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UC-NRLF 


THE  Al.TERNATING-CURRi: 


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W.  PERKEN   HAYCOCK,  M.I.E.E. 


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REESE  LIBRARY 

OF  THK 

UNIVERSITY  OF  CALIFORNIA. 

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Deceived  st^/ct^L-  ,i8Q%.< 

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No.    7/3~AJ.   .   C/.M.s-M>. 


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THE 

ALTEENATING-CUEEENT 
CIRCUIT. 


WORKS    BY    THE    SAME    AUTHOR. 


ELECTRIC  LIGHTING  AND  POWER  DISTRIBU- 

TION.      An   Elementary  Manual    on  Electrical  Engineering.      THIRD 
EDITION.    Rewritten  and  considerably  Enlarged  and  brought  up  to  date. 
Tn  two  volumes.    Vol.  I.,  cloth  boards,  crown  8vo.,  430  pp.,  231  Illustra- 
tions, and  ruled  pages  for  Notes.  6s.    Vol.  II.  (in  preparation). 
SOME  OPINIONS  OF  THE  PRESS  (THIRD  EDITION). 
ELECTRICAL  REVIEW.—'  A  vast  improvement  on  the  last  edition.  .  .  .  The 
work  will  no  doubt  become  a  standard  text-book  for  schools  and  classes  on  this 
subject ;  as  such  it  has  few  rivals.' 

ELECTRICITY. — 'One  of  the  best  and  most  up-to-date  educational  electrical 
engineering  manuals  now  before  the  public.' 

A  FIRST  BOOK  OF  ELECTRICITY  AND  MAGNE- 

TISM.  For  Elementary  Science  and  Engineering  Students.  SECOND 
EDITION.  Entirely  Rewritten  and  considerably  Enlarged.  Cloth  boards, 
crown  8vo.,  233  pp.,  107  Illustrations,  Index,  Priced  List  of  Apparatus, 
and  ruled  pages  for  Notes.  2*.  M . 

SOME  OPINIONS  OF  THE  PRESS  (SECOND  EDITION). 
ELECTRICIAN. — 'The  whole  book  bears  evidence  that  its  writer  has  had 
considerable  experience  in  the  teaching  of  elementary  students.' 

ELECTRICAL  REVIEW. — '  This  book  is  deserving  of  warm  commendation.' 
ELECTRICITY.—'  Teachers  of  science  will  welcome  the  book  as  one  of  the 
best  that  can  be  recommended  to  their  pupils.' 

PRACTICAL  ELECTRICAL  NOTES  AND  DEFINI- 

TIONS.    SECOND  EDITION.    Pocket  size,  4f  in.  by  3  in.,  286  pp.,  79  Illus- 
trations, French  morocco,  gilt  edges,  3s.  6rf. ;  red  cloth,  2s. 
ELECTRICAL  REVIEW.— -'  The  whole  work  contains  much  useful  matter  in  the 
shape  ©f  notes,  tables,  diagrams,  rules,  &c.' 

THE  ALTERNATING-CURRENT  CIRCUIT.    An  In- 

troductory  and  Non-Mathematical  Book  for  Engineers  and  Students. 
Crown  8vo.,  102  pp.,  51  Illustrations,  Index,  and  ruled  pages  for  Notes.  2s. 


IN    PREPARATION. 


ELECTRIC   LIGHTING  AND  POWER  DISTRIBU- 

TION.  Vol.  II.  THIRD  EDITION.  This  will  contain  Chapters  on  :-  The 
Theory  and  Working  of  Dynamos,  Alternating  Currents,  Alternators, 
Motors,  Meters,  Lamps,  Accumulators,  Transformers,  Central  Stations, 
Switches  and  other  Accessories,  Methods  of  Wiring,  Calculations,  etc.,  as 
well  as  a  complete  Index  to  both  Vols. 

PRELIMINARY  ELECTRIC  LIGHTING.    An  Extract 

(with  Additions)  of  those  parts  of  the  Author's  larger  work  which  relate 
to  the  Preliminary  Grade  Syllabus  of  the  City  and  Guilds  of  Londoii 
Institute.  

WHITTAKER    &    CO. 
1897. 


THE 

ALTERNATING  -  CUBBENT 
CIKCUIT. 

AN  INTRODUCTORY  AND  NON- MATHEMATICAL  BOOK 
FOR  ENGINEERS  AND  STUDENTS. 

BY 

W.  PEKEEN  MAYCOCK,  M.I.E.E. 


WITH    51    ILLUSTRATIONS,    INDEX,    AND 
RULED    PAGES    FOE    NOTES. 


WHITTAKEE    &    CO. 

WHITE  HART  STREET,  PATERNOSTER  SQUARE,    LONDON. 
AND    66   FIFTH   AVENUE,   NEW   YORK. 

1897. 


PR1NTED    BY 

SPOTTISWOODE    AND    CO.,    NEW-STREET  SQUARE 
LONDON 


PREFACE. 


THIS  little  book  is  an  attempt  to  convey  some  idea  of 
the  phenomena  of  the  ordinary  or  single-phase  alter- 
nating-current circuit  to  the  minds  of  those  new  to  the 
subject,  by  means  of  plainly  worded  and  non-mathe- 
matical language. 

The  matter  is,  of  course,  a  very  important  one  to 
electrical  students  ;  and  it  also  abounds  in  difficulties, 
many  of  which  arise  from  the  fact  that  authorities  are 
by  no  means  agreed  as  to  the  explanation  of  the  various 
phenomena  here  dealt  with. 

In  a  work  written  from  a  practical  or  engineering 
standpoint,  such  as  this  is,  it  is  almost  absolutely 
necessary  to  look  upon  electricity  as  a  something  which 
flows  along  the  conductor,  as  both  explanation  and 
comprehension  are  rendered  much  easier  thereby.  The 
student  of  advanced  theories  will  consequently  find  little 
to  interest  him  herein,  except  in  a  technical  sense. 

I  have  been  greatly  assisted  by  my  friend  Mr.  C.  H. 


vi  PEEFACE. 

Yeaman  (Chief  Assistant  Engineer  at  the  Islington 
Electricity  Works),  in  the  preparation  of  this  volume, 
and  it  is  with  great  pleasure  that  I  here  acknowledge 
my  indebtedness  to  him. 

The  book  forms,  in  fact,  the  substance  of  a  chapter 
in  the  forthcoming  Vol.  II.  of  my  Electric  Lighting  and 
Power  Distribution ;  it  being  thought  expedient  to 
publish  it  in  advance,  instead  of  delaying  its  appear- 
ance till  the  completion  of  the  larger  work. 

Notwithstanding  the  smallness  of  the  book,  a  very 
great  amount  of  time  and  labour  has  been  spent  upon 
its  evolution  ;  with  the  result — it  is  hoped — that  much 
additional  information  concerning  alternating  currents 
has  been  brought  within  the  ken  of  the  non-mathematical 
reader. 

W.   PERREN   MAYCOCK. 

'  MILBER,'  WADDON,  SURREY. 
April,  1897. 


CONTENTS. 


PAR.  PAGE 

1.  THEORY  OF  ELECTRICITY      .  1 

2.  ALTERNATING  CURRENT  .  .2 

3.  ALTERNATING  CURRENT  (continued)                                     .  .        5 

4.  INDUCTANCE  OR  SELF-INDUCTION             .  9 
5.'  ALTERNATING  CURRENT  (continued)             .            .  10 

6.  CAPACITY  IN  ALTERNATING-CURRENT  CIRCUITS  .            .  .13 

7.  CAPACITY  IN  ALTERNATING-CURRENT  CIRCUITS  (continued)  .       18 

8.  EFFECT  OF  CAPACITY  IN  THE  CIRCUIT    .            .            .  .21 

9.  INDUCTANCE,  CAPACITY,  ETC.,  IN  A  DIRECT-CURRENT  CIRCUIT  .      24 

10.  INDUCTANCE,    CAPACITY,    ETC.,    IN    AN    ALTERNATING-CURRENT 

CIRCUIT  .            .            .            .            .            .  29 

11.  INDUCTANCE  IN  A  CIRCUIT           .            .            .            .  .34 

12.  EFFECTS  OF   AN   ALTERNATING   CURRENT  AND   OF  INDUCTANCE 

AND  CAPACITY  ON  THE  INSULATION  OF  A  CIRCUIT           .  .       35 

13.  ELECTRIFICATION  OF  CONDUCTOR  DIELECTRIC     .             .  .41 

14.  EXPERIMENTS  ON  INDUCTANCE           .             .             .  42 

15.  GRAPHICAL  EEPRESENTATION  OF  AN  ALTERNATING  CURRENT  .       47 

16.  FREQUENCY          .                         .            .            .            .  .55 

17.  FREQUENCY  OF  ALTERNATORS     .     .     .  56 


viii  CONTENTS. 

PAR.  PAGE 

18.  VIRTUAL  VOLTS  AND  AMPERES  .  .  .  .  .57 

19.  AMPLITUDE  AND  PHASE         .  .  .  .  59 

20.  LAG  AND  LEAD    .......      59 

21.  EEACTANCE    .  .  .  .  .  .  63 

22.  EEACTANCE  AND  IMPEDANCE        .  .  .  .  .64 

23.  DIFFERENT  ACTION  OF  RESISTANCE  AND  EEACTANCE  ox  CURRENT  . 

CHOKING  COILS  .            .            .            .  .  65 

24.  PRACTICAL  FORMS  OF  CHOKING  COILS     .            .  .  .68 

25.  USE  OF  CHOKING  COILS       .            .            .  .  75 

26.  '  SKIN  RESISTANCE  '  OR  CONDUCTOR  IMPEDANCE  .  .       76 

27.  CONDUCTORS  FOR  ALTERNATING  CURRENTS  .  .  78 

28.  ELECTRICAL  EESONANCE  .             .            .             .  .80 

29.  EFFECTIVE  VOLTS  AND  AMPERES     .             .  .  83 

30.  CONNECTION    BETWEEN    INDUCTANCE,   EEACTANCE,   IMPEDANCE, 

IMPRESSED  VOLTS,  AND  VIRTUAL  CURRENT              .  .      84 

31.  POWER  IN  ALTERNATING-CURRENT  CIRCUITS            .  89 

32.  POWER  IN  ALTERNATING-CURRENT  CIRCUITS  (continued]  .       93 
33     CONCLUSION  .            .            .            .            .            .  96 

INDEX      ........       99 

EULED  PAGES  FOR  NOTES. 


THE 

ALTERNATING-CURRENT  CIRCUIT. 


1.  THEORY  OF  ELECTRICITY. — To  start  with,  it  is 
necessary  to  adopt  some  theory  of  electricity.  Now 
there  are  many  theories  for  electrical  action,  but  it  is 
impossible  to  pick  out  any  one  and  say  that  is  the 
right  one,  though  those  in  which  electricity  is  looked 
upon  as  a  movement  or  vibration  of  the  ether  are 
seemingly  the  most  plausible.  But  to  explain  the 
phenomena  with  which  we  are  about  to  deal,  in  the 
light  of  any  of  the  advanced  theories,  would  be  extremely 
difficult,  if  not  impossible,  in  a  book  of  this  character : 
and,  moreover,  it  should  be  remembered  that  w<  are 
here  concerned  not  so  much  with  what  is  vaguely  called 
'  electricity,'  as  with  certain  of  its  effects.  Hence  we 
must  choose  some  simple  practical  theory,  at  the  same 
time  remembering  that  it  is  adopted  to  facilitate  ex- 
planations, and  keeping  our  minds  ready  for  the  recep- 
tion or  conception  of  some  better  one  at  a  future  time. 

