4.  POWER  AND  EFFECTIVE  VALUES 

20.  The    power   of  the  continuous  e.m.f.  E  producing  con- 
tinuous current  /  is  P  =  El. 

The  e.m.f.  consumed  by  resistance  r  is  EI  =  7r,  thus  the 
power  consumed  by  resistance  r  is  P  =  72r. 

Either  EI  =  E,  then,  the  total  power  in  the  circuit  is  con- 
sumed by  the  resistance,  or  EI  <  E}  then  only  a  part  of  the 
power  is  consumed  by  the  resistance,  the  remainder  by  some 
counter  e.m.f.,  E  —  EI. 

If  an  alternating  current  i  =  I0  sin  6  passes  through  a  resist- 
ance r,  the  power  consumed  by  the  resistance  is, 

i*r  =  702r  sin2  0  =  ^r  C1  ~  cos  2  0), 
& 

thus  varies  with  twice  the  frequency  of  the  current,  between 
zero  and  70V. 

The  average  power  consumed  by  resistance  r  is, 

avg. 
since  avg.  (cos)  =  0. 


16       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Thus  the  alternating  current  i  =  IQ  since  0  consumes  in  a  resist- 
ance r  the  same  effect  as  a  continuous  current  of  intensity 


The  value  /  =  —7=  is  called  the  effective  value  of  the  alter- 

V2 

nating  current  i  =  IQ  sin  0;  since  it  gives  the  same  effect. 

ET 

Analogously  E  =  —i  is  the  effective  value  of  the  alternating 

V2 

e.m.f.,  e  =  EQ  sin  6. 

Since  E0  =  2  irfn$,  it  follows  that 


J' 

; 


=  4.44  fn&    ; 

is  the  effective  alternating  e.m.f.  generated  in  a  coil  of  turns  n 
rotating  at  a  frequency  of  /  (in  hundreds  of  cycles  per  second) 
through  a  magnetic  field  of  <E>  megalines  of  force. 

This  is  the  formula  of  the  alternating-current  generator. 

21.  The  formula  of  the  direct-current  generator, 

E  = 


holds  even  if  the  e.m.fs.  generated  in  the  individual  turns  are 
not  sine  waves,  since  it  is  the  average  generated  e.m.f. 
The  formula  of  the  alternating-current  generator, 

E  =  V2  *fn$, 

does  not  hold  if  the  waves  are  not  sine  waves,  since  the  ratios  of 
average  to  maximum  and  of  maximum  to  effective  e.m.f.  are 
changed. 

If  the  variation  of  magnetic  flux  is  not  sinusoidal,  the  effective 
generated  alternating  e.m.f.  is, 

E  =  7  \/2 


7  is  called  the  form  factor  of  the  wave,  and  depends  upon 
its  shape,  that  is,  the  distribution  of  the  magnetic  flux  in  the 
magnetic  field. 

Frequently  form  factor  is  defined  as  the  ratio  of  the  effect- 
ive to  the  average  value.  This  definition  is  undesirable  since  it 
gives  for  the  sine  wave,  which  is  always  considered  the  standard 
wave,  a  value  differing  from  one. 


POWER  AND  EFFECTIVE  VALUES  17 

EXAMPLES 

22.  (1)  In  a  star-connected  20-pole  three-phase  machine,  re- 
volving at  33.3  cycles  or  200  rev.  per  min.,  the  magnetic  flux 
per  pole  is  6.4  megalines.  The  armature  contains  one  slot  per 
pole  and  phase,  and  each  slot  contains  36  conductors.  All  these 
conductors  are  connected  in  series.  What  is  the  effective  e.m.f. 
per  circuit,  and  what  the  effective  e.m.f.  between  the  terminals 
of  the  machine? 

Twenty  slots  of  36  conductors  give  720  conductors,  or  360 
turns  in  series.  Thus  the  effective  e.m.f.  is,. 


=  4.44  X  0.333  X  360  X  6.4 
=  3400  volts  per  circuit. 


The  e.m.f.  between  the  terminals  of  a  star-connected  three- 
phase  machine  is  the  resultant  of  the  e.m.fs.  of  the  two  phases, 
which  differ  by  60  degrees,  and  is  thus  2  sin  60°  =  -\/3  times 
that  of  one  phase,  thus, 


E  = 

=  5900  volts  effective. 

23.  (2)  The  conductor  of  the  machine  has  a  section  of  0.22 
sq.  cm.  and  a  mean  length  of  240  cm.  per  turn.  At  a  resistivity 
(resistance  per  unit  section  and  unit  length)  of  copper  of  p  = 
1.8  X  10~6,  what  is  the  e.m.f.  consumed  in  the  machine  by  the 
resistance,  and  what  the  power  consumed  at  450  kw.  output? 

450  kw.  output  is  150,000  watts  per  phase  or  circuit,  thus 

150  000 
the  current  /  =     omn     =  44.2  amperes  effective. 


The  resistance  of  360  turns  of  240  cm.  length,  0.22  sq.  cm. 
section  and  1.8  X  10~6  resistivity,  is 

360  X  240  X  1.8  X  10~6 
r  =  -  —  -rT^  —  "  =  0.71  ohms  per  circuit. 

44.2  amp.  X  0.71  ohms  gives  31.5  volts  per  circuit  and  (44.2)2 
X  0.71  =  1400  watts  per  circuit,  or  a  total  of  3  X  1400  =  4200 
watts  loss. 

24.  (3)  What  is  the  self-inductance  per  wire  of  a  three- 
phase  line  of  14  miles  length  consisting  of  three  wires  No.  0 
(Id  =  0.82  cm.),  45  cm.  apart,  transmitting  the  output  of  this 
450  kw.  5900-  volt  three-phase  machine? 


