  ——•;-••<-• — — — — — — — — — — —  be  subsequent frequencies  » x  Q  . ^  Equa-  become  —  O O O  5.10  Soil  reactances  f  has  at  frequency  t  gi  and  define  ks k  one  ,  2  parameter,  Thus  can  l  V  1  and  .  frequen-  ..  T  f  negative  restricted  this  and  condition for  expressions to  5„1  voltages,  motor.  order  general,  JL  The  positive  i n the  4.12.  Equation  corresponding  resultant  expressions  ^>  Q  parameters.  s a t u r a t i o n occurs  expressions,  5.5  motor  to  voltages  Equations  ( l -  i n terms  and  them t o g e t h e r  tions  torque  expressed  of  is +  Each  sequence v o l t a g e  ~z  t  ™"  1  f o r the  JJ  k  3L  fundamental  eoo  torque  component  5o  12  41 where  a  c  As  seen  from  Equations -  V  i  f  c  +  =  f  s  C ±  high  f  — - =. j p.Uop i = - l  Equations  frequency  c  s  4rcf  5„  .  5.14  o o o  O  5 . 1 1 , 5.8 a n d 5.14 g i v e s  torque  N  f o r the posi-  components:  r r2  _ mil  3,  9  '  Combining tive  4.11 a n d 4.12  (1)2 J L 3' 3k+l  2  (3k +  2(k+l) + + 1) ^r ( 3 ^k -r 2o  1) s(k-l)  V  0  0  0  2(k+lks(k-l)> +<«**> 2  ( x  2  1  +  x  2  )  2  5.15  r (7k-l) 2  - pall (I)  T  7f.  - f  A  c  ~ 4itf  K  l  2  }  _L_  6k~T *• o  7k-1  6k  . s  +  \ • »•—A I  ,(k- 1 ) > r (9k 2  jH  c  s  ,1)2  4Ttf .. \ 9 s•  ;  n  +  2(4k+l) + r„(,9k+lJ  1 9k+T < l r  +  o  O +  (  7  k  °  1  <y*2-r  }  O  1)  O  the f i r s t  5.11 c o m b i n e d  with  three  2  2(4k+l)+s(k-l)>  negative  Equations  5.9,  torque  o  2  +  (  9  k  +  l  )  (  2  2(2k-l) x  3f  -f C  " 4-rcf 3' - s v  S  components,  5.12 a n d 5.14  r (3k  w o  slk-T]  O  For  5.16  O  l  x  O  O  +  x  2  )  5.17 V o  Equation  gives:  -.1) ~  s(k-l)  3k= (  r  l  +  2(2k-l)-  ( k - 1 )  }  2  +  (  3  k  =  1  }  2  (  X  .  1  +  X  2  }  5.18  2  42 r T  +f  5f  = frtr* ^  (5k +  2  6k - s ( k - l ) r „ ( 5 k + 1) 2  5k+T ( r  l+  6k - s ( k - 1 )  -  9  9f  c  -f  s  (  s  2(5k-l)  ^ 9k~T  4 i f ~ 9> s  r  ( l r  +(  }  r (9k T  1)  -  O (9k—1 )  2 5 k  2  +D  ( l+ x  « o  1)  1) pared  Higher  to  torques be  these  expressions harmonic  £ except c" s =  tend  characterized 2)  The a v a i l a b l e  increased varied  almost  T1  +  0  » 5 e20  o  components  other,  are very  zero.  t h e motor  Since  s  >o  com-  these  performance  by E q u a t i o n  f o r a given  small  will  5.13. decreases  with  r e f e r e n c e f r e q u e n c y , i „e. i n c r e a s e d k , b u t i t c a n b e  over  At  r> 2  one c a n c o n c l u d e :  exclusively  torque  5e 19  9  2  k  torque  each  )  2(5WlKs(k-l)) + ( 9 k - l ) ( lx " "x2 ) 2  +  w h e n ,s a p p r o a c h e s  to cancel  2  s(k-l) n  •  From  e  x  a wide  range.  a given reference frequency  teristic  takes  t h e form  latively  low r o t o r  Fig.  indicated  the torque-slip  i n F i g . 5.3  (assuming r e -  resistance).  5.3  Typical  torque-slip  charac-  curve  2  43 Neglecting for  which  rived  higher  maximum  from  harmonic  electromagnetic  Equation  f  T  -  slip  s  t  peed.  given  Equation  T  near A  m,k  i  —  that  1.  But  to  In'order  to  quency  f  and  s m  be  de=  m  m  s  5.21  2 or  as  T_  a  f u n c t i o n o f k, Thus  one  1 :  +^r  X  relation  torque  torque  s  can  increased frequency  m  1  from  ...  s .  T—r*  this  ranging  this  torque  with, s _  k-1  corresponding  how  with  2 1  +(k-l) (x +x ) 1  to  not  be  . s give °  40  to  is  . o o  5.22  :  for values accurate  i s therefore available about  ^,  2  hold  fairly  syn-  findss  —* 2  does  i t i s expected  substantial  speeds  5.13  2  8Ttf  Note  ,N2/„  r^+dt-irU^)'  Maximum  V  occurs,  m  s =  ' 2./,  decreases  m  chronous by  T  slip  for s J m  m  The  torque  the  f  c  gives • **  components,  5.13s  d_ ds  This  torque  200$ of the  at  of  for  k  k>1.5.  synchronous  synchronous  speed  f  an  idea  o f how  compares  with  maximum t o r q u e ,  1.0  p.u.  voltage  T  input,  , varies m,n consider  T_  with g  ,  the  k,  at  and  fre-  ratio  s pp-*— m, s  «  Recall  reference  T  m,k  m  »  frequency  ,2,2  T ~ s  =  H)  K  that V  1  f  r  i n Equation  5.22  equals  —  corresponds  to  =  Hence  l \ / +  r  l  2  +  (*i+*2>  k  r  x  +\/r %(k-l)' 1  : J  ^  (x +x )' 1  and  that  2  r-o——-~2~r-  k~T  2.  p.u.  2  0 0 0 2  '  44 l '  r  l  x  on  a  the  n  d  2  x  a  right  familiar  r  e  no  "k  hand  kn° side  w n  «  But  of  Equation  quantity Q =  write T  motor,  Equation  m,k  ,2 2  Q has  divided For  a  by  r^„  the  conventional  l  a value  1+Vl  +_g[  of  about  5.  Thus  one  can  2.47  2  1+ \ / l ( k - l ) Q 2  2  k  +  Equation  are  and^ d e n o m i n a t o r  5.23:  1  N  5.23  appears. r  induction  i f numerator  5.24  i s presented  ~  5.24 v/ o  o o o  1+ \ / l + 2 5 ( k - l )  1  graphically  2  in Fig.  5.4.  m,k  1.0  5Q =  1.5  1.75  2.0  2.25  2.5  5  2.75 f  Fig.  5.4  The slip  s,  Maximum  o n l y way i s to  requires  the  efficiency  factor  decreases impedance. with  torque  maintain  on  will  because  of  as  constant  applied  e x t r a equipment  and  crease  to  vary  factor  input  relative  voltage.  the,supply drop a  with  larger  H y s t e r e s i s and  increased frequency.  a  function  torque But  at  a  this  side.  Both  i n c r e a s e d k. inductive  eddy-current But  of  although  k  given of  course  power The  power  component losses these  i n -  in  45 drawbacks rule  the  may  be  important  the  oscillate  an  external  source.  in  the  inductive  load  split-phase  can  one  assume  windings.  ductive  load  primary  current.  