B 


2  ALTERNATING-CURRENT   CIRCUIT.  [§  2. 

The  theory  advocated  by  the  Author  is  that  known 
as  the  '  surplus  and  deficit  theory,'  and  it  was  first  fully 
treated  and  extended  by  him  in  a  series  of  articles  in 
the  Electrical  Engineer,  which  articles  were  subsequently 
embodied  in  the  Author's  First  Book  of  Electricity  and 
Magnetism.  * 

It  must  here  suffice  to  indicate  the  mere  outlines  of 
this  theory.  All  things,  conductors  and  insulators  alike, 
are  supposed  to  be  imbued  with  electricity  normally  dis- 
tributed—i.e.  at  even  pressure  or  potential.  Electri- 
fication is  the  act  of  heaping  up  electricity  on  one  body  or 
bodies,  leaving  a  corresponding  deficit  on  another  body 
or  bodies ;  the  former  is  or  are  then  said  to  be  positively 
electrified,  and  the  latter  negatively  electrified.  A 
battery,  or  dynamo,  or  alternator  is  simply  an  electric 
pump,whose  electro-motive  (electro-pumping)  force  alters, 
or  tends  to  alter,  the  even  distribution  of  electricity  in 
the  circuit :  these  apparatus  must  consequently  not  be 
looked  upon  as  '  generators '  of  electricity.  In  most  cases 
where  an  uneven  distribution  of  electricity  exists,  there 
will  be  a  tendency  for  it  to  flow  so  as  to  regain  a  general 
level  or  distribution :  when  there  is  such  a  tendency, 
there  is  said  to  be  a  difference  of  pressure  or  potential, 
or  a  potential  difference  (abbreviated  P.D.). 

2.  ALTERNATING  CURRENT. — The  simplest  kind  of 
current  is  that  derived  from  a  battery  ;  this  is  a  steady 
direct  current,  and  its  principal  properties  are  presumed 
to  be  well  known  to  the  reader.  A  well- designed  and 
properly-constructed  direct-current  dynamo  gives  a 

*  Second  Edition. 


§  2.]  REVERSING  SWITCH.  3 

current  which  is  very  nearly  similar  in  its  effects  to 
that  of  a  battery ;  and  for  practical  purposes,  the  laws 
which  apply  to  the  current  from  a  battery  may  be 
equally  well  applied  to  that  from  a  dynamo. 

If  a  reversing  switch  R  *,  inserted  in  the  circuit  of  a 
battery  or  direct-current  dynamo,  as  shown  in  Fig.  1,  be 


FIG.  1. 

operated  at  regular  intervals,  alternating  E.M.Fs.  will 
be  impressed  on  the  outer  circuit  C,  and  an  alternating 

*  The  construction  and  action  of  this  form  of  reversing  switch  are 
as  follows  :—  On  an  insulating  base,  e,  pivoted  at  p,  and  provided  with  a 
handle,  h,  are  mounted  the  U-shaped  piece  of  metal,  +  +,  and  the 
straight  piece,  —  ,  to  which  the  +  and  —  poles  of  the  battery  are 
respectively  connected.  When  the  switch  handle  is  in  the  position 
shown,  the  metal  tongues,  T  T',  connected  with  the  extremities  of 
the  outer  circuit,  C,  rest  on  +  +  ,  and  no  current  flows  from  the 
battery.  If  the  switch  handle  is  moved  to  the  right,  the  right-hand 
leg  of  the  U  -piece  remains  in  contact  with  T',  and  the  straight  piece 
touches  T,  a  current  consequently  flowing  round  C  in  a  counter- 
clockwise direction.  If  the  switch  handle  is  moved  to  the  left,  the  left- 
hand  leg  of  the  U  -piece  is  in  contact  with  T,  and  the  straight  piece  with 
T',  and  a  current  flows  round  C  in  the  opposite  direction.  Thus,  if  h  is 
constantly  worked  to  and  fro,  an  alternating  current  will  beset  up  in  C. 


UNIVERSITY 


4  ALTERNATING-CURRENT   CIRCUIT.  [§  2. 

current  will  be  set  up  therein,  as  conveniently  repre- 
sented by  the  double-headed  arrows  «— >  <— >. 

Supposing  the  circuit  C  had  no  inductance  or  other 
disturbing  effect,  the  current  or  rate  of  flow  of  electricity 
in  it  would  always  be  the  same,  but  would  be  reversed 
in  direction  at  regular  intervals,  as  shown  by  the  '  curve  ' 
in  Fig.  2. 

The  explanation  of  this  curve  is  as  follows : — Time 
is  represented  along  the  horizontal  line  (say,  in  one- 
second  intervals),  starting  from  the  left.  Current  in 


CURRENTS  IN 
ONE  DIRECTION 
OR 
RIGHT  HAND 
CURRENTS. 
CURRENTS 
IN  OPPOSITE 
DIRECTION  OR 
LEFT  HAND 
CURRENTS. 

a 
10 

CO    _ 

uj    5 

QC 

UJ 

1 
3             ( 

1        f        \ 

;                       X        J 

TIME  IN 

SECOND 

0.    0 

5  _ 

<  5 

10 

c 

F 

2 

3 
1          6 

4 

5 
I 

< 

IG.   2. 

1 

one  direction  is  shown  by  vertical  distances  above  this 
line,  and  current  in  the  other  direction  by  vertical 
distances  below  it.  It  is  usual  to  style  currents  in  one 
direction  +  (positive),  and  those  in  the  opposite  direc- 
tion —  (negative) ;  but  these  terms  are  confusing  to 
the  beginner,  who  would  probably  assume  that  a '  +  cur- 
rent '  was  different  in  its  properties  from  a  '  —  cur- 
rent.' We  shall  therefore  refer  to  them  as  right-  and 
left-hand  currents  respectively,  these  terms  well  con- 
veying the  idea  that  they  flow  in  opposite  directions 
round  the  circuit.  Suppose  at  the  time  of  commencing 


§  3.]  ALTERNATING   CURRENT.  5 

the  *  curve,'  a  'right-hand  current'  was  flowing,  and 
that  its  value  was  10  amperes,  and  suppose  also  that  the 
direction  was  reversed  every  second ;  our  curve  would 
then  start  at  the  point,  &,  and  would  run  in  a  horizontal 
direction  for  1  second — i.e.  from  a  to  b — when  it  would 
suddenly  drop  to  c,  the  current  having  been  reversed  : 
the  '  left-hand  current,'  c  d,  would  continue  for  1  second, 
as  shown,  and  would  then  immediately  change  to  the 
'  right-hand  current,'  ef.  During  the  fourth  second 
the  current  would  be  '  left-handed,'  g  h,  during  the  fifth 
second  '  right-handed,'  i  j,  and  so  on. 

The  above  is  a  purely  imaginary  condition  of  things, 
for  a  current  cannot  really  change  suddenly  from  one 
direction  to  another  at  its  full  value  :  but  it  is  useful, 
as  it  gives  the  student  a  preliminary  idea  of  an  alter- 
nating current. 

3.  ALTERNATING  CURRENT  (CONT.). — It  is  convenient 
to  liken  a  steady  direct  current  to  a  steady  flow  of 
water  in  one  direction  through  a  pipe :  an  alternating 
current  may  then  be  compared  with  the  movement  of 
water  in  the  pipe  when  the  direction  of  flow  is  changed 
more  or  less  rapidly.  Fig.  3  represents  a  pipe  bent 
round  so  as  to  form  a  complete  circuit,  which  includes 
a  pump,  P,  the  whole  being  filled  with  water.  The 
water  represents  electricity,  the  pipe  the  conductor,  and 
P  the  dynamo  or  alternator — according  to  its  method 
of  working.  P  is  represented  as  a  kind  of  small  water- 
wheel,  actuated  by  a  pulley  or  handle  outside.  If  P  is 
rotated  continuously  in  one  direction,  it  represents  the 
action  of  a  battery  or  direct-current  dynamo,  the  water 


ALTERNATING-CURRENT   CIRCUIT. 


Ba- 


in the  pipe  (electricity  in  the  conductor)  being  set 
flowing  in  one  direction.  If  P  rotates  first  in  one 
direction  and  then  in  the  other,  at  regular  intervals,  it 
represents  the  action  of  an  alternator,  for  there  will  be 
a  flow  of  water  in  the  pipe  (electricity  in  the  conductor) 
first  in  one  direction  and  then  in  the  other.  Now 
electricity— -like  water — may,  for  the  purposes  of  this 

argument,  be  assumed  to 
be  incompressible  ;  so  that 
with  a  given  flow  (cur- 
rent), the  number  of  gal- 
lons of  water  or  coulombs 
of  electricity  passing  any 
point,  a,  in  the  pipe  or 
circuit,  is  the  same  as 
the  number  passing  any 
point,  b.  Thus,  let  the 
shaded  part,  C,  represent 
one  gallon  of  water  or  one 
coulomb  of  electricity :  when 
C  moves  in  either  direction, 
all  the  water  or  electricity  in  front  or  behind  it,  i.e. 
all  round  the  circuit,  moves  at  exactly  the  same  rate, 
irrespectively  of  the  size  of  the  pipe  or  conductor,  which 
may  vary  at  different  parts  of  the  circuit.  In  other 
words,  the  flow  of  electricity,  in  coulombs  per  second  (am- 
peres), is  the  same  at  all  parts  of  a  closed  series  circuit.* 
When  the  circuit  is  not  of  this  description,  i.e.  when  it 
has  branches,  the  current  may  vary  in  different  parts. 

*  Provided  it  has  negligible  capacity  (§§  6,  7,  etc.). 


FIG.  3. 


FBEQUENCY.  7 

Eeferring  still  to  Fig.  3,  let  us  consider  the  action  of 
alternating  flow  at  different  frequencies — i.e.  at  different 
rates  of  alternation  (§  16) .  The  faster  P  works  the  greater 
its  water-motive  force,  and  the  more  rapid  will  be  the  flow 
with  a  given  length  and  size  of  pipe  (circuit  conductor) ; 
it  being  presumed  that  there  is  very  little  waste  of 
energy  in  the  useless  carrying  round  of  water  in  the 
spaces  s  s  s  s  :  this  water-motive  force  is  clearly  analo- 
gous to  the  electro -motive  force  of  an  alternator.  The 
frequency  of  the  flow  (of  water  or  electricity)  does  not 
depend  on  the  value  of  water-motive  or  electro-motive 
force,  but  on  the  rate  at  which  the  latter  change  their 
direction.  Thus,  if  P  rotates  in  the  direction  of  the  top 
arrow  for  half  a  minute,  and  then  in  the  direction  of  the 
dotted  arrow  for  half  a  minute,  the  current  will  change 
its  direction  twice  a  minute.  Now,  with  a  given  length 
and  size  of  pipe  (circuit  conductor),  any  particular 
gallon  or  coulomb,  Ct  may  make  10,  20,  30,  or  more 
'  laps '  (journeys  round  the  circuit)  before  the  reversal 
of  flow  takes  place  :  if  the  direction  of  flow  is 
changed  at  lesser  intervals,  i.e.  if  the  frequency  is 
increased,  our  gallon  or  coulomb  may  only  succeed  in 
making  two  or  three  journeys  round  in  one  direction 
before  the  reversal  of  flow  occurs.  It  is  thus  con 
ceivable  that,  with  a  high  frequency,  our  unit  of  water  or 
electricity  may  only  traverse  a  part  of  the  circuit  (say, 
from  a  to  b)  before  it  has  to  turn  back,  and  that  the 
greater  the  frequency  the  less  the  distance  actually 
travelled  over.  Thus  this  path  may  decrease,  as  the 
frequency  increases,  to  a-a',  a-a",  or  a-a'"t  it  being 


8  ALTEBNATING-CUREENT   CIRCUIT.  [§  3. 

remembered  that  there  is  a  similar  movement  in  the 
other  parts  of  the  circuit.  The  motion  of  water  or 
electricity  in  the  circuit  depicted  in  Fig.  3,  may,  when 
the  water-  or  electro-motive  force  has  medium  fre- 
quency, be  compared  with  that  of  the  balance-wheel 
of  a  watch.  The  current  in  a  given  circuit  is  thus 
proportional  to  the  distance  traversed  at  each  alter- 
nation by  any  given  coulomb,  (7,  multiplied  by  the 
number  of  alternations  per  second  :  so  that  if  the  current 
is  kept  constant,  when  the  frequency  is  doubled,  the 
path  traversed  by  any  given  coulomb  will  be  halved,  and 
vice  versa.  It  will  be  remembered  that  current  is  de- 
fined as  the  number  of  coulombs  passing  any  given 
point  in  a  circuit  per  second :  and  in  the  case  of 
alternating  current  we  consider  the  actual  number  of 
coulombs  passing  by,  irrespective  of  their  direction  of 
flow.  Thus,  with  a  very  high  frequency,  it  is  conceivable 
that  the  coulomb  C  (Fig.  3)  will  merely  oscillate  in  front 
of  the  point  a'" ,  the  number  of  times  it  passes  this  point 
in  one  second  being  a  measure  of  the  current. 