18       ELEMENTS  OF  ELECTRICAL  ENGINEERING 


450  kw.  at  5900  volts  gives  44.2  amp.  per  line.  44.2  amp. 
effective  gives  44.2\/2  =  62.5  amp.  maximum. 

14  miles  =  22,400  m.  The  magnetic  flux  produced  by  / 
amperes  in  1000  m.  of  a  transmission  line  of  2  wires  45  cm. 
apart  and  0.82  cm.  diameter  was  found  in  paragraph  1,  example 
3,  as  2  $  =  0.188  X  106/,  or  $  =  0.094  X  106/  for  each  wire. 

Thus  at  22,300  m.  and  62.5  amp.  maximum,  the  flux  per 
wire  is 

$  =  22.3  X  62.5  X  0.094  X  106  =  131  megalines. 
Hence  the  generated  e.m.f.,  effective  value,  at  33.3  cycles  is, 

E  =  A/2  */  $ 

=  4.44  X  0.333  X  131 
=  193  volts  per  line; 
the  maximum  value  is, 

#o  =  E  X  \/2  =  273  volts  per  line; 
and  the  instantaneous  value, 

e  =  E0  sin   (0  -  0i)  =  273   sin   (0  -  0i) ; 

or,  since  0  =  2  irft  =  210  t  we  have, 

e  =  273  sin  210  (t  -  h). 

25.  (4)  What  is  the  form 
factor  (a)  of  the  e.m.f.  gene- 
rated in  a  single  conductor  of 
a  direct-current  machine  hav- 
ing 80  per  cent,  pole  arc  and 
negligible  spread  of  the  mag- 
netic flux  at  the  pole  corners, 
and  (6)  what  is  the  form  fac- 
tor of  the  voltage  between  two 
collector  rings  connected  to 
diametrical  points  of  the  arm- 
ature of  such  a  machine? 

(a)  In  a  conductor  during 
the  motion  from  position  A, 

T?     *,     T^-  shown  in  Fig.  7,  to  position 

FIG.  7. — Diagram  of  bipolar  generator. 

15,    no    e.m.f.    is    generated; 

from  position  B  to  C  a  constant  e.m.f.  e  is  generated,  from 
C  to  E  again  no  e.m.f.,  from  E  to  F  a  constant  e.m.f.  —  e, 


POWER  AND  EFFECTIVE  VALUES 


19 


and  from  F  to  A  again  zero  e.m.f.     The  e.m.f.  wave  thus  is  as 
shown  in  Fig.  8. 
The  average  e.m.f.  is 

ei  =  0.8  e; 

hence,  with  this  average  e.m.f.,  if  it  were  a  sine  wave,  the  maxi- 
mum e.m.f.  would  be 

ez  =  |  ei  =  0.4  ire, 
and  the  effective  e.m.f.  would  be 


C    D 


FIG.  8. — E.m.f.  of  a  single  conductor,  direct-current  machine 
80  per  cent,  pole  arc. 

The  actual  square  of  the  e.m.f.  is  e2  for  80  per  cent,  and  zero 
for  20  per  cent,  of  the  period,  and  the  average  or  mean  square 
thus  is 

0.8  e2, 

and  therefore  the  actual  effective  value, 


The  form  factor  7,  or  the  ratio  of  the  actual  effective  value 
e4  to  the  effective  value  e3  of  a  sine  wave  of  the  same  mean 
value  and  thus  the  same  magnetic  flux,  then  is 

e4      VT6 
T  =  e3  =  ^T 

=  1.006; 
that  is,  practically  unity. 

(6)  While  the  collector  leads  a,  b  move  from  the  position  F, 
C,  as  shown  in  Fig.  6,  to  B,  E,  constant  voltage  E  exists  between 
them,  the  conductors  which  leave  the  field  at  C  being  replaced 


20       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

by  the  conductors  entering  the  field  at  B.  During  the  motion 
of  the  leads  a,  b  from  B,  E  to  C,  F,  the  voltage  steadily  decreases, 
reverses,  and  rises  again,  to  —  E,  as  the  conductors  entering  the 
field  at  E  have  an  e.m.f.  opposite  to  that  of  the  conductors 
leaving  at  C.  Thus  the  voltage  wave  is,  as  shown  by  Fig.  9, 
triangular,  with  the  top  cut  off  for  20  per  cent,  of  the  half  wave. 


FIG.  9.  —  E.m.f.  between  two  collector  rings  connected  to  diametrical  points 
of  the  armature  of  a  bipolar  machine  having  80  per  cent,  pole  arc. 


Then  the  average  e.m.f.  is 

e1  =  0.2  E  +  2  X 


=  0.6  E. 


The  maximum  value  of  a  sine  wave  of  this  average  value  is 
e2  =  2  ei  —  0-3  irE, 

and  the  effective  value  corresponding  thereto  is 

e-2        0.3  irE 


63   = 


7= 

V2 


The  actual  voltage  square  is  E2  for  20  per  cent,  of  the  time,  and 
rising  on  a  parabolic  curve  from  0  to  E2  during  40  per  cent,  of 
the  time,  as  shown  in  dotted  lines  in  Fig.  9. 

The  area  of  a  parabolic  curve  is  width  times  one-third  of 
height,  or 

OAE2 


hence,  the  mean  square  of  voltage  is 

0 


and  the  actual  effective  voltage  is 


_,4       1/280 

~  e,  ~  TT  V  27        L°25' 


SELF-INDUCTANCE  AND  MUTUAL  INDUCTANCE    21 

hence,  the  form  factor  is 

7    r 

or,  2.5  per  cent,  higher  than  with  a  sine  wave.