current,  of  at  be  feed an  at  destroying  but  of  power  the  of  the  converter  and  converter  operation  on  an  inductance  disregarded,  load,  free  machine  switching  Leakage be  an  power must be  shows t h a t  not  motor,  and  in  neither  'secondary  primary  transformer in-  requires  instantaneous  changes  in  the  charging  e f f e c t of  transistors  as  a  no  the  compensation a  load.  for  is  lagging  possibility  Loading  r e a c t i v e VA  only  to  without  possible  will  load run  addiat  the  transistors.  to  into  This  induction  motors  to use  the  load  terminals.  at  become  i s to  real  induction  available.  an  according  possibility  control  over-  i n t e r r u p t i o n , whereas  for the  necessarily  between primary  without  nearly  appears  approximately  several  can  compensation  both  reactive  output  the  t h e o r e t i c a l l y i t presents  motors  adjusted  Another  The  compensation  arranged,  cps  insufficient  Capacitive  be  60  speed  difficulties.  resistive  not  circuits  chapter  coupling  component  but  induction  to  ideal  flows  to  do  e x i s t s : Reactive  transformers  current  the  magnetic  previous  With  of  Examination  leads  the  they  supply  always  between the  described  generally  power  condition  to  risk  enough,  p o t e n t i a l advantages  Concerning  tional  serious  the  costly.  load a  motor  and  terminals The  load  system,  impedance  the  to  establishing converter  s o l u t i o n i s recommendable  in  are  same  to  be  operated  at  must  speed.  condenser  thus  across  be  capacitors  synchronous  synchronous  can  the  case speed  46 and  fed  from  excitation the  next  5.2  of  well  the  the  as  degree  mined c  as  -  by f  s  torque tion  the  only.  negative and  this  lagging  under  problem  will  be  given  motor  of  proper  discussed  output  mmf  to  the The  higher  sequence  converter.  condition  on  the  in  harmonic  voltages i n the  Excitation  of  From  can  poles. be  rotor the  the  increased excitation  synchronous  is  have  a  currents  factor  is  can  be  voltages  be  deter-  converter, produce  fixed  useful  space  rela-  synchronous of  both  to  avoid  point  of  view  air-gap  of  and  by  for  re-  fre-  positive  extra  impedance  further  on  directly  motor  low  input  the  Compensation at  re-  speed  applied  the not  small  The  Its  the  will  place  kept  a  from  is  depending  and  with  is  power  load.  rotor.  takes  indicates  rotor  by  be  unity  i t s speed w i l l  not  harmonic  should  fitted  current,  load),  4.16,  do  at  Assuming  currents  armature  best  shaft  frequency  waves  converter  conditions,  (no  to  harmonic  their  armature  a  the  operates  stable  4.14  from  motor  frequency.  induction  windings  inductance  motor  f l u x wave p r o d u c e d  quency  sign,  or  input  Higher  the  motor  This  lowest  power  three-phase  Equations  power  in  The  condenser  reactive  of  load  to  an  active  and  unit.  load  e x c i t a t i o n at  of  since  to  type  of  by  .  of  leading  proportional  given  motor  motor.  independent  started  converter  synchronous  supply  synchronous  f  the  Synchronous  stricted,  as  single  section.  Since  as  a  losses  motor no  damper  for  adding  de-  higher  series  circuit. motor  so  as  to  obtain  unity  47 power  factor at various  speeds  technical  problems  which w i l l  the  features  will  basic  and loads  presents  n o t be d e a l t w i t h  be d i s c u s s e d ,  under  a number o f  here.  Only  simplified  condi-  tions , The  following  assumptions  a)  The motor  b)  No  c)  The a r m a t u r e  d)  Armature  e)  has a uniform  saturation  gap  resistance  inductance  phase v o l t a g e  when  simplify relations  compared  to,air-  frequency t o some  and, l o a d  extent  i n t h e machine  one  only  considers  and  neglects  perfectly  but cannot out.  drop  components  justified  well  f and  of current  be a c c e p t e d  Assumption  part  de-  c ) and d) and v o l t a g e  i f a more  e) i m p l i e s  component  i s produced only,  by c a r e f u l  are specified;  i n the converter.  the u s e f u l torque  and c u r r e n t  ranges  the fundamental  voltage  quite  the presentation  i s t o be c a r r i e d  that  i s s i n u s o i d a l , of frequency  b) can be s a t i s f i e d  analysis  cerned.  i s negligible  amplitude.  Assumption  age  i s negligible.  inductance.  constant  fact  air-gap.  occurs.  leakage  The,applied  sign  a r e made:  of input In view  exact  that voltage, of the  by t h e fundamental  of t h i s  as f a r as t h e subsequent  volt-  assumption i s a n a l y s i s i s con-  48  Fig.  The Fig.  5.5  basic  5.5,  Phasor  phasor  diagram  diagram  for  for  synchronous  one  phase  terminal voltage,  E^  =  internal  induced  1^  =  armature  current,  s  L ...= m a g n e t i z i n g  inductance.  =  flux  per  pole  due  to  =  flux  per  pole  at  0 ^  =  flux  per, p o l e  In  accordance  0  Q  rms  k^  voltage  rms  are E.  with 0  =  <*A  «  k  k  A  rotor  no to  the  above  i  X  reaction.  assumptions  o o o  f  Fig.  5 25 0  5.26  A  i s given  to  mmf.  armature  constants.  according  value.  load.  due  f  rms  value.  i ^=  field  current  (DC).  The  by s=  Also,  in  value.  voltage,  m  and induced  i s shown  where  =  0  motor  5,5,  constant  o o «  5.27  49 2  E Substitution k Combine  2  =. of E^  2 E  w  2  0  2  (toL^I^)  from = V  2  Equations  +  Equation +  2 t  5.29  . o . 5.28 5.27  (wL I ) d  and  into  5.28  yieldss ...  2  A  5.25  and  solve  for  5.29  i ^ :  5.30 ( k _ k  Equation applied  frequency,  factor. if »  shows  ) V  how  voltage  (k k )' E  Equation  and  5.30  f  the f i e l d load  Let excitation current From  0  5.30  f  current  current,  a t no  i t follows  load  depends  at unity (l.