The  greater  the  frequency  the  sharper  the  to-and-fro 
movement  of  electricity ;  and  the  comparatively  non- 
dangerous  character  of  extremely  high  frequency  currents, 
such  as  are  sometimes  used  in  experimental  work,  may 
be  roughly  accounted  for  by  supposing  that  the  electricity 
in  the  circuit  moves  over  so  minute  a  path  that  the 
current  is  indefinitely  small,  certain  retarding  effects 
increasing  with  the  frequency  (§§21,  22,  27,  29). 

The  hydraulic  analogue  of  an  alternating-current 
circuit  is  often  illustrated  as  in  Fig.  4  ;  the  pulley,  p, 


§4.] 


ALTERNATING   CURRENT. 


representing  the  rotating  part  of  the  alternator  ;  the  force 
of  the  pump  piston,  P,  the  electro-motive  force ;  and  the 
up-and-down  movement  of  the  piston,  the  reversals  in 
the  direction  of  the  electro-motive  force.  Good  as  this 
analogy  is  in  some  respects,  it  is  rather  a  faulty  one,  inas- 
much as  there  is  no  actual  passage  of  water  through  the 
pump  ;  and  the  student  might  from  this  infer  that  there 
was  no  passage  of  electricity  through  the  alternator :  and 
we  assume  that  the  electricity 
flows  through  the  alternator,  or 
dynamo,  or  battery,  just  as  it  does 
through  the  other  parts  of  the 
circuit.  ^ 

An  alternating  current  might 
be  described  as  a  '  continual  os- 
cillation '  of  electricity  in  the  cir- 
cuit, just  as  the  movement  of  the 
balance-wheel  of  a  watch  is  a 
'  continual  oscillation.'  It  must 
be  borne  in  mind,  however,  that  the  use  of  the  term 
'  electrical  oscillation '  is  applied  to  the  movement  of 
electricity  when  a  condenser  is  discharged,  a  rapid  to- 
and-fro  movement  in  an  incomplete  circuit,  which  dies 
away  to  nothing :  this  movement  is  similar  to  that  of 
the  prong  of  a  tuning-fork,  or  of  one  end  of  a  compass- 
needle  coming  to  rest  in  a  strong  magnetic  field.  The 
term  oscillation  should  therefore  be  confined  to  the  case 
of  condensers,  to  prevent  confusion. 

4.  INDUCTANCE  OE  SELF-INDUCTION. — When  a  direct 
current   begins   to  flow  along  a  circuit,  it   sets   up   a 


FIG.  4. 


10  ALTERNATING-CURRENT   CIRCUIT.          [§  5. 

magnetic  field  around  the  conductor.  This  magnetic 
field,  in  being  set  up,  reacts  upon  or  cuts  the  conductor, 
and  induces  a  momentary  reverse  E.M.F.  therein.  When 
the  current  flowing  along  a  conductor  is  stopped,  the 
magnetic  field  collapses,  and  in  collapsing  cuts  the  con- 
ductor, and  in  consequence  another  momentary  E.M.F. 
is  induced  in  the  conductor,  which  is  *  direct,'  i.e.  in 
the  same  direction  as  the  inducing  current. 

This  action,  which  is  due  to  inductance  or  self- 
induction,  momentarily  opposes  the  setting  up  of  a 
current  in  a  circuit  by  reason  of  the  opposing  '  reverse  ' 
E.M.F.,  and  momentarily  retards  its  *  breaking'  or 
cessation,  because  of  the  momentary  '  direct '  E.M.F. 

The  effect  of  inductance  is  not  very  noticeable  in 
straight  conductors,  as  the  conductor  cannot  be  so  effec- 
tively cut  by  the  lines  as  when  it  is  coiled  up  (§  26). 
Also  because  the  lines  of  force  set  up  in  a  circuit  are  more 
crowded  if  the  circuit  is  coiled  up,  and  are  increased  in 
number  if  the  coils  have  iron  cores  ;  inductance  is  always 
greatest  in  circuits  containing  electro-magnetic  apparatus 
such  as  magnets,  transformers,  and  the  like. 

The  effects  of  inductance  are  noticeable  in  a  circuit 
not  only  when  a  current  is  set  up  or  st9pped,  but  also 
when  it  is  increased,  or  diminished,  or  reversed ;  such 
increase,  or  diminution,  or  reversal,  altering  the  number 
of  lines  of  force  passing  through  or  interlinked  with  the 
circuit,  and  their  direction,  and  therefore  giving  rise  to 
momentary  induced  E.M.Fs. 

5.  ALTERNATING  CURRENT  (CONT.). — From  what  was 
said  in  the  preceding  paragraph,  it  should  be  clear 


5.] 


DISTURBING  EFFECTS. 


that  it  is  impossible  to  suddenly  start  a  current  at  its 
full  value,  and  equally  impossible  to  suddenly  stop  it ; 
because  of  the  effects  of  inductance  or  self-induction, 
etc.,  the  current  taking  time  to  '  grow  '  and  time  to  die 
away.  It  is  thus  even  more  out  of  the  question  to 
suddenly  reverse  a  current  in  a  circuit- 

Although  it  is  possible  to  arrange  a  simple  circuit  or 
to  wind  a  coil  so  that  it  shall  have  little  or  no  inductance, 
as  shown  in  Fig.  5,  where  each  half  of  the  circuit  or  coil 
neutralises  the  other's  magnetic  effect,  the  conductor 
will  still  have  capacity  *  and  this  also  exercises  a  dis- 
turbing effect  on  the  cur- 
rent. Moreover,  a  coil  such 
as  that  shown  in  Fig.  5 
would  be  useless  for  sole- 
noids or  electro-magnets,  as 
it  would  have  no  magnetic 
field.  It  therefore  follows 
that  every  working  circuit  exercises  more  or  less  dis- 
turbing effect,  and  also  that,  in  the  case  of  an  alter- 
nating current,  this  disturbing  effect  is  continuous. 
Consequently,  the  '  curve '  in  Fig.  2  does  not  represent 
a  real  alternating  current,  for  such  not  only  varies  in 
direction,  but  is  also  constantly  varying  in  strength. 
With  a  given  circuit,  the  changes  in  direction  and 
strength  take  place  at  regular  intervals,  and  an  alter- 
nating current  is  thus  often  called  a  periodic,  harmonic, 

*  See  the  Author's  First  Book  of  Electricity  and  Magnetism,  Second 
Edition,  §  159  :  or  his  Electric  Lighting  and  Poiver  Distribution,  Third 
Edition,  vol.  i.  §  37. 


FIG.  5. 


12  ALTERNATING-CURRENT   CIRCUIT.  [§  5. 

or  icave  current.  In  fact,  the  curve  of  a  real  alternating 
current  is  a  series  of  waves,  which  may  be  roughly 
likened  to  those  set  up  in  a  rope  which  is  fixed  at  one 
end,  while  its  other  end  is  rapidly  moved  up  and  down 
(Figs.  31,  34,  and  36). 

The  simplest  case  in  which  an  alternating  current 
is  set  up  is  when  two  Bell  telephones  are  used  as  trans- 
mitter and  receiver  respectively.  The  iron  plate  or 
diaphragm  of  the  telephone  used  as  transmitter,  is 
caused  by  the  voice  to  perform  motions  to  and  fro  in 
front  of  a  magnet,  on  the  end  of  which  a  coil  of  wire  is 
placed.  The  changes  in  the  strength  of  field  caused  by 
the  motions  of  the  iron  disk,  induce  E.M.Fs.  in  the  coil 
of  wire,  and  as  these  motions  are  to  and  fro,  the  field  is 
alternately  strengthened  and  weakened,  the  result  of  the 
movement  of  its  lines  being  an  alternating  E.M.F.  in 
the  coil,  which  is  cut  by  those  lines.  As  the  transmitter 
is  in  this  case  connected  to  an  exactly  similar  telephone 
by  a  couple  of  wires  forming  the  circuit,  a  current  alter- 
nates in  the  circuit  and  coils  of  both  instruments,  and 
the  magnet  of  the  second  telephone  (or  receiver),  being 
correspondingly  strengthened  and  weakened,  its  dia- 
phragm is  caused  to  perform  movements  of  a  similar 
character  to  those  of  the  transmitter  diaphragm,  and  it 
sets  up  sound  waves  in  the  air  in  front  of  it.  The  trans- 
mitter and  receiver  thus  really  act  as  a  miniature 
alternating-current  dynamo  and  motor  respectively. 

The  magneto-machine  and  bell  so  much  used  in 
telephone  and  other  work  afford  another  example.  The 
magneto-machine  (sometimes  called  the  '  ringer '  or 


§6.] 


CONDENSEES  IN   CIECUIT. 


13 


'generator,')  is  a  simple  form  of  alternating-current 
dynamo,  the  alternating  current  being  induced  by  the 
rotation  of  a  coil  of  wire  in  a  two-pole  magnetic  field 
(§  15) :  while  the  magneto  bell  may  be  likened  to  an 
alternating-current  motor,  for  its  hammer  will  only 
move  continuously  when  an  alternating  current  is  passed 
through  its  coils.* 

6.  CAPACITY  IN  ALTERNATING- CURRENT  CIRCUITS. — 
One  very  and  increasingly  important  difference  between 
the  action  of  direct  and  alternating  currents  is  shown  by 


O 

D 

Fm.  6. 


the  experiments  illustrated  in  Figs.  6  and  7.  Here  two 
circuits  are  depicted,  each  containing  a  source  of 
E.M.F.,  a  glow-lamp  L,  and  two  condensers  C  C  ;  but  in 
the  one  the  E.M.F.  is  due  to  a  direct-current  dynamo 
D,  and  in  the  other  to  an  alternator  A.  Now,  in 
Fig.  6  it  is  clear  that  no  current  can  flow  through  the 
lamp,  even  if  one  of  the  condensers  be  removed,  for 
each  interposes  a  break  in  the  continuity  of  the  circuit. 

*  See  the  Author's  Electric  Lighting  and  Power  Distribution,  Third 
Edition,  vol.  i.  §  84. 


14 


ALTERNATING-CURRENT   CIRCUIT. 


[§6. 


In  Fig.  7,  if  the  condensers  are  suitable  in  capacity,  the 
lamp  L  will  light  up,  and  at  first  sight  this  result  seems 
most  inexplicable ;  but  when  we  consider  the  action  of 
the  condenser,*  and  the  fact  that  the  alternator  is 
keeping  up  a  constant  surging  of  electricity  backwards 
and  forwards  between  the  plates  a  and  b,  it  becomes 
evident  that  there  must  also  be  a  corresponding  flow  of 
electricity  in  the  lamp  circuit,  between  the  plates  c  and  d. 
The  results  would  be  precisely  the  same  if  one  con- 
denser only  were  employed  in  each  experiment ;  but  the 


c 

\  „  f  L       -< 

% 

a 

c                                    d 

b 

.-,       „    ^ 

i 

A 

FIG.  7. 

use  of  two   makes  the   effect  in   Fig.  7  all  the  more 
remarkable. 

It  will  be  noticed  that  in  the  experiments  above 
described  the  capacity  is  in  series  with  the  circuit — i.e. 
there  is  no  through  conducting  path.  This  state  of 
things  effectually  prevents  the  continual  flow  of  a  direct 
current,  but  does  not  stop  the  *  action  '  of  an  alternating 
one. 

*  See  the  Author's  First  Book  of  Electricity  and  Magnetism,  Second 
Edition,  §  166. 


§  6.]  CONDENSERS  IN  CIRCUIT.  15 

A  fuller  explanation  of  the  second  experiment 
(Fig.  7)  is  as  follows.  Before  the  alternator  is  working, 
the  whole  circuit  is  filled  with  electricity  evenly  distri- 
buted, and  at  zero  potential  or  pressure  (§  1).  Now 
suppose  the  alternator  to  work.  During  the  first 
alternation,  i.e.  while  its  E.M.F.  is  in  one  direction 
(§  16),  it  pumps  electricity  from  a  to  6,  causing  a  P.D. 
between  a  and  b  about  equal  to  its  own  E.M.F.  b  is 
consequently  +  ly.  electrified  and  a  —  ly.  electrified,  as 
indicated  by  the  signs  +  and  — .  Influence  (electrostatic 
induction)  takes  place  across  the  condenser  dielectrics, 
causing  a  rush  of  electricity  through  the  lamp  from 
right  to  left,  so  that  c  is  +  and  d  —.  During  the 
second  alternation,  that  is  when  the  reversal  of  the 
alternator  E.M.F.  occurs,  electricity  is  pumped  from  b 
to  at  so  that  a  becomes  +  and  b  —  ;  a  rush  consequently 
takes  place  at  the  same  time  from  c  to  d,  c  becoming 
—  and  d  -f,  and  so  on  ;  the  reversal  and  flow  of 
electricity  in  the  alternator  circuit  causing  a  correspond- 
ing reversal  and  flow  in  the  lamp  circuit. 

It  has  been  stated  that  the  same  results  would  have 
been  obtained  with  one  condenser  only  in  circuit ;  and 
this  will  be  understood  from  what  follows.  In  Fig.  8,  A 
is  an  alternator,  with  two  wires  joined  to  its  terminals  ; 
one  of  the  wires  being  severed  and  a  lamp,  L,  inserted. 
The  ends  of  the  wires  approach  very  closely,  as  at  a  and  b, 
but  are  not  in  contact,  a  sheet  of  glass  or  other  dielec- 
tric, d,  being  interposed  to  prevent  sparking  across  :  the 
alternator  circuit  is  consequently  not  complete.  Now 
the  ends  of  the  wires  a  and  b,  and  the  dielectric  d. 


16 


ALTERNATING-CURRENT   CIRCUIT. 


[§6. 


virtually  form  a  condenser  of  extremely  small  capacity, 
and  the  alternator  pumps  electricity  backwards  and 
forwards  between  a  and  b.  But  in  this  case  very  little 
electricity  passes  at  each  reversal  of  the  E.M.F.,  owing 
to  the  small  capacity  of  the  ends  of  the  circuit,  and  an 
ordinary  lamp  will  consequently  show  no  indication  of  a 
current.* 

When  the  alternator  is  pumping  in  one  direction,  say 
from  a  to  b,  a  quantity  of  electricity  will  pass  sufficient 
to  make  the  P.D.  between  a  and  b  equal  to  the  E.M.F. 


s* 

A 


^ 


Fm.  8. 

of  the  alternator ;  or,  in  other  words,  the  condenser  a  b 
will  be  charged  to  the  potential  of  the  alternator.  Now, 
the  smaller  the  capacity  of  a  condenser,  the  less  is  the 
displacement  of  electricity  necessary  to  raise  the  P.D. 
between  its  coatings  to  a  given  amount :  in  the  present 
case,  because  of  the  extremely  small  capacity  of  the  ends 
of  the  circuit,  only  a  very  minute  quantity  of  electricity 

*  The  wires  are  supposed  to  be  suspended  in  mid-air,  and  not 
running  side  by  side  or  near  other  bodies,  as  we  wish  to  consider  the 
circuit  as  only  having  appreciable  capacity  at  its  ends, 


§-6.j  CONDENSERS  IK    CIRCUIT.  17 

will  pass  from  a  to  b.  When  the  alternator  reverses  its 
E.M.F.,  another  small  quantity  of  electricity  will  be 
pumped  from  b  to  a,  and  so  on  backwards  and  forwards 
with  every  alternation  of  the  E.M.F. 

By  putting  metal  plates  on  each  side  of  the  dielec- 
tric, d,  as  shown  by  the  dotted  lines  p  p,  the  capacity  of 
the  adjacent  ends  of  the  circuit  (i.e.  of  the  condenser) 
will  be  greatly  increased,  and  a  much  greater  quantity 
of  electricity  will  pass  to  and  fro  through  the  lamp  ;  but 
the  current  will  still  be  insufficient  to  light  it  with  a 
simple  two-plate  condenser  such  as  this,  unless  of  very  un- 
wieldy dimensions,  or  unless  an  enormously  high  E.M.F. 
is  employed.  It  will  be  seen,  however,  that  by  using 
a  large  or  multiple-plate  condenser  of  sufficient  capacity, 
an  ordinary  E.M.F.  will  cause  enough  electricity  to  pass 
to  and  fro  to  light  a  lamp,  or,  if  need  be,  a  number  of 
lamps. 

It  has  been  explained  how  what  is  practically  an 
alternating  current  can  be  kept  up  all  round  the  circuit, 
even  if  one  or  two  condensers  be  inserted  therein  (Figs. 
7  and  8)  ;  and  the  reader  should  now  be  able  to  under- 
stand that  the  fanciful  arrangement  of  things  depicted 
in  Fig.  9  is  possible  ;  any  number  of  lamps,  L,  and 
condensers,  C,  being  joined  consecutively  in  the  circuit 
of  an  alternator,  A  ;  the  lamps  burning  brilliantly  if 
the  condensers  are  of  sufficient  capacity,  and  the  E.M.F. 
high  enough. 

As  will  be  presently  pointed  out,  every  electric 
lighting  circuit  possesses  more  or  less  capacity,  owing  to 
the  proximity  of  the  conductors  to  each  other  and  to 

o 


18  ALTERNATING-CURRENT   CIRCUIT.  [§  7. 

the  Earth.  Whether  capacity  can  be  extensively  made 
use  of  in  ordinary  methods  of  alternating-current  electric 
lighting  and  power  supply,  is  an  open  question,  though 
Mr.  James  Swinburne  has  maintained  that  it  can,  and 
has  constructed  tinfoil  condensers  with  thick  paper 
dielectric  compressed  between  metal  plates,  and  placed 
in  a  solid  air-tight  iron  box  filled  with  special  insulating 


material  of  an  oily  nature  to  maintain  the  insulation. 
We  cannot  enter  into  the  consideration  of  the  circum- 
stances under  which  condensers  have  been  or  are  being 
applied,  as  they  are  used  only  in  isolated  cases  at 
present  (§  32).  At  all  events,  the  results  depicted  in 
Figs.  8  and  9  are  closely  related  to  many  beautiful  experi- 
ments with  alternating  currents  of  extra  high  pressure 
and  frequency,  which  certainly  seem  to  foreshadow  great 
advances  on  the  methods  of  electrical  distribution  and 
lighting  as  at  present  carried  out. 

7.   CAPACITY    IN     ALTERNATING  -  CURRENT     CIRCUITS 
(CONT.). — The  reader  will  probably  have  been  puzzled  by 


7.] 


CAPACITY  OF  CABLES. 


19 


the  statement  made  at  the  end  of  the  preceding  para- 
graph, to  the  effect  that  every  ordinary  electric  lighting 
circuit  possesses  more  or  less  capacity.  Such  is  the 
case,  but  the  capacity  is  in  parallel  with  the  circuit,  not 
in  series  with  it  as  in  Figs.  6,  7,  8  and  9. 

In  Fig.  10,  C  is  an  electric-light  cable  laid  direct  in 
the  ground,  or  in  a 
conduit ;.  the  con- 
ductor   forms   one 
coating  of  the  con-  c( 
denser,  the  insula- 
tion  of    the   cable 
the  dielectric,  and  FlG>  10> 

the  outer  sheathing, 

material  of  the  conduit  (if  metal),  or  the  Earth,  the  other 
coating.  This  state  of  things  may  be  diagrammatically 
represented  as  in  Fig.  11,  where  we  may  imagine  the  con- 

EARTH 


EARTH 


CABLE 

FIG.  11. 

ductor  of  the  cable  as  joined  at  intervals  to  the  coatings 
of  condensers,  the  other  coatings  being  connected  with 
Earth :  from  this  it  is  clear  that  the  capacity  is 
parallel  with  the  cable. 

Or  suppose  there  are  two  cables  running  side  by  side 


in 


20  ALTERNATING-CURRENT  CIRCUIT.  [§  7. 

in  a  pipe  or  conduit,  or  in  the  ground,  as  represented  in 
Fig.  12,  which  cables  may  or  may  not  form  part  of  the 
same  circuit :  we  may  then  look  upon  the  two  cable 
conductors  as  the  respective  coatings  of  the  condenser, 


FIG.  12. 

and  the  two  insulating  coverings,  etc.,  in  between  as  the 
dielectric.  The  conception  of  this  state  of  things  as  a 
condenser  is  not  so  easy  as  in  the  case  of  a  single 
cable  laid  in  the  ground ;  but  it  is  made  clearer  in  Fig.  13, 
which  represents  a  section  of  what  lies  between  one 
conductor  and  the  other.  The  break  in  the  condenser 

{w////y///////yy/////////////^i COMDUCTOR 

J^iM, 




OR  AIR      I 1T3  CONDUCTOR 

FIG.  13. 

dielectric  (cable  insulation)  caused  by  the  presence  of 
the  cable  sheathing  or  containing  pipe,  the  earth  or  air 
practically  makes  little  difference  in  the  'condenser 
action '  between  the  two  cable  conductors,  as  the  cables 
usually  lie  close  together. 

The  greater  the  length  of  the  cables,  and  the  closer 
together  or  to  Earth  they  are,  the  greater  their  capacity. 


§8.] 


CAPACITY  OF  CABLES. 


21 


The  capacity  of  underground  mains  varies  from  about 
•3  to  -6  microfarads  per  mile.  It  depends  somewhat 
on  size  and  construction,  and  is  reduced  by  employing 
paper  instead  of  india-rubber  as  dielectric.  The  paper 
being  sometimes  wrapped  comparatively  loosely  round 
the  conductor,  especially  in  the  case  of  telephone  cables, 
a  certain  amount  of  air  is  imprisoned  between  the  folds, 
and  air  and  paper  allow  influence  (§  6)  to  take  place 
across  them  to  a  less  extent  than  india-rubber. 

8.  EFFECT  OF  CAPACITY  IN  THE  CIRCUIT. — The  effect 
of  capacity  upon  the  current  in  an  alternating-current 
circuit  is  exactly  opposite  to  that  of  inductance,  for  it 
assists  or  tends  to  assist  the  current  to  rise  to  its 
maximum  value  sooner  than  it  would  otherwise  do, 
whereas  inductance  retards  or  tends  to  retard  the  current 
(§  4).  This  effect  is  the  same  whether  the  capacity  is 
in  series  or  in  parallel  with  the  circuit. 

In  Fig.  14,  A  is  an  alternator,  the  mains  from  which 
run  for  a  long  distance  side  by  side,  and  feed  a  number 


M 


M 


FIG.'  14. 

of  transformers,  etc.  For  convenience  we  place  the 
transformers,  T  T,  at  the  latter  end  of  the  circuit,  and 
think  of  the  condenser  effect  of  the  first  portion. 