=0)  current  must  relation, when a air The  vary  however,  constant,  as t h e i n v e r s e does  t h e no  load  of applied  not hold  s u b s t a n t i a l amount  be  0  *E f  V^. i s a s s u m e d  f o r high  of torque  power denoted  that  t ^•fo ~ k - l LE u u> Since  on  0  0  5  o  3  1  excitation  frequency.  synchronous  i s required  to  This speeds  overcome  friction. relation  V «VA> i  The  _  between I  S  /  total  i ^ ,  to a n d  input  power  per  phase,  S  V  L  2  "t  .  2  d  electromagnetic  T =  P  p  h  ,  (q =  P  2  ph  torque  3)  c  T  i s given  o o  by  ...  5.33  50 Thus t h e r e l a t i o n  between i „ , m and T c a n be w r i t t e n :  4  to  (k k ) E  According  2  f  V  2 W  2 t  to Equation I  minimum when t h e two t e r m s equal  d  9p v(k k ) 2  B  f  5.34 2  5.34, i „ a s a f u n c t i o n o f to h a s a  under  the square-root  sign are  i n magnitude: 2 V  t  3  P  L  d  V f  T  ~  The  and  W  la):  as  L  hence  3p 2L T  f , min  c u r r e n t i ^ a s a f u n c t i o n o f OJ w i t h  a parameter  i s indicated  *  5o35  d  constant  i n F i g . 5.6, w h e r e  torque  T  T^<Tg<Tg„  0  Fig.  5.6  The maximum  Excitation  c u r r e n t as a f u n c t i o n o f frequency torque as parameter  maximum v a l u e temperature  o f i ^ depends  rise  i n the field  on d e s i g n windings.  and  with  specified  The  available  51 torque able  at a p a r t i c u l a r frequency  armature  current.  i s l i m i t e d b y maximum  The t o r q u e  T  allow-  (&)) i s a p p r o x i m a t e l y max  proportional  to ~  (Equation  ture  resistance  increases  must  be r e d u c e d  unless  speeds.  Additional  5.33).  with  better  Since  increased cooling  restrictions  t h e e f f e c t i v e arma-  frequency,  i s provided  may b e p l a c e d  1^  at  with  a prescribed  Theoretically, three best  quantities approach  angle the at  control  t h e output  motors motor  voltage  f o r correct  o f synchronous  6 = 0 i s a necessary  power  factor  ration ted  operation,  problem.  on L  i n max  and a d d i t i o n a l  angle  (6 < 0 )  be  increased,  be  decreased. As  applied teristic  indicates  voltage  improve  of the induction  Then  motor.  a n d make  of  The  i n this  synchronous or as a  Note  condition  that  f o r unity  a unique  key t o the f i e l d  and l o a d  variations,  condition.  the f i e l d  angle  be  satuaccoun-  Lagging  current  indicates  section,  will  i ^ should  that  control  the torque-frequency  motor.  6 = 0  induction  case,  single that  the  i f 6 i s measured  components  i n the previous  would  current,  load  phase  But the  and condenser.  o f speed  by t h i s  t o to.  be t o c o n s i d e r  and s u f f i c i e n t  inductive  and l e a d i n g  mentioned  motor  on any o f t h e  can consist  condenser  i tprovides  Effects  f o rautomatically  phase  load  to the synchronous  since  control  excitation.  as a synchronous  combination  probably  and l i n e  of the converter,  i n addition acts  o f i ^ c a n be based  i n p r a c t i c e would  x  margin.  1^, P ^ a n d T, i n a d d i t i o n  6 between phase  criterion  stability  a  higher  A  connection  m  I f f o r instance  i t could  of the characthe applied  52 voltage  i s made p r o p o r t i o n a l  practical  T_  ^  =  limits),  K  .  '  r  which  unity  less  i^vAi  2  +  indicates that  dependent at  Equation k  «=» 1  ^  *  no  load,  the  or  independent  field of  a  _ ^  ^  2  i  x  the  +  r e g u l a t i o n would  view  machine  design,  as  , K  —  be  be  f  (within  form  ooo  consto  i s much l e s s motor,  when t o r q u e  can  since  the  -  5 35 0  be seen  ratio  speed  approximately from  desirable  i t leads  and  frequency  operating  voltage/frequency  would  also  ^  f  2  torque  constant  current  speed,  x  on  synchronous  current  Voltage of  :  To  factor,  frequency  5 „22 t a k e s  ;  variation in field  At  the  the maximum  than before. power  i  to  to  varies. constant  Equation  from  the  nearly  means  5.31.  point  of  constant  flux  densities. On tional must  the  other  voltage-control  perform well  voltage.  This  verter  described  age  the  of  hand, v a r i a b l e equipment  regardless  appears  ratio  to  i n Chapter  of  be 4,  transformers  i s necessary. actual  possible as  long  power t r a n s i s t o r s i s n o t  The  variations with as  the  the  exceeded.  and  addi-  converter  i n input  type  of  break-down  rms  convolt-  53 6.  E X P E R I M E N T A L FREQUENCY CONVERTER AND  6.1  SQUARE-WAVE  Introduction It  quency  was  speed nous  decided  converter  monstrate  of  i t s basic  speed 110  i s of volts  the  described  type  features  rms  rated voltage  mum  output  sistors  from  2E.  means t h a t  The  interest.  converter  BV^_  specified  E must  be  have  A volts  small  rms  value  across  less  than  line-to-line  out  High are  at  so  this  power  that  low  available. at  stand  300  nearly with  transformers  the  or  load,  volts.  each branch  by  use  by  (20)  commonly the  maxi-  the  break-  power  tran-  As  can  each t r a n s i s t o r 20  volts. at  be  is  This the  load  V3  —  speed-tests  of  =  15.6  rated could  at  volts. 20.  eventually  be  voltage.  But  present  then  is  volts.  voltage  i n d u c t i o n motor  transistors  line-to-line  achieved  less  three-phase  was,available  carried  volts  an  40  and  voltage  Ordinary as  de~r'  synchro-  But  is limited  transistors.  and  of  output  motors.  fre-  supply  desirable since this  2 should  power  4,  control  An  three-phase  fundamental  three-phase  i n Chapter  combined  maximum v o l t a g e  Hence  the  a  motors„  the  with  F i g . 4.4,  =  max  small  experimental  as  be  (BVQJJ) o f t h e  were used,  from  would  of  voltage  down v o l t a g e  (2e)  an  particular  the  seen  to b u i l d  c o n t r o l " d e v i c e f o r AC  about  in  GENERATOR  with  values  i n order the  power  Although  transistor of  ( F i g . 