22  ALTERNATING-CURRENT   CIRCUIT.  [§  8. 

The  alternator  is  constantly  pumping  electricity  back- 
wards and  forwards  between  the  mains  MMM  and 
M'  M'  M',  and  these  may  be  looked  upon  as  the  oppo- 
site coatings  of  a  condenser.  Let  us  suppose  the 
alternator  first  pumps  from  M  to  M',  electricity  will 
be,  so  to  speak,  heaped  up  on  Mf,  and  a  deficit  left  on  M, 
M'  being  +  and  M  — .  Now,  neglecting  for  the  moment 
the  latter  end  of  the  circuit,  suppose  the  alternator  were 
suddenly  stopped  :  there  would  then  be  a  momentary 
return  flow  of  electricity  from  M'  to  M  through  the  alter- 
nator ;  in  other  words,  the  condenser  would  discharge  itself. 
If  the  alternator  goes  on  working,  however,  it  is  obvious 
that  the  electricity  heaped  up  on  M'  helps  or  increases 
the  flow  when  the  alternator  begins  to  pump  from  M'  to 
M.  M  then  becomes  +  and  M'  — ,  and  when  the 
alternator  again  reverses  its  E.M.F.,  the  +  charge  on 
M  flows  round  to  M',  and  helps  the  ordinary  current. 
This  auxiliary  current,  if  we  may  so  call  it,  is  generally 
termed  the  condenser  current,  and  is  clearly  greater  the 
greater  the  capacity  of  the  mains.  In  the  above  ex- 
planation we  have  to  think  as  if  the  alternator  were 
pumping  to  and  fro  very  slowly,  whereas  the  reversals  of 
E.M.F.  really  take  place  several  times  a  second  (§  16). 

When  the  '  go  '  and  *  return '  mains  do  not  run  side 
by  side,  the  condenser  action  may  be  pictured  as  fol- 
lows : — Suppose  the  alternator  to  pump  from  left  to  right 
(Fig.  15),  a  surplus  is  heaped  up  on  the  right-hand  cable, 
and  a  deficit  created  in  the  left-hand  one ;  influence  takes 
place,  and  +  and  —  charges  are  respectively  influenced 
(or  induced)  on  the  outsides  of  the  cables,  as  shown  by 


§8.]  EFFECT   OF  CAPACITY.  23 

the  signs.  If  the  alternator  E.M.F.  suddenly  stopped, 
there  would  be  a  momentary  current  from  right  to  left 
through  the  alternator  :  it  is  clear,  therefore,  that  when 
the  alternator  reverses  its  E.M.F. ,  there  will  be  a 
greater  transference  of  electricity  from  right  to  left  than 
there  was  when  the  alternator  first  started  and  pumped 


FIG.  15. 

from  left  to  right.  The  left-hand  cable  now  becomes 
-fly.  charged,  and  the  right-hand  one  —  ly.  charged, 
and  the  discharge  helps  the  alternator  when  it  again 
reverses  its  E.M.F. 

There  is  one  difficulty  which  will  probably  have 
occurred  to  the  reader,  and  that  is,  that  the  two  cables 
in  Fig.  14  being  connected  across  at  various  points  by 
transformers,  &c.,  are  not,  consequently,  strictly  analo- 
gous to  the  insulated  plates  of  a  condenser.  In  Fig.  16, 
for  instance,  C  C  (7,  &c.  are  condensers  representing  the 
capacity  of  the  two  cables,  T  T  T,  &c.  the  primary  coils 
of  transformers  connected  between,  and  A  the  alternator. 
Now  of  course,  any  metallic  cross-connection  would  pre- 
vent the  charging  of  the  condensers  with  a  steady  pres- 
sure ;  but  it  is  conceivable — and,  indeed,  is  proved  by 
practice — that  with  a  rapidly  alternating  pressure  the 
condenser  action  is  not  perceptibly  affected  if  the  cables 


24  ALTEBNATING-CUBPENT   CIECUIT.  [§ 9- 

be  connected  across  by  some  non-inductive  resistance — 
glow  lamps,  for  example.  When  inductive  resistances, 
such  as  transformers,  are  joined  to  the  cables  (Fig.  16), 
the  capacity  effect  will  be  reduced  in  consequence  of  the 
inductance  thus  put  in  circuit,  though  when  a  trans- 
former is  fully  loaded  with  glow-lamps  or  other  non- 
inductive  work,  its  inductance  becomes  negligible. 


FIG.  16. 

Capacity  and  inductance  only  tend  to  neutralise  each 
other  when  both  are  distributed  along  the  whole  length 
of  the  circuit,  as  in  Fig.  16.  In  Fig.  14,  the  capacity  of 
the  first  part  of  the  circuit  would  be  little  affected 
by  or  have  little  effect  on  the  inductance  at  the  far  end 
(§§  10,  20).* 

9.  INDUCTANCE,  CAPACITY,  ETC.,  IN  A  DIRECT-CURRENT 
CIRCUIT.— In  direct-current  work  it  is  generally  suffi- 
cient to  liken  a  current  to  a  steady  flow  of  water  through 
a  pipe,  the  rate  of  flow  representing  current,  the  pressure 
on  the  water — E.M.F.,  and  the  resistance  of  the  pipe 
—resistance  in  the  electrical  circuit.  But  here  there  is 

*  The  oscillatory  nature  of  the  discharge  of  a  condenser  has  been 
alluded  to  in  §  3  ;  but  it  is  not  taken  into  account  in  the  rudimentary 
explanations  of  the  capacity  effect  given  in  this  paragraph  and  here- 
after. 


§9.J  MECHANICAL  ANALOGIES.  25 

no  good  analogy  for  inductance,  or  for  capacity ;  which 
two  quantities  are  nearly  always  present  in  an  alternating- 
current  circuit.  Consequently,  some  other  help  is  neces- 
sary to  enable  us  to  picture  in  our  minds  the  phenomena 
of  an  alternating  current,  and  in  comparing  it  with  a 
direct  current.  In  a  course  of  lectures  delivered  at  the 
Eoyal  Institution,  in  1895,  Professor  Forbes  employed 
various  mechanical  analogies  to  illustrate  electrical  phe- 
nomena, and  these  we  shall  here  make  use  of,  with  certain 
extensions  and  modifications.* 

In  Fig.  17  (a),  T  W  is  a  short  length  of  thick  wire, 
which  is  supposed  to  be  held  vertically  by  its  upper 
end,  T,  between  the  fingers  and  thumb  of  the  left  hand. 
Twist  the  top  of  the  wire  with  the  fingers  and  thumb  of 
the  right  hand  continuously  round  in  the  direction  indi- 
cated by  the  curved  arrow,  and  assume  the  twisting  force 
applied  to  correspond  with  the  E.M.F.  in  the  electric 
circuit,  and  the  rate  of  rotation  of  T  W  to  represent 
the  current.  Then,  assuming  that  the  wire  is  merely 
steadied  by  the  left  hand  while  it  is  being  twisted  by 
the  right,  T  W  corresponds  with  an  electric  circuit  in 
which  there  is  practically  no  resistance,  inductance,  or 
capacity  ;  for  it  may  be  set  rotating,  kept  rotating,  and 
stopped  without  appreciable  effort — i.e.  the  current  may 
be  started  or  stopped  at  once,  or  kept  up  with  a  very 
small  expenditure  of  energy. 

*  The  teacher  or  student  should  not  be  content  with  merely  ex- 
plaining or  reading  through  the  account  of  the  following  experiments, 
but  should  himself  experiment  with  the  simple  contrivances  depicted  in 
Figs.  17  and  18. 


26  ALTERNATING-CURRENT  CIRCUIT.  [§  9. 

In  Fig.  17  (b),  a  large  paper  vane,  V,  is  fastened  to  the 
wire.  The  effect  of  this  is  to  oppose  continuous  air 
resistance  to  the  rotation  of  T  W,  although  it  does  not 
appreciably  retard  the  setting  up  or  stopping  of  that 
rotation.  This  air  resistance  must  be  compared  with 
electrical  resistance,  and  the  arrangement  then  corre- 
sponds with  a  circuit  in  which  there  is  appreciable 
resistance,  but  practically  no  inductance  or  capacity.  If 
the  same  twisting  force  be  applied  as  in  case  (a),  the 
rotation  of  the  wire  will  not  be  so  rapid  ;  in  other  words, 
with  a  given  E.M.F.  the  increase  of  resistance  diminishes 
the  current. 

In  Fig.  17  (c),  a  flat  circular  lump  or  disk  of  lead,  L,  or 
other  heavy  body,  is  tightly  fixed  to  the  end  of  T  W.  Now, 
while  the  air  offers  little  or  no  resistance  to  the  turning 
of  I/,  on  account  of  its  shape ;  the  latter,  because  of  its 
inertia,*  opposes  considerable  momentary  resistance  to 
the  setting  up  of  motion  in  T  W,  and  it  also  tends  to 
prevent  the  sudden  stopping  of  T  W.  The  addition  of 
L  therefore  has  the  effect  of  adding  inertia  to  the  con- 
trivance as  a  whole,  and  it  will  be  found  to  require 
appreciable  effort  to  set  T  W  rotating ;  and  when  in 
motion  it  will  resist  any  sudden  stoppage.  This  mechani- 
cal inertia  is  comparable  with  the  inductance  (sometimes 

*  Inertia  is  that  property  of  a  body  in  virtue  of  which  it  resists 
being  set  in  motion,  having  its  motion  changed,  or  being  stopped  when 
in  motion.  The  inertia  of  a  body  depends  upon  its  weight  (or,  more 
strictly,  its  mass),  and  also,  to  some  extent,  upon  its  shape.  Force  is 
necessary  to  overcome  inertia,  for  it  requires  considerable  force  to  set  a 
heavy  body  (a  flywheel,  for  instance)  in  motion,  and  also  considerable 
force  to  stop  it.  When  a  body  is  in  motion,  it  is  said  to  have  momentum. 


§9.] 


MECHANICAL  ANALOGIES. 


27 


called  electric  or  electromagnetic  inertia)  in  the  electric 
circuit,  the  effect  of  which  is  to  momentarily  oppose  the 
starting,  change,  or  stopping  of  a  current  (§  4) ;  and 
Fig.  17  (c)  thus  presents  the  mechanical  analogy  of  a  cir- 
cuit with  resistance  and  inductance,  but  without  capacity. 


mm.   .iniiimiiii. 

(9) 

In  the  above  examples  we  have  likened  E.M.F.  to  a 
twisting  or  rotating  force,  current  to  rotation,  electrical 
resistance  to  air  friction  or  resistance,  and  inductance  to 
inertia.  We  must  now  get  something  to  represent 
capacity.  In  Fig.  17  (d),IR  is  an  india-rubber  or  other 
flexible  rod  or  tube  rigidly  fixed  at  the  bottom,  say,  to  a 
block  of  wood,  B,  which  cannot  move.  P  is  a  pointer 
(such  as  a  pin)  stuck  into  the  upper  end  of  IR,  to  indicate 
its  movement.  It  will  now  be  shown  that  this  arrangement 
is  typical  of  a  condenser.  If  E.M.F.  is  applied  to  a  con- 
denser, there  will  be  a  momentary  current  due  to  the  rush 


28  ALTERNATING-CURRENT   CIRCUIT.  [§  9. 

of  electricity  into  one  of  its  coatings  (or  set  of  coatings) 
and  out  of  the  other,  which  in  amount  will  depend  upon  its 
capacity,  and  the  displaced  electricity  will  represent  the 
charge  in  the  condenser.  If  the  E.M.F.  be  removed, 
and  the  condenser  left  insulated,  it  will  retain  its  charge  : 
but  when  the  condenser  terminals  are  connected  by  a 
conductor,  it  will  discharge  itself,  there  being  a  sudden 
rush  of  electricity  (momentary  current)  in  the  opposite 
direction.*  Turning  now  to  our  mechanical  analogue 
(Fig.  17  (d)) ;  on  applying  a  twisting  force  (E.M.F.)  to  the 
top  of  7  R,  there  will  be  a  certain  rotation  of  P  (current) 
until  the  force  with  which  I R  tends  to  untwist  equals 
the  twisting  force.  The  amount  of  twist  (charge)  that 
can  be  put  upon  I R  depends  on  its  flexibility  (capacity), 
and  on  the  twisting  force  (charging  E.M.F.)  applied. 
When  I R  has  been  twisted  as  much  as  possible,  let  its 
top  be  fixed  (insulated)  by  means  of  a  clamp ;  it  will 
then  represent  an  insulated  charged  condenser.  Now 
release  the  clamp,  and  IR  will  fly  round,  as  indicated 
by  P,  this  being  equivalent  to  discharging  the  condenser, 
the  momentary  movement  of  P  representing  the  momen- 
tary current  of  discharge.  It  is  evident  that  IR  might 
be  replaced  by  a  coiled  spring,  as  shown  in  Fig.  17  (e). 
These  experiments  (d  and  e)  only  serve  to  show  the 
effect  of  capacity  in  a  condenser  circuit,  as  distinguished 
from  one  which  is  completely  closed  to  allow  of  the  con- 
tinuous passage  of  electricity. 

Fig.  17(/)  illustrates  a  circuit  with  capacity  (due  to 
the  light  coiled  spring  £),  and  resistance  (due  to  V), 
*  See  footnotes,  pp.  14,  24. 


§  10.J  MECHANICAL  ANALOGIES.  29 

but  practically  no  inductance.  S  has  two  pointers,  P 
and  P',  fixed  one  at  each  end,  and  when  S  is  untwisted 
P  and  Pf  should  be  exactly  in  line  as  viewed  from  the 
top.  Now  begin  to  twist  the  top  of  the  wire,  T  W, 
keeping  the  eyes  fixed  on  the  pointers.  It  will  be  found 
that  P  moves  round  a  little  in  advance  of  P'  (if  the 
spring  is  not  too  thick,)  before  V  begins  to  rotate, 
this  representing  the  preliminary  charging  of  the  con- 
ductor :  P  keeping  in  advance  of  P'  all  the  time  the 
rotation  is  continued  (the  permanent  charge  in  the  con- 
ductor). If,  now,  the  twisting  force  (E.M.F.)  is  suddenly 
stopped  at  T  W,  V  will  continue  its  motion  through  a 
short  distance,  until  P'  catches  up  to  P,  this  being 
representative  of  the  discharge  from  the  conductor, 
which  tends  to  prolong  the  current. 

In  Fig.  17  (g),  the  disk  of  lead,  L,  is  added  to  represent 
inductance  in  the  circuit.  On  applying  a  twisting  force 
(E.M.F.)  to  the  top  of  the  wire,  P  first  moves  round 
slightly  in  advance  of  P',  then  the  inertia  (inductance) 
of  L  has  to  be  overcome,  and  at  last  V  gets  up  full 
speed  (current).  On  trying  to  stop  the  rotation  (current), 
the  momentum  of  L  (E.M.F.  due  to  inductance)  and  un- 
twisting of  S  (discharge  due  to  capacity),  but  principally 
the  former,  tend  to  prolong  the  rotation  (extra  current) ; 
this,  be  it  remembered,  being  the  case  of  a  direct-current 
circuit. 

10.  INDUCTANCE,  CAPACITY,  ETC.,  IN  AN  ALTERNATING- 
CURRENT  CIRCUIT. — In  the  preceding  paragraph,  mecha- 
nical illustrations  of  the  direct-current  circuit  were  given. 
In  his  lectures  (p.  25)  Professor  Forbes  followed  up 


30  ALTERNATING-CURRENT   CIRCUIT.         [§  10. 

these  analogies  still  further ;  but,  like  most  others,  they 
must  not  be  carried  too  far.  The  last  portion  of  the 
preceding  paragraph  paves  the  way  for  their  application 
to  the  alternating- cur  rent  circuit. 

In  Fig.  18  (a),  T  W  is  a  piece  of  thick  wire.  Hold  it 
vertically  at  the  top  in  the  right  hand,  and  steady  it 
lightly  with  the  left.  Twist  it  rapidly  to  and  fro,  giving 
a  turn  first  in  one  direction  and  then  in  the  other,  as 
indicated  by  the  double-headed  arrow.  This  represents 
the  application  or  *  impression '  of  an  alternating 
E.M.F.  to  or  on  the  circuit.  If  we  suppose  that  T  W 
has  no  inertia  or  flexibility,  and  that  no  resistance  is 
opposed  to  its  rotation,  it  may  be  taken  to  represent  a 
circuit  with  no  inductance,  capacity,  or  resistance :  and 
the  direction  of  twist  (E.M.F.)  and  rotation  (current) 
may  be  changed  immediately ;  and  one  might  almost 
say  that  the  rate  of  rotation  (strength  of  current)  is 
uniform,  though  rapidly  alternating  in  direction.  This 
case  may,  therefore,  be  taken  as  an  analogy  for  the  ima- 
ginary alternating  current  represented  by  the  *  curve ' 
in  Fig.  2. 

In  case  (b)  (Fig.  18),  a  paper  vane  V  is  put  on  to 
represent  resistance  in  the  electrical  circuit ;  but  it  must 
be  supposed  that  it  does  not  materially  add  inertia  to  the 
arrangement.  Then,  if  the  same  alternating  twisting 
force  (E.M.F.)  be  applied  to  the  wire  as  before  (case  a), 
the  rate  of  rotation  (strength  of  current)  will,  be  less 
than  in  the  first  instance,  in  consequence  of  V\  but 
there  being  no  inertia,  as  we  suppose,  the  rotation 
(current)  will  change  directly  the  twisting  force  changes, 


§10.] 

and  will  be  always  at  the  same  rate.  Here  we  have  the 
representation  of  an  alternating-current  circuit  in  which 
there  is  practically  only  resistance,  and  no  appreciable 
inductance  or  capacity.  In  Fig.  18  (c)  is  shown  the 
mechanical  analogue  of  a  circuit  with  resistance  (due 
to  F),  and  inductance  (due  to  L)t  Apply  an  alternating 


T.W. 


(a)          (b) 


^ 

TW. 

V 

(c) 

FIG.  18. 


(d) 


urn '"I'll  in  nnmL 

to 


twisting  force  (E.M.F.)  to  the  top  of  T  W\  the  result 
will  be  that  the  rotation  (current)  will  be  far  from 
uniform,  it  taking  appreciable  time  to  set  up,  stop,  or 
reverse.  Furthermore,  the  rotation  (current)  will  '  lag 
behind  '  the  twisting  force  (E.M.F.)  ;  that  is  to  say,  the 
rotation  (current)  will  start,  stop,  and  reverse,  after  the 
twisting  force  (E.M.F.)  has  been  started,  stopped,  or 
reversed. 

In  Fig.  18  (cl),  the  spring  S  is  introduced  to  represent 
capacity  in  the  circuit.  On  applying  an  alternating 
twisting  force  to  the  top  T,  the  effect  of  this  flexibility 
(capacity)  in  the  wire  (circuit)  will  be  opposite  to  that  of 
inertia  (inductance)  :  for  it  will  be  found  to  assist  the 
setting  up  and  stopping  of  the  rotation  (current),*  the 

*  This  seems  to  contradict  what  was  said  at  the  end  of  the  preceding 


32  ALTEENATING-CUERENT   CIRCUIT.          [§  10. 

movement  of  the  pin-head  Pr  being  in  advance  of  the 
twisting  force  (E.M.F.).  There  is  this  important  differ- 
ence between  the  effects  of  capacity  and  inductance,  that 
capacity  in  an  alternating-current  circuit  apparently 
increases  the  current,  and  inductance  decreases  it.  That 
such  is  the  case  has  already  been  pointed  out  (§§  4,  8). 
Thus,  if  inductance  and  capacity  be  both  present,  and  in 
their  right  amounts,  no  evidence  of  either  will  be  found  : 
in  other  words,  capacity  and  inductance  neutralise  each 
other's  effects  (§§8,  20). 

In  Fig.  18  (e)  is  shown  the  mechanical  analogy  for  a 
circuit  with  both  inductance  and  capacity,  as  well  as 
resistance.  Here  it  should  be  noticed  that  the  flexibility 
(capacity)  increases  the  amount  of  turning  that  can  be 
done  by  a  given  force  through  a  fixed  distance  ;  that  is 
to  say,  it  points  to  the  fact  that  the  addition  of  capacity 
to  a  circuit  will  decrease  the  effective  inductance.  Any- 
how, it  will  be  clearly  seen  that  on  applying  an  alternat- 
ing twisting  force  to  T,  the  effect  of  both  flexibility 
(capacity)  and  inertia  (inductance)  will  be  to  alter  the 
rotation  (current)  with  a  given  twisting  force  (E.M.F.) 
from  what  it  would  be  were  these  properties  absent. 

In  the  mechanical  analogies  given  (Figs.  17  and  18), 
the  effect  of  inductance,  as  represented  by  inertia,  is 
considerably  greater  than  the  effect  of  flexibility  or 
capacity :  but,  of  course,  in  some  circuits  there  may  be 
greater  capacity  than  inductance,  as  might  be  repre- 

paragraph ;  but  it  should  be  remembered  that  here  we  are  dealing  with 
an  alternating-current  circuit,  whereas  §  9  refers  to  the  direct-current 
circuit. 


§  10.]  MECHANICAL  ANALOGIES.  33 

sented  by  using  a  stronger  spring,  S,  and  a  lesser  weight, 
L  (Fig.  18  (e)).  As  a  general  rule,  however,  the  effects 
of  inductance  preponderate. 

With  a  given  wire  (circuit),  in  which  there  is  prin- 
cipally inertia  (inductance),  the  average  rate  of  rotation 
(current)  will  be  much  greater  in  the  case  of  a  unidirec- 
tional twisting  force  (E.M.F.)  than  in  the  case  of  an 
alternating  twisting  force.  In  other  words,  a  direct 
constant  E.M.F.  will  set  up  a  greater  current  in  a  given 
circuit  than  will  an  alternating  E.M.F.  of  the  same 
equivalent  value  ;  for  in  the  first  case  inductance  exerts 
its  effect  only  on  making  and  breaking  circuit,  or  when 
the  current  strength  is  suddenly  changed :  whereas  in 
the  latter  its  effects  are  observable  the  whole  time. 

With  a  constant  direct  E.M.F.  the  current  is  uniform, 
but  with  an  alternating  E.M.F.  it  is  wavy  or  undulatory, 
i.e.  constantly  varying  in  strength.  Even  if  the  impressed 
alternating  E.M.F.  were  constant  in  value,  as  in  the  ex- 
periment described  in  §  2,  this  would  be  the  case,  because 
of  inductance,  &c. ;  but  as  the  E.M.F.  of  an  alternator 
constantly  varies  in  strength  as  well  as  in  direction  (§  15), 
the  waves  of  current  are  much  more  accentuated. 

Perhaps  it  is  hardly  necessary  to  point  out  that  one 
particular  in  which  the  above  mechanical  analogies  do 
not  fit  the  true  condition  of  things,  is  that  the  resistance 
(air  friction),  inductance  (inertia),  and  capacity  (flexi- 
bility), are  contained  in  separate  parts  of  the  circuit 
(wire) ;  whereas  in  the  real  electric  circuit  these  pro- 
perties are,  as  a  rule,  more  or  less  intermingled  along 
the  whole  of  its  length. 

D 


34 


ALTERNATING-CURRENT  CIRCUIT.         [§  11. 


11.  INDUCTANCE  IN  A  CIBCUIT. — It  has  been  several 
times  stated  that  the  effect  of  inductance  in  an  alternat- 
ing-current circuit  is  to  cut  down  the  current.  The 
following  experiment  conclusively  proves  this.  The 


circuit  LKB  (Fig.  19)  is  fed  at  a  virtual*  alter- 
nating pressure  of,  say,  100  volts,  from  the  mains  M. 
B  is  a  laminated  iron  bar,  built  up  of  thin  wires, 
on  which  are  coiled  several  turns  of  thick  wire  of 
negligible  resistance,  which  may  be  short-circuited 
by  the  key  K.  B  obviously  possesses  considerable 
inductance  (§  4),  whereas  the  rest  of  the  circuit  has 
very  little.  The  light  given  by  L  depends  upon  the 
strength  of  the  current  passing  through  it,  and  is  a 
convenient  indicator  of  it.  Suppose  when  K  is  de- 
pressed so  as  to  cut  out  B,  the  lamp  is  fully  lighted ; 
when  B  is  put  in  circuit  by  opening  the  key  K,  the 
lamp  will  burn  dimly,  proving  that  the  effect  of  induct- 
ance is  to  permanently  reduce  the  current.  This  effect 
is  the  same  as  if  a  back  E.M.F.  had  been  introduced 
into  the  circuit,  which,  in  fact,  is  the  case,  the  back  or 
counteracting  E.M.F.  being  that  due  to  the  inductance 
of  B. 