4.4),  to  o f BV^_ provide  around e.g.  transistors this  must  either  by  s e v e r a l power t r a n s i s t o r s the  converter  110  V  rms  with-  load voltage  ratings,  100  can  be  step-up in  becomes more  series expensive  54 and  less  reliable.  transistors this  which  converter The  square  suitable  load  of  difficulties  of  this  author Only  chapter.  stages  become  cuits the  were  a  was  not  final  power  will  make  most  could  that  this  will  with  pulse be  and  A  arrangement  when i n c r e a s e d power  can  number  the to  design the  fact  digital  described  additional  used  unit,  above.  them r e l a t e d  familiar with  A  concerned.  control  mentioned  the  have been  motor were  of  from  transistors.  i n connection  arrangement  necessary  the  the  purpose  too  i s controlled  transistorized  generator,  diagram  four  in  cir-  this  still  be  amplification output  of  x  frequency  from  control  blocks  a,  b  converter  units.  Each  of  these  power  transistors, 4.  The  115  V  13.5  x  2 =  arranged  supply  unit  across  38.2  V.  and  has  delta-connected  Maximum v o l t a g e 2  and  The  transformers,  secondary.  converter 6.1.  i n Chapter  therefore  of  encountered  i s shown i n P i g .  of  future,  power  the  description  block  plained phase  build  particular  the  single-phase  sists  for  i s required.  General A  ratings  near  oscillator  Some m o d i f i c a t i o n s a n d  converter 6.2  tests  It is realized  improved.  i n the  triggers  frequency  to  square-wave  cuits.  voltage  frequency  which  speed  decided  f o r the  the  expected  fundamental  variable  no  i t was  that  be  wave g e n e r a t o r  f a r as  But  can  higher  more v e r s a t i l e , ,  output  three-phase as  However,  c  represent  units  operated  three  primary  each power As  and  cir-  conas  singleand  2x  transistor  indicated  ex-  i n Pig.  13.5 is 4.4,  V  Three-phase 60 c p s supply-  DC  Three-phase parasitic load  power  _ supply Three-phase load  A=stable multivibrator  Fig.  Pulse forming circuits  6d  Block  diagram  Gates  of  — - BP  1  Flip-flops  experimental  B l o c k s a, b and c r e p r e s e n t switching units.  converter.  single-phase  55 inductive  c o u p l i n g i s used  base-emitter The  six  The  how  output  changing In  two  two  generator  erate as  two  "and t h r e e  these  circuits  fundamental BC  t h e two  illustrated  time  one g a t e  the  i s open,  corresponding  that  flip-flop  tive  pulses  and  block  6.3,  C)and  i f one o r b o t h  plained  1, 2 a n d 3.  the multivibrator  These v o l t a g e s  are used  t o gen-  p u l s e s , d i s p l a c e d 180° i n time,  D).  Each pulse  side.  sequence  pulse  This pulse  on.  i s then  G-^ ^ a n d G |  negative  The g a t e s  have  Fig.6.2.  will  nega-  voltage  a v o l t a g e v.  process  on  effect i f  transmit  ouput v o l t a g e waveforms a t The wave-shaping  turn  h a s no  terminals are a t zero  the period of o s c i l l a t i o n  noted  t  C).  Assume  m  , and l e t t h e f i r s t  Gates i n at D  pulse  that the f l i p - f l o p s  and 2 a r e o f f , 3 i s on.  comes  by  ,level,  Fig.  flip-=fl©p  w i l l be ex-  next.  Let  off.  from  g-terminals  shows  shows i n  i n the multivibrator.  voltages  an incoming  when t h e c o n t r o l  P i g . 6.2  c a n be c o n t r o l l e d  s e t of gates,  flip-flop  inverters,  arranged.  constants  side i s already  E ) , F ) a n d G)  terminals  are  of negative  to the corresponding  If  to t h e  multivibrator  a n d two  flip-flops.  b y A) a n d B ) .  sequences  o f one  circuits  frequency  output  s h o w n i n F i g . 6.3,  fed  consists  differentiating  equal  F i g . 6.3,  are  1  wave  t h r e e - p o r t gates  more d e t a i l  signals  terminals.  square  (a-stable),  to f e e d t r i g g e r i n g  G-^ a n d Gg  i n the multivibrator appear  are i n i t i a l l y  Consequently  are then  a t t = o.  open.  be d e -  ( F i g . 6.3,  s e t so t h a t  1' a n d 2' a r e o n , 3' I f the f i r s t  ( F i g . 6.2), i t can only pass  through  pulse  gate G i ,  A-stable multivibrator  A  B  Pulseforming circuit  Pulseforming circuit  FF,  FF„ ?2  (  g'2 FF  D  \ d  r  ?3' g l c  Pig.  6.2  Block FF G 1,2,3, c,d g o g ' -  diagram  of three-phase  square-wave  flip-flop gate a n d 1} 2\ 3' - output terminals gate input t e r m i n a l s gate c o n t r o l terminals  5  generator  A)  V V V  B)  h-t  u  »\  r i  i r  5  C)  D)  B  2  -  E)  F)  G)  Fig.  6.3  Voltage  waveforms  i n t h e square-wave  A),B) t m u l t i v i b r a t o r outputs C),D) s triggering pulses E)s,F),G)s s q u a r e — w a v e o u t p u t signals  generator  56 but  i t has  The  first  at  D  and  1)  t -  no  effect  pulse so  2)  and  Pulse V  t = i t .  the  output  the No.  X"""' » V  Pulse  T  effects 1 passes  i7"*~Q°  No.  2  t =  t  f f l  .  Pulse  No.  passes  t =  2^m°  Pulse  3 passes  V|-*-0„  *-v, 5)  t =  2t  .  Pulse  No.  Gg  g^"t .  Pulse  m  No.  Tgj—^v, At  t  again.  trol tem  of  3t  of  gate  sequence, ferent  of  /  , 6t  that  to  one  Gj  as  follows?  G^ Gg  G  triggers  F F  Gg  Gg  on  Yg—••Oo , 9t  the  FF^.  closes. FF . o 0  than  need  other  pulse  not  closes. 1)  -  does  not  6)  cause  i t enters. be  but  flip-flops  starts  over  Therefore,  delayed. the  must  With  sequence  i n order be  opening  to  the of  start  preset  to  or con-  sys-  output this a  dif-  two.  available  The  process  i n F i g . 6.2,  three  the  Gg  opens,  s i d e where  Vg i s fixed,  power  low.  G^  ;  shown  the  closes.  Gg a n d t r i g g e r s F F ^ „  the  m  gates  V g and of  6 passes  Gj  o  closes.  triggers  opens,  0  FFg.  triggers  and  PF-^"  closes.  triggers  and  Q  and  next  closes.  G^  opens,  Gg  a triggering  the  output  course  m  control  v^,  state  The  m  gates  signal  voltages  is  =  Note  closing  summarized  o V g — T g , — e - O .  t =  the  opens,  m  6)  C,  triggers  and  on.  at  opens,  Gg  5 passes  and  Gl  0^  4 passes  No.  be  i s already  appear  through  Gg  Vg—Vg,—""-Go 4)  can  to  Gg'opens,  Vg-»-0o 3)  s i n c e 2"  i s t h e r e f o r e assumed  on,  Oo  on  from  the  external load to  square-wave each  generator  flip-flop  side  Pig.  6.4  Circuit  of converter  Supply transformer: Driver transformers: Power t r a n s i s t o r s % Diodes: R R  = 1.5k.ohm = 6 8 0 ohm  9  phase  a.  1 1 5 / 2 x 1 3 5 v o l t s , 130 w a t t s 4 0 0 / 1 0 0 ohms, 0,5 w a t t s R C A T y p e No 2 N 3 0 1 A (PNP) RAYTHEON T y p e N o . C K 8 4 8  (parasitic  FF,flip-flop a - output terminal  unit  y  load)  57 consists  of a r e s i s t a n c e  base-emitter c i r c u i t s basic is  i n s e r i e s w i t h two  i n the  a r r a n g e m e n t f o r one  s i m i l a r f o r the The  output  amplified  phase  o t h e r two  signals  currents  (phase  flip-flop  a).  the  require  A—stable  multivibrator  Fig.  shows- t h e  section. Current  6.5  A l l four gain  o f f mode, and The  6 =  of  amp.  arrangement  The  50  has  of o s c i l l a t i o n  voltage the  signals  i s equally  and  low,  circuits  dealt with i n  of t h e  one  this  same t y p e , 2N224.,  t r a n s i s t o r s operate i n the been used f o r  i n the  i s only  the  2Ve and  at  available."  multivibrator,  of  on-  6.  t i m e c o n s t a n t T = BC.  drop eraitter-to-base  t r a n s i s t o r i s on,  voltage  the  The  booster' a m p l i f i e r s  y  l a r g e l y d e t e r m i n e d by  be at  peak v a l u e .  pulse forming  c i r c u i t r y t o be  120.  a value  period  and  t r a n s i s t o r s are  60  the  t e s t s r e q u i r i n g more power w o u l d  have been p o s s i b l e w i t h s t r o n g e r c o n t r o l 6.3  The  shows  power-transistors properly  l a r g e r t h a n 0.2  outputs, but  6.4  square-wave g e n e r a t o r must  simple motor speed t e s t s d i d not the  Pig.  connected  phases.  from the  i n o r d e r to; d r i v e  converter load  converter.  parallel  t  „ is m  Since  o r d e r 0.1  th©  v o l t when  emitter-to-colleetor  saturation  finds ~2T  V  = 2T  In  hence t The  m  r e l a t i o n between t f  and  f  2  6.1  i s indicated  in Fig.  6.3. o ,o o 6.2  Ill =  65  Fig,  0  A-stable  Transistors.s  B B  B  T  _ 10  . = mxn max  koOhm  47 ko ohm  = 500koohm =  . O l uF  multivibrator  P H I L C O T y p e No.  and p u l s e  forming  2N224 R  l  =  Bg =  68  k„ohm  1„8koohm  C, = 500  pF  circuits  +12  volts  Combining  Equations f  Equation resistance selected  f  570 c p s .  61  B  ^  B ^  load  t h e base  f o r T.  current  6.3  input  The v a l u e s  f = 70 c p s a n d c min * •  J  i s given  ( P i g . 6.5), E ^ should  .  by 6 and be  selected  that . "V  Q  If  the circuit  not start  arranged cuit  when t h e v o l t a g e V  unsymmetrical  capacitors. starts  normally C  o f ways.  n  A  i s applied.  oscillations Starting  can be  f o r i a s h o r t moment b y s h o r t i n g o n e o f t h e  consider  circuit  set-up,  I I , F i g . 6.5  The o p e r a t i n g p o i n t  equals  5k„ohm.  the multivibrator  dissymmetry.  i n the saturated region.  ( a n d D)  max K  One way i s t o make t h e c i r -  due t o s l i g h t  identical):.  i  symmetrical,  I n the experimental  by i t s e l f  Next are  i s completely  i n a number  . Tj  TJ  max  at  true unless  yield max  V-  do  .. o  In 2  and E  Maximum  Since  forf :  1  i s included i n the expression  ^base^min" so  =  c  6.3 i s n o t s t r i c t l y  f o r C, E . ' mm  =  6.1 a n d 6.2 y i e l d s  (circuits  I I andI I I  of the transistor i s  Minimum  output  E g = 1.8 k.ohm.  impedance  Therefore  9 B ^  =  A ,  thevoltage  50 -g—g- =  with off at put is  a time  68 k.ohm. at A '  constant  f o ra period C.  A negative  rises  a step  B^C^.  rise  The t r a n s i s t o r  = T-^ I n 2, t h u s voltage  voltage  occurs a t  t o 2V i n s t a n t a n e o u s l y and decays =  =  a t C s i n c e whenever i n t h e on s t a t e .  When  step  this  producing  a negative  a t A h a s no e f f e c t  occurs,  The n e g a t i v e  the inverter  pulses  i s turned  produced  pulse  on t h e o u t transistor by c i r c u i t  II  59 and  I I I are used  two  conditions concerning:  The  first  condition  According ing  to trigger  will  t h e f l i p - f l o p s , a n d must 1) d u r a t i o n t ^ a n d 2)  b e c o n s i d e r e d now,  t o r e f e r e n c e 3,  fulfill  amplitude.  t h e second  c a n be e s t i m a t e d  from  later.  the follow-  relation: *d for  With  these  u  =  the transistor  values,  a] )  -  Equation  6.5  n  short  (about  time  o f f - t i m e x ^ .= t ^ .  the sec.  -...6.5  b  a s 0.98, f ^ =  510 k c .  yields microsec.  f o rthe inverter  2 microsec.)  a  c a n be t a k e n  t, = n o J. o_—cTTT = 1 5 . 6 d 0.02 x 2TC 510 The t u r n - o f f  2ir f  h  transistor  i s relatively  due t o t h e s t r o n g b a c k - b i a s .  With  a n d t ^ = 20 m i c r o s e c .  = 600 p F , x ^ becomes  (measured),  which  i s more  22  Hence micro-  than  suffi-  cient. 6.4  Gates The  and f l i p - f l o p s  s i x gates  -flops.. gates in  F i g . 6.6 s h o w s  G^ a n d G^.  F i g . 6.2.  load  o f about  details to  Since will  explain  and output  1 k.ohm i n e a c h  transistor  k.ohm.  Input  arid a l s o  the three  flip''  t h e c i r c u i t r y o f f l i p - f l o p FF^,,  The f l i p - f l o p s were  specifications, off  are identical,  the flip-flop  f o r an e x t e r n a l  According  to  transistor  a t V ^ g = -12 v o l t s .  