*  See  §  18, 


§  12.]  INSULATION.  35 

If  a  direct  current  is  used,  the  insertion  or  cutting 
out  of  B  will  make  no  appreciable  difference,  as  its  re- 
sistance is  small,  except,  perhaps,  a  faint  flicker  of  the 
lamp  at  the  moment  of  depressing  or  releasing  K ;  but 
this  would  be  hardly  noticeable. 

12.  EFFECTS  OF  AN  ALTERNATING  CURRENT  AND  OF 
INDUCTANCE  AND  CAPACITY  ON  THE  INSULATION  OF  A 
CIRCUIT. — In  a  conductor  carrying  a  given  virtual  alter- 
nating current,  there  is  a  greater  tendency  to  leak 
through  or  break  down  the  insulation  than  in  the  case 
of  a  direct  current  of  the  same  value  ;  for  the  reason 
that  in  the  first  case  the  electricity  is  moving  rapidly 
backwards  and  forwards,  and  the  impressed  virtual 


Fm.  20. 

E.M.F.  is  only  about  seven-tenths  of  its  maximum 
values  (§  18) ;  while  in  the  latter  case  it  is  flowing 
steadily  in  one  direction,  and  the  E.M.F.  is  also  steady. 
Let  us  illustrate  this  by  an  analogy.  Consider  a  pipe, 
P  (Fig.  20),  with  water  flowing  in  it,  as  resembling  a 
conductor  carrying  current ;  and  suppose  the  material  of 
the  pipe  to  represent  the  insulation  round  the  conductor  : 
then,  if  the  pressure  of  water  causes  a  fracture  of  the  pipe, 
it  is  clearly  analogous  to  the  breaking  down  of  the  insula- 
tion of  the  conductor.  Now  it  must  be  evident  that  there 
is  a  greater  strain  on  the  sides  of  the  pipe  when  the  water 
is  rapidly  moving  to  and  fro  (alternating  current),  than 
when  it  is  flowing  steadily  in  one  direction  (direct  current). 

D   2 


36  ALTERNATING-CURRENT  CIRCUIT.  [§12. 

The  effects  of  inductance  and  capacity  in  a  direct- 
current  circuit  are  only  observable  on  making  or  breaking 
the  circuit,  or  on  suddenly  changing  the  strength  of 
current  therein  (§  4)  ;  but  in  an  alternating-current 
circuit  they  exert  a  continual  influence  on  the  current, 
and,  indirectly,  on  the  insulation  of  the  circuit. 

Fig.  21  presents  an  analogy.  Here  S  is  a  stand 
carrying  two  bearings  BB',  in  which  is  mounted  an 
upright  glass  tube  G,  which  may  be  rotated  by  the 
handle  H.  A  portion  of  the  glass  is  cut  away,  and  a 
short  length  of  rubber  tubing,  E,  inserted,  to  represent 
capacity  in  the  circuit.  Down  the  centre  of  both  glass 
and  rubber  tubes  passes  a  metal  wire  which  stands  for 
the  conductor,  the  glass  tubing  being  looked  upon  as  the 
insulating  covering.  The  friction  of  the  bearings  and 
of  the  vane  V  corresponds  with  electrical  resistance,  and 
the  inertia  of  the  lead  disk  L  represents  inductance. 
The  strain  on  the  glass  tubing,  to  the  outside  of  which 
V  and  L  are  fixed,  may  be  taken  as  analogous  to  the 
strain  on  the  insulation  in  an  electrical  circuit.  We 
will  first  take  a  case  in  which  L  is  removed — i.e.  where 
there  is  no  inductance  in  the  circuit.  Now,  if  H  be 
rotated  steadily  in  one  direction  (steady  direct  current), 
the  strain  on  the  glass  tubing  (insulation  of  the  circuit) 
will  be  comparatively  small ;  but  if  H  is  sharply  and 
continuously  turned,  first  in  one  direction  and  then  in 
the  other,  clearly  a  good  deal  of  strain  is  thrown  on  (2, 
but  this  is  lessened  in  proportion  to  the  flexibility  of  E 
(capacity  of  the  circuit).  This  seems  to  point  out  that 
if  a  circuit  has  capacity  but  no  inductance,  the  presence 


§12.] 


INSULATION. 


37 


of  the  former  will  not  increase  the  strain  on  the  insula- 
tion, but  rather  the  reverse.  If  L  be  now  put  on  (in- 
ductance put  in  circuit),  with  continuous  rotation  (direct 
current),  an  extra  strain  will  be  thrown  on  G  at  the 
moment  of  starting,  stopping,  or  altering  the  speed  of 
rotation  (current)  ;  and  will  be  the  greater  the  more 
suddenly  the  starting,  stopping,  or  alteration  of  the 
speed  is  brought  about.  The  reason  of  this  is  that  the 
flexibility  (capacity)  exists  at  the  near  end  of  the  circuit, 
while  the  inertia  (inductance)  is  all  at  the  far  end  ;  a 
condition  of  things  which  obtains  in  electricity  distribu- 
tion work  when  the  *  feeders  '  supplying  the  distribution 
network  have  great  length.  If  the  inductance,  as  repre- 
sented by  the  inertia  of  L,  were  more  distributed  along 
the  circuit,  the  extra  strain  on  the  insulation  would  be 
correspondingly  lessened ; 
while  if  it  were  all  at  the 
near  end — i.e.  if  L  were 
placed  in  the  position  of 
R — there  would  be  prac- 
tically no  strain  on  the 
insulation  directly  due  to 
inductance  or  capacity, 
except  when  the  current 
(rotation)  was  suddenly 
stopped. 

In  the  case  of  alter- ' 
nating  rotation  (alternat- 
ing current)  the  strain  on   Gr  will  be  continuous  and 
very  considerable  if  the  condition  of  things  be  as  repre- 


38  ALTERNATING-CURRENT  CIRCUIT.  [§  12. 

sented  in  the  figure ;  so  much  so,  in  fact,  that  if  G  is 
not  thick  enough  or  has  any  flaws  in  it,  it  will  be 
fractured  (insulation  broken  down).  But  if  the  induct- 
ance and  capacity  be  more  intermingled,  the  strain  will 
be  lessened;  and  if  the  inductance  be  all  brought  to  the 
near  end  of  the  circuit,  it  will  be  still  further  reduced. 
This  mechanical  analogy  affords  a  capital  illustration 
of  the  opposite  effects  of  capacity  and  inductance  in  the 
circuit,  and  the  fact  that  one  neutralises  the  other. 

The  presence  of  dynamos  and  other  electro-magnetic 
apparatus  in  direct-current  circuits  is  the  main  cause  of 
inductance  (§  4) ;  but  alternating-current  circuits  gene- 
rally have  much  more  inductance  than  direct-current 
ones,  because  of  the  numerous  transformers  therein  ; 
though,  by  the  way,  the  inductance  of  these  latter 
decreases  as  their  load  increases,  when  the  load  is  non- 
inductive,  as  in  the  case  of  glow-lamp  lighting.  Now, 
thinking  of  the  mechanical  analogy  (Fig.  21),  it  would 
appear  that  the  more  suddenly  the  full  E.M.F.  (whether 
direct  or  alternating)  is  thrust  upon  a  circuit,  with 
capacity  at  the  near  end  and  inductance  at  the  far  end, 
the  greater  will  be  the  extra  strain  on  the  insulation. 
In  most  circuits,  whether  direct  or  alternating,  but  par- 
ticularly in  those  with  inductance — however  distributed 
-considerable  strain  is  thrown  on  the  insulation  if  the 
circuit  be  broken  rapidly.  As  an  example  of  this  latter 
effect  in  direct-current  circuits,  if,  while  the  water  is 
flowing  (Fig.  20),  the  cock,  <7,  is  suddenly  turned  off,  a 
great  strain  will  be  thrown  on  the  pipe.  An  illustration 
of  this  may  be  found  in  some  houses  where  water  is 


§  12.]  SLOW  MAKE-AND-BEEAK  SWITCH.  39 

supplied  direct  from  the  main,  and  therefore  at  consider- 
able pressure,  and  old-fashioned  taps  are  used.     On  sud- 
denly turning  the  tap  off,  the  momentum  of  the  water 
expends  its  energy  on  the  pipe.    Screw  taps  are  designed 
to  prevent  this  sudden  strain  being  thrown  on  the  pipes. 
The  strain  due  to  inductance  is  not  very  noticeable 
on  '  making '  the  circuit,  though  its  analogue  is  observ- 
able in  the  mechanical  illustrations.*     On  breaking  the 
circuit,  inductance  is  very  manifest,  as  it  shows  itself 
in  the  form  of  an  '  extra -cur  rent '  arc  or  spark.     Now 
it  is  a  common  but  very  erroneous  idea  that  the  more 
suddenly  any  circuit  is  broken  the  better,  as  the  extra- 
current   spark   tends   to   destroy  the    ordinary   switch 
contacts.     This   spark,  or   arc,  represents  energy,  and 
if  this   energy  is   prevented   from   expending  itself  in 
the  form  of  a  spark,  it  will  wreak  its  force  somewhere 
else — viz.   on    the    insulation    of    the    circuit.      Main 
switches  for   circuits  having  large   inductance   should 
therefore  be  so  designed  that  the  circuit  is  both  made 
and  broken  gradually.     A  small  switch  fulfilling  these 
conditions  is  depicted  in  Fig.  22.f     On  one  of  the  fixed 
contact  pieces,  and  on  one  end  of  the  movable  contact 
arm,  there  is  a  short  cylinder  of  carbon.    When  putting 
the  switch  on,  the  carbons  come  into  contact  first  of  all, 
then  the  one  on  the  movable  arm,  being  mounted  on  a 
spring-hinged  pivot,  gives  way  to  allow  the  arm  to  go 
into  place.     When  the  switch  is  put  off,  the  carbon  con- 

*  Experiments  which  illustrate  the  effect  of  inductance,  both  on 
making  and  breaking  circuit,  are  described  in  §  14. 
f  Made  by  Messrs.  Siemens  Bros.  &  Co. 


40 


ALTERNATING-CURRENT  CIRCUIT. 


[§12. 


tacts  and  the  slight  arc  formed  between  them  momen- 
tarily prolong  the  connection  of  the  circuit.  Though 
the  switch  was  probably  designed  for  the  single  purpose 
of  preventing  or  minimising  the  '  spark-wear  '  of  the 
contacts,  it  will  be  seen  that  it  also  eases  the  strain  on 
the  insulation  of  the  circuit. 

Some  electricians  will  probably  disagree  with  the 
statement  that  it  is  necessary  in  practice  to  gradually 
*  make  '  as  well  [as  *  break  '  alternating-current  circuits, 


FIG.  22. 

and  will  maintain  that  there  is  really  no  excessive  stress 
on  the  insulation  of  a  circuit  when  the  full  pressure  is 
suddenly  thrust  upon  it ;  but  theoretical  considerations, 
and  the  mechanical  illustrations  (Figs.  17,  18,  and  21) 
seem  to  indicate  otherwise ;  the  amount  of  extra  strain 
(if  any)  thrown  on  the  circuit  at  the  moment  of  making 
it  appearing  to  depend  upon  the  relative  distribution  of 
the  inductance  and  capacity,  as  already  pointed  out. 

The  matter  of  slow  *  making  '  probably  only  becomes 
of  really  practical  importance  when  pressures  above,  say, 


§  13.]  ELECTEIFICATION.  41 

2,000  volts  are  used ;  or  when  the  circuits  consist  of  a 
number  of  miles  of  cables  of  large  capacity.  Professor 
Forbes  has  arranged  for  slow  '  making  '  and  '  breaking  ' 
on  the  Niagara  power  circuits,  and  it  is  now  being  done 
at  Deptford,  and  elsewhere. 