0.1 v o l t circuits  n o t be i n c l u d e d h e r e ,  the characteristic  designed  branch.  = 10 m i c r o a m p i s back-biased  t e r m i n a l s a r e marked as  by s e l e c t i n g  56  are conventional,  nor should  flip-flop  i t be  operation.  The  design  necessary  K V  = + 13.5 v o l t s  V  =+12  volts 4  1—Wv-  'b "  " " C5 c  v  <  -w-  5  G-l  5 '5  |  r,  VVV  A A A r — j  (-  XwvL R  3 3  B  VVr  H  L.  J Pig.  6.6  Gates G ^  Transistors: Diodes: 1.0  G^ a n d f l i p - f l o p  P H I L C O T y p e No HUGHES T y p e No  k.ohm  2N224 1N191  R  L  =  R  3  = 1 8 . 0 k.ohm  Tg  =  5.6 k . o h m  R  4  =  C  5  r  .01 u F  56.0 k.ohm  C„ = 200 uF  R  c  PF1  = 68.0 k.ohm  r  = 10.0  k.ohm  = 15.0  k.ohm  .02  uF  60 Next, ferring  to blocks  connected  c  Vg  G^ a n d K,  to flip-flop  and V v o l t s ,  should  If  and t r i g g e r i n g  i n F i g . 6o2»  dicated zero  t h e gate  only  or  gate.  H e n c e Vg  through  or both  Let the pulse  sider  t h e case  applied,  i ^ drops  Since  Vg  v  pc  < 12  i s sufficient  remains  constant  — X0  volts,  = 0.  = Vg, =r 0,  i . ^= 0  conditions.  these  amplitude  a t the base,  tive).  Base  selected  5.6  input  w h e n TR1  Cg  a n d Cg, t h e t r i g g e r i n g r ^ remains  D  constant  ii  transmits  c  i s  by  a  606  >0.4  mA, hence  b e 0.3  larger  as r e q u i r e d . D^ h a s n o a p p r e c i a b l s drive v^  a negative o n TR1,  to turn  on.  P  1  applied at C will  voltage during  i ^ given  forward-biased,  g  blocks  a maximum  i s turned  When v  shows t h a t  resistance i s taken  0.4  volts  pulse  v ^, s h o u l d  of  and c o n -  000  and diode  In order  k.ohm t o g i v e  0<  not pass the  = Vg„ = V , i ^ b e c o m e s  i f Vg  under  6.6  t h e diode  t h e gate  Vg, =  nxAt  =d  Now  Therefore,  volts.  to a value  t h e gate  negative.  should  initially.  *  I f Vg  A negative  12  =0  g  Therefore,  bias.  when V g =  With resistance  mA.  forward  pulse at  V =  Equation  t o keep *  A negative  v  0.8  than  ing  v  i n -  a t c be d e n o t e d  instantaneously  x^  which  mA,  0  as  a l t e r n a t e between  o f TR1  V, t h e p u l s e  o r Vg, =  w h e n Vg  2 a n d 3° a r e  respectively,  a n d Vg, w i l l  amplitude  be e x p l a i n e d , re-  Terminals  a n d 3'  t o t h e base  equal  a s g i v e n , i , = 1.2 ' a .  values  2  will  but not simultaneously.  pass  Vg,  6060  Pig.  outputs  system  -  0.4  pulse  the pulse  volts  a s 2 k.ohm.  emitter-to-base  (negarg i s voltage  Due t o t h e c a p a c i t o r s  a t the base  the pulse  decays.  period,  Assum-  Cg a n d Cg  61 are  chosen  voltage  to  sistor as  large less  enough t o than  i s turned  on.  reaches  signal  i s necessary.  Cg  Cg  and  The large  has  been  value  of  for turn-on  microsec  on  level Cg  0  +  D^  be  b  vance  based,  turned  on  When t h e  which  i n the  Equation  the  base  triggering  reaches  triggering o f f , v^  tran-  selection  of r ^ ,  Appendije-.  6.5  appears  to  voltage v^  pulse the  to  be  time  i s only  from  rising  disappears.  presence  approximately  a value  rise  too 5-6  o f Rg.  equal  to  to V =  to  cut-off  level  condition  charge  Tg  = ,  f  13.5  1  cut-  max volts  and  D^  in  TRI is  f o r the  adis  to  still flip-flop  properly.  is  shown t o g e t h e r F i g . 6,8  waveforms  with  one one  shows  relation.  of of  two The  phase.  i s shown  Output  in Fig,  are  the the  of  shown i n F i g u r e s  output pulse  the  output  i s s h o w n i n F i g , 6.9,  i n that  o  to  The  A l s o , when t r a n s i s t o r  i s a necessary  I n F i g . 6.7,  time  close  action.  can  This  6.10.  neutral,  far  triggering  * c  to  gering.  tran-  i s s a f e as  external  Actual turn-on  r e s t o r e d due  Vg  function  verter  i s given  no  triggering  once t h e  assumption  state,  analysis  action.  Some p h o t o g r a p h e d  nal  above  the  decreases  5  of  proper  r ^  of  i s concerned.  t ^ g i v e n by  prevents  back-biased. to  The  Rj.) C(- i s c h o s e n  Therefore,  be  action  conducting  when t h e  can  6  the  decay  (measured).  Diode off  the  the  volts,  But  external triggering  sistor  (r  0,1  limit  trains  three  together  used  signals  for  square-waves  voltage,  line-to-line  6,10,  square-wave  6.7  with  line the  voltage  to  trigin  converter  control at the  sigcon-  Pig.  6.8  Output  voltages  Scaling:  10 .5  at  flip-flop  volts/div. msec./div„  terminals  1 and  2  P i g . 6«9  Output l i n e - t o - n e u t r a l v o l t a g e a t c o n v e r t e r , and square-wave s i g n a l S c a l i n g ? 10 volts/div„ 5 msec./diVo  P i g . 6010  Output l i n e - t o - l i n e v o l t a g e a t c o n v e r t e r S c a l i n g ? 20 v o l t s / d i v . 5  msec/diVo  62 7,  The it  NO  low output  i t s application  motors. a  LOAD S P P E D T E S T S  However,  principle  and,the motor  It  amount  small  speed  these  The  test  Although ted,  creased the  air-gap. creases not  this  across  At high  t h e power  threq-phase  i s to verify conversion,  to the input  between  the torque  frequencies. drop  Switching  falls  by the t e s t  speed  r.p.m. A.  attemp-  low f r e -  At low f r e q u e n c i e s , i n resistance available  electromagnetic  torque  6.13, a n d f i n a l l y  without  The l o a d f  reduces at the  dei t i s  windage.  i n operating  place  results  limits.  has been  o f fat very  a l s o t h e power  of frequency  No  on which  i n F i g . 7,1, c u r v e  the increased  transistors.  available.  3 5 0 a n d 5600  of torque  Equation  takes  was  the theory  i n t h e armature  and hence  with  volts  within reasonable  was e n c o u n t e r e d  wave-form,  Encouraged  stage  lim-  o f 1800 r.p,m. a t 60 c p s .  measurement  t o overcome  load.  a t 20  confirmed  frequencies,  i n accordance  sinusoidal  motor  voltage  No d i f f i c u l t y on  that  voltage  sufficient  small  on f r e q u e n c y  a r e shown g r a p h i c a l l y  and a t h i g h  internal  the converter  i s equivalent  was a c h i e v e d  no d i r e c t  per unit  at this  rated  speed  are based,  i t i s obvious  quencies  motor  on t h i s  variation  results  from  load.  experiments  Continuous  necessary  synchronous  tests  purpose  r e g u l a t i o n based  three-phase  has a r a t e d  load  t h e main  o f power  a t no  and current  AC^MOTORS  to rather unconventionally  o f speed  power A  voltage  ON T H R E E - P H A S E  the inverter  voltage  current  spikes  approches  - f . obtained  f o r t h e 20  volt  f  6000-1  5000  4000  3000  2000  1000  30 Fig.  7o  150 Experimental  speed-frequency  A)  20 v o l t  motor  B)  50 v o l t  motor  180  210  characteristics  240  63 motor,  one  decided to t r y a  tunately,  the  able,  i t s rated  and  but  rated  ratings  power  distributed neutral.  built  tests  on  control, only  be  but  7.1,  B)  the  the  The  Hence  drive  and  be  converter  with  isolated  has  been  by  a  the  basic  phase v o l t a g e , 500  to  frequency  i s 1800  speed-frequency h i g h , as  limit  must  At  signals  and  speed could  (Rated  syn-  for  this  expected  when  frequencies the switching i n are  power t r a n s i s t o r s reduced  speed  curve  f o r proper  control  of  r.p.m.) | | F i g .  be  low  not  into  suffi-  saturation.  distorted,  and  power  increases.  volt  machine w i t h  salient  proved  less  the  o t h e r means, Synchronous  2000 r.p.m. by  the  3000 r.p.m.  50  torque.  for re-  The  principle  made t o  by  designed  squirrel-cage  also  Although  volts  i n I-connection  was  p r e v i o u s two.  chronizing  the  50  poles,  voltage.  the  about  avail-  E n g i n e e r i n g Department.  from  i s , the  experiment  up  motor were not  motor  replaced  input  reduced  Unfor-  four  which  low  range  voltage will  This  when s t a r t e d  to  exceeds  i n the  motor.  has  verified  cps  That  using the  rotor.  rotor,  is relatively  with  attempt  The  winding  experimental  strong to  An  1200  the slip  output  motor,  60  current  dissipation  the  at  converter.  ciently  due  i n the  runs  load  50 w a t t s .  also  mainly  shows  motor  peak  motor  speed  motor.  about  o p e r a t i o n , was  varied  chronous  particular  i n the E l e c t r i c a l  this  induction  v o l t a g e i s e s t i m a t e d t o be  original  luctance-motor rotor  of t h i s  three-phase The  larger  changing  run  a test  t o be motor  on  the  synchronous  p o l e DC  successful  d i d run  i t exhibited speed  a  could  in  very be  frequency  excited than  synchronism low  varied slowly.  synbetween  64 There not  behave  are  two  well*  major The  reasons  first  why  one  i s that  voltage  (fundamental)  was  too  voltage  i s limited  the  converter.  and thus a  c o r e s were  by  made o f  simplifying  the  solid  low.  iron  mechanical  As  work  low  negative  Consequently  the  torque  pulsations  c u r r e n t s can  become  sufficiently  chronizing chronism.  torque  and  sequence  thereby  pull  synchronous  motor  the  phase  applied  e x p l a i n e d above, Secondly,  instead  relatively  put  the  of  due  the  to  motor  iron,  This leads i n the  to higher  large  this  shoes  laminated  involved.  inductance  pole  to  machine.  harmonic i n -  exceed out  did  of  the syn-  syn-  65 8.  The phase  particular  motors  which  CONCLUSIONS  principle  o f speed r e g u l a t i o n o f t h r e e -  i s introduced  here,  has been  c o n f i r m e d toy  experiments-. Practical is  limited  for  application of the switching  by break-down v o l t a g e  t h e power t r a n s i s t o r s .  fundamental  line-to-line  break-down v o l t a g e . quires The  power  idea  converter appear  is  Output  system - f  a higher  converter  voltages  output  volts.  voltage,  systems  , and c a r r i e s  than  this  motor  operation  does n o t  the  presence  salient  most  of other  and give  rise  from  harmonic  motor  sequences  positive  the point  the load  currents  load.  appear  sequence frequency  A l l sequence  t o p u l s a t i n g torques  pole- synchronous  by a s e r i e s of  systems  of view o f  The z e r o - s e q u e n c e  by i s o l a t i n g  higher  voltages  or fundamental  o f t h e power.  entirely  dissipa-  i n output  a l l three  t h e lowest,  and performance.  be s u p p r e s s e d  efficiency  where  one a r e u n d e s i r a b l e  can  content  The dominating  i s associated with g  has low i n t e r n a l  c a n be r e p r e s e n t e d  frequencies.  other  a  s a f e l y a t 300  motor r e -  t r a n s i s t o r s i n s e r i e s i n each  but t h e harmonic  component  different  £  type  requirements,  symmetrical  f  output  i s 55$ o f t h e t r a n s i s t o r  operate  several  current  of  a p p l i c a t i o n t o a 110 v o l t  to obtain  switching  high.  at  collector  t o be p r a c t i c a b l e .  The tion  voltage  transistors that  branch,  a n d maximum  converter  T h e maximum p e a k v a l u e  Thus,  of connecting  type  currents  neutral.  reduces  But  motor  i n t h e case of  A synchronous  motor  66 supplied  from  this  sequence  inductance.  Compensation sirable.  The  chronous mental  tation  rate  case The  applied  method which  and  of  must  speed  be  control  i n the  motor  will  by  d e s i g n of motors  be  range  teristics sesses  of motor  speed v a r i a t i o n c a n be  some  power  synchronous  investigated  at  and  generally  modified  control  by  and  equipment,  voltage desirable  the  I  development power  (  of  ratings.  and  speed  possible  speed-torque this  features.  that  efficiency  Since the  control,  solid-state  thesis  It is felt  on  i  exciAlso,  p r o v i d e good  strongly  and  con-  further.  i t s realization for practical application  voltages  in  condenser  however,  higher  syn-  funda-  arise  i n this  efficiency.  i s large,  i n t e r e s t i n g and  the  a  de-  frequency i s r e s t r i c t e d i n  maintained at a reasonable l e v e l . of  is  i s t o use  problems  principle outlined  at the  can  load  controlled.  regulation careful  or  of applied  experiments cost  at the  reactive  technical  s h o u l d be  change  a high negative  of compensation  Several  which  volt-amps  supplies  synchronous  control,  time  this  best  condenser  with  should exhibit  for reactive  frequency.  nection  the  converter  charac-  system I n any  poscase,  depends  components  towards  67 APPENDIX.  The 6.6,  Pig.  a  state  of  the  i s characterized  V, by  quiecent  SELECTION  v  = V',,  5  negative of  v  =  by  step voltage  the  base  TBI  r ^  is actually  can  be  can  be  at  based  shunted  neglected since  by  C  gate  and  triggering  3  C, on  are  AND  2  following  changes  Bg,  they  r ,  the  For  0,  OP  the B^  this  of  circuit  and  Bg,  circuits,  voltages.  from  analysis  Cg  state,  the  response  shown  but  a l l ££>r.^ -  at  in Fig. A - l .  these  2  caused  k ohm 0  resistors The  0  To branch gates  c o n t a i n i n g 2Cg Gg  Cg voltage  and  mately  Cg  v ^  from  must  be  sufficiently  decaying  period.  0.4  •g—• r e p r e s e n t s t h e  two  (closed)  Gg.  and  switching  and  The  to  a  initial  large  value value  to  below  prevent  0.3  v ^  volts  i s chosen  p  the  trigger  over  the  approxi-  voltso  "(V)  t=o  +V  A  2  3 -WW*r  pb  (V) 2C, .R,  •r  l-j+lg+lg  "iP  2  Fig.  A-l  Since  the  actual  time  Simplified  time  circuit  interval  constants  i n the  ib  "2  for analysis  considered i s short circuit,  i t becomes  of v  ^  compared  to  unnecessary  68 to  carry  stead, into by  t h e change  account  The  equations  network.  i s opened  2 2~"  *"1^2 l 2 B  +  i  the capacitors  A t t = 0,  A small  importance  voltage  C g a n d 2Cg  are charged  1  2  In-  be  taken  i s there-  here.  at point  Con-  C  to V  equals volts.  the following  t  2(3 ^ .2  ^2 2  ^3 2  R  R  t  2^ 2 + 2^ + 2 R  r  C~^  +  + i (R 2  2  + r  2  ^  =  oo.  Aol  t  ^3^2  +  +  R  2  (T^ ~  +  V  2 i R  will  error  a t t = 0 a n d one c a n w r i t e  ^ i  o f t h e system.  f o r 0 =• t = t ^ :  r  R  across  of t h e form ^ i .  and the c a p a c i t o r s  switch  analysis  b u t i t i s o f no p r a c t i c a l  t h e above  volts  transient  i n voltage  by terms  committed,  sider V  out a complete  000A  -  °  2  2  + £-) Ct  +  i  3 ib r  =  i (R 3  v  2  + r  2  + r  3  + r .  b  + § - •+  =  pb  . . . A.3  0  Substituting  numerical  values  f o r V, R , 0  d in  V  Equations  A . l , A.2,  A.3,  and s o l v i n g  180 + i ^ 10~ 2  r  n  d  , r  0  0  A  o  4  a n d jr.,  g  ID  f o rv ^ y i e l d s  3  °2  v  i  P  "'  =  1  i_i  III  +  II  i  o o  o A. o  5  O  232.8 + A ( r + ~ - 10"°) + B 3 5 0  where A  1  D  =  U  2 39.9 + 4.24 ~~ 6~ + — °2 C  10"  3  5  ...  A„6  2  2  B =.17.7-*- 10~ + 0,5.4 2 C,  10~  3  C  r„  i n k.ohm,  t i n sec.  6  2  and C i n Farad.  ooo A.7  69  and  = 0.4 v o l t s o  t a= 0,  At  A„7, A =  Equation  3 9 . 9 a n d B = 0.  A. 5 y i e l d s  select  In  order  A.5,  to find  the other  which for  .= 5.6  5„45  appears  one o f t h e c a p a c i t o r v a l u e s ,  one must  be  chosen.  t o be a r e a s o n a b l e  A = B  gering  larger  value  voltage.  into  from  Choose  value.  A.6 a n d A.7  i sprescribed.  With  Equation  = 0.02 u P these  values  yields  44.24  = 18.2.  A . 5 c a n now Cg  A  values  k.ohm.  s e c , v ^ = 0.3 v o l t s  t , v ^ and Cg, E q u a t i o n s  Equation  these  A„6  k.ohm.  t ss t ^ , = B 2 0 x 10""  At  Substituting  to Equations  f o r r^s  7pf232.8 r„ = --~ _ 3 39.9 Hence  According  =  be s o l v e d  6.25  o f Cg w i l l  f o r Cg, and t h e r e s u l t i s  l O ^ F reduce  To be on t h e s a f e  the decay  side,  select  of the t r i g C g .=. 0.01 u F „  70 BIBLIOGRAPHY  1.  Card,  W.H.,  2.  Jewett,  3.  Shea,  4..  Fitzgerald,  " T r a n s i s t o r - O s c i l l a t o r Induction-Motor Drive", A I E E T r a n s a c t i o n s , v o l „ 77 p a r t I I , S e p t e m b e r , 1958, pp 531-534.  W.E., a n d S c h m i d t , P . L . , "A M o r e S t a b l e 3 - P h a s e T r a n s i s t o r - C o r e Power I n v e r t e r " , A I E E T r a n s a c t i o n s " , v o l . 78 p a r t I I , N o v e m b e r . 1 9 5 9 , pp 6 8 6 - 6 9 1 .  R.F., T r a n s i s t o r John Wiley  5..,, A l g e r ,  C i r c u i t E n g i n e e r i n g , New and Sons, I n c . , 1957.  A.E., and K i n g s l e y , C , E l e c t r i c New Y o r k , M c G r a w - H i l l , 1 9 5 2 „  York,  Machinery,  P.L.,Nature o f t h e Polyphase I n d u c t i o n Machine, New Y o r k , J o h n W i l e y a n d S o n s , - I n c . , 1 9 5 6 .  6.  Manley,  J.M., a n d Rowe, H.E.., "Some G e n e r a l P r o p e r t i e s of N o n l i n e a r E l e m e n t s - P a r t I . " G e n e r a l Power R e l a t i o n s " , P r o c e e d i n g s o f t h e I.R.E., v o l . 4 4 , No 7, J u l y 1 9 5 6 , p p 9 0 4 - 9 1 3 .  7.  Chirgwin,  K.M., " A i r c r a f t S w i t c h i n g - T y p e S t a t i c Inverter S u p p l y i n g R o t a t i n g Machines'" , A I E E T r a n s a c t i o n s , v o l . 77 p a r t I I I , N o v e m b e r 1 9 5 8 , pp 334-338. 1  8„  Millman,  J . , and Taub, H „ , P u l s e and D i g i t a l New Y o r k , M c G r a w - H i l l , 1 9 5 6 .  Circuits,  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

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.831.1-0105063/manifest

Comment

Related Items