It  is  interesting  to  note  that  on  breaking  a  high- 
pressure  alternating-current  circuit  the  switch  may  show 
a  large,  small,  or  no  spark  at  all,  according  to  the  point 
in  the  current  wave  where  separation  occurs.  Thus,  if 
the  circuit  happens  to  be  broken  at  the  moment  the 
wave  is  at  its  peak  or  maximum  (Fig.  31),  the  largest 


CONDUCTOR 


FIG.  23. 


spark  will  be  obtained,  while  if  the  current  is  just  re- 
versing there  will  be  no  appreciable  spark. 

13.  ELECTRIFICATION  OF  CONDUCTOK  DIELECTRIC.— 
There  is  one  particular  point  in  which  the  mechanical 
analogies  fail,  and  that  is,  they  furnish  no  good  example 
for  the  action  taking  place  across  the  dielectric  or 
insulation  —  viz.  the  electrification  due  to  condenser 
action.  In  Fig.  23  is  given  a  section  of  a  cable  con- 
ductor, its  insulation,  and  the  surrounding  sheathing, 
pipe,  or  Earth.  Let  the  conductor  be  carrying  a  steady 
direct  current,  and  suppose  that  particular  portion  of  it 
under  consideration  is  at  a  higher  potential  than  the 


42  ALTERNATING-CURRENT   CIRCUIT.  [§  14. 

Earth  ;  *  it  will  then  have  a  steady  +  charge.  Influence 
will  take  place,  and  a  —  charge  will  be  created  on  the 
inner  surface  of  the  metallic  sheathing  of  the  cable,  or 
other  surroundings,  the  system  acting  like  a  condenser 
(§  7).  Now  these  charges  will  mutually  attract  each 
other,  and  will  soak  into  the  dielectric  and  tend  to 
approach  nearer  to  one  another,  and  so  in  a  sense 
lessen  the  thickness  of  the  insulation  surrounding  the 
conductor.  With  an  alternating  current,  on  the  other 
hand,  there  is  little  or  no  electrification  of  the  dielectric 
('  soaking-in '  action),  as  the  charge  of  the  cable  is  con- 
stantly and  rapidly  alternating  in  sign. 

14.  EXPERIMENTS  ON  INDUCTANCE. — Inductance  was 
briefly  dealt  with  in  §  4,  and  its  effects  have  been  de- 
scribed in  the  immediately  preceding  paragraphs,  prin- 
cipally by  means  of  mechanical  analogies.  The  following 
electrical  experiments  further  show  its  effects,  and 
should  be  considered  in  conjunction  with  that  depicted 
in  Fig.  19. 

In  Fig.  24,  L  is  a  glow  lamp,  connected  through  the 
switch  or  key,  and  wires,  +  and  — ,  with  a  source  of 
direct  E.M.F.  C  is  a  coil  of  fairly  fine  wire,  with  a 
removable  iron  core,  and  is  connected  as  a  shunt  to  the 
lamp.  The  resistance  of  C  should  be  such  that  when 
the  current  is  flowing  steadily  the  lamp  filament  is 
just  perceptibly  red,  or  thereabouts.  At  the  instant  of 

*  There  is  a  gradual  fall  of  pressure  or  potential  along  a  conductor 
carrying  a  direct  current ;  but  there  is  almost  bound  to  be  a  P.D. 
between  the  conductor  and  the  Earth,  and  the  conductor  will  be  either 
+  ly  or  -  ly  charged. 


§14.]  EXPERIMENTS  ON  INDUCTANCE.  43 

making  the  circuit,  the  lamp  will  momentarily  glow 
more  brightly  than  when  the  current  is  steady;  on 
breaking  the  circuit,  the  lamp  will  momentarily  flash 
with  great  brightness.  In  the  first  case  the  counter 
E.M.F.  due  to  inductance,  as  indicated  by  the  small 
dotted  arrow,  will  momentarily  oppose  the  main  E.M.F. 
in  the  shunt  circuit,  C,  so  that  the  latter  is  enabled 
to  send  a  momentarily  stronger  current  through  the 
lamp.  On  breaking  the  main  circuit,  the  field  of  C  will 
collapse,  generating  a  momentary  and  much  greater 


E.M.F.  than  in  the  first  instance,  in  the  direction 
shown  by  the  larger  dotted  arrow  :  a  momentary 
current  will  flow  through  C  and  L  in  a  counter-clock- 
wise direction,  and  the  lamp  will  flash  up  brightly  in 
consequence. 

Now  remove  C,  with  its  iron  core,  and  insert  instead  a 
coreless  coil  having  the  same  resistance  as  <7,  but  wound 
as  in  Fig.  5,  so  that  it  shall  have  no  inductance  :  pass  an 
alternating  current  through  the  lamp  and  coil,  of  such  a 
strength  that  the  filament  of  L  is  perceptibly,  but  dimly, 
heated.  Now  insert  the  former  coil,  (7,  with  its  core, 
and  it  will  be  found,  in  consequence  of  the  inductance 


UNIVEBSITY 


44  ALTERNATING-CURRENT   CIRCUIT.  [§14. 

of  0,  that  L  is  increased  in  brilliancy.  The  explanation 
of  these  different  effects  on  the  lamp  is  as  follows.  In 
the  first  case,  the  non-inductive  coil  shunts  a  certain 
amount  of  current  from  the  lamp  circuit,  but  otherwise 
exerts  no  effect.  In  the  second,  where  the  coil  has  the 
same  resistance,  and  also  considerable  inductance,  the 
back  E.M.F.  due  to  the  latter  constantly  opposes  the 
working  E.M.F.,  offering  a  kind  of  extra  resistance  in 
addition  to  the  ordinary  resistance  of  the  coil  (§§  21,  22, 
23) ;  so  that  the  total  apparent  resistance  of  the  coil  is 
increased,  and  less  current  is  shunted  off  the  lamp, 
which  consequently  glows  more  brightly. 

The  difference  in  the  result  of  this  experiment,  as 
compared  with  that  illustrated  in  Fig.  19,  is  accounted 
for  by  the  fact  that  in  one  case  the  inductive  coil  is  in 
series  with  the  lamp,  while  in  the  other  it  forms  a  shunt 
thereto.  It  must  be  pointed  out  that  if  the  ends  of  the 
circuit  in  Fig.  24  be  kept  at  a  constant  potential  differ- 
ence, the  insertion  of  C  will  not  affect  the  current 
passing  through  L. 

The  following  experiment,  due  to  Edlund,  which, 
however,  is  only  performable  with  direct  currents, 
serves  to  show  in  a  marked  manner  the  effect  of  induct- 
ance both  on  completing  and  breaking  the  circuit. 
G  (Fig.  25)  represents  diagrammatically  a  differential 
galvanometer,  of  which  C'  and  C"  are  the  two  coils ; 
B  is  a  battery,  and  K  a  key.  R  and  C  are  two  coils,  equal 
in  resistance,  but  R  is  wound  so  that  it  shall  be  non- 
inductive,  while  C  is  wound  in  the  ordinary  way,  and 
provided  with  an  iron  core.  C  and  G  must  be  so  far 


14.] 


EXPERIMENTS  ON  INDUCTANCE. 


45 


removed  that  the  magnetism  of  the  former  cannot  act 
directly  on  the  galvanometer  needle.  The  galvanometer 
coils  Cf  and  C"  being,  of  course,  equal  in  resistance,  it 
follows  that  after  K  has  been  depressed  equal  steady 
currents  will  flow  round  the  circuits  C'  R  K  and  C"  C 
K)  and  the  galvanometer  will  be  unaffected,  as  the  coils 
C"  and  C"  will  exercise  equal  and  opposite  effects 
upon  the  needle.  But  at  the  moment  of  depressing  K, 
the  opposing  counter  or  back  E.M.F.  of  (7,  as  indi- 


FIG.  25. 

cated  by  the  small  dotted  arrow,  opposes  the  battery 
E.M.F. ,  and  the  current  in  C"  is  consequently  momen- 
tarily weaker  than  that  in  Cr,  so  that  the  needle,  N, 
moves,  but  afterwards  returns  to  its  zero  position. 
On  breaking  the  circuit,  the  momentary  induced  E.M.F. 
in  C,  which  is  now  in  the  direction  of  the  large  dotted 
arrow,  i.e.  in  the  same  direction  as  the  battery  E.M.F., 
for  the  instant  increases  the  current  in  C",  and  the 
galvanometer  needle  makes  a  momentary  deflection  in 
the  opposite  direction  to  that  of  its  deflection  on  the 
closing  of  the  circuit. 


46 


ALTERNATING-CURRENT  CIRCUIT. 


[§14. 


There  will  be  little  or  no  spark  at  the  contact  points 
of  K  on  breaking  circuit,  for  the  reason  that  it  is  shunted 
by  R  and  C'.  This  latter  fact  introduces  an  error  in  the 
experiment,  for  the  momentary  induced  currents  due  to  C, 
on  both  making  and  breaking  circuit,  travel  vidR  through 
both  coils  C'  and  C"  of  the  galvanometer,  so  that  both 
exercise  the  same  directive  effect* on  the  needle,  and  the 
inductance  in  the  circuit  thus  appears  to  be  greater 
than  it  really  is. 

Another  experiment,  due  to  Maxwell,  is  illustrated 
2  in  Fig.  26.     a,  b, 

c,  and  d  are  four 
coils  of  exactly 
equal  resistance, 
*a,  b,  and  c  being 
wound  so  as  to 
be  non-inductive, 
while  d  is  wound 
in  the  ordinary 
way,  and  provided 
with  an  iron  core. 
The  four  coils  are 

FIG.  26.  joined  up  with   a 

galvanometer  (G),  battery,  and  key,  exactly  as  in 
the  Wheatstone  Bridge  method  of  measuring  resist- 
ance.* When  a  steady  current  is  flowing  through  the 
system,  the  fall  of  potential  along  the  path  1  a  2  d  4  will 
be  equal  to  that  along  the  path  1  b  3  c  4 ;  and  because 

*  See  the  Author's  Electric  Lighting  and  Power  Distribution,  Third 
Edition,  vol.  i.  §  112. 


§  15.]        ALTERNATING  E.M.F.  AND   CURRENT.         47 

the  coils  are  all  equal  in  resistance,  the  potentials  at 

2  and  3  will  be  equal,  and  there  will  be  no  current 
through    the   galvanometer.      When    the   key   is   first 
depressed,  the  back  E.M.F.  of  inductance  due  to  d  will 
cause  the  potential  at  2  to  rise  more  slowly  than  the 
potential    at   3,   and    there    will    consequently    be    a 
momentary    current    through   the   galvanometer   from 

3  to   2.      When   the    battery    circuit    is  broken,   the 
momentary  direct  E.M.F.  in  d,  due  to  the  collapsing  of 
its  lines,  will  prolong  the  potential  at  2,  and  there  will 
consequently  be  a  momentary  current  from  2  to  3. 

15.  GRAPHICAL  EEPRESENTATION  OF  AN  ALTERNATING 
CURRENT. — It  has  been  demonstrated  that  because  of 
the  inductance  and  capacity  in  an  alternating-current 
circuit,  the  current  is  in  the  form  of  waves,  even  if 
the  impressed  alternating  E.M.F.  is  constant  in  value, 
as  in  the  experiment  shown  in  Fig.  1.  In  an  alternator, 
however,  the  E.M.F.  itself  is  constantly  altering  in 
value,  as  well  as  in  direction — i.e.  it  is  in  the  form  of 
waves,  and  this  further  accentuates  the  waves  of  cur- 
rent. 

To  show  approximately  what  an  alternating  current 
is  like,  one  may  draw  a  picture,  in  the  form  of  a  curve, 
of  the  changes  which  take  place  in  the  strength  and 
direction  of  the  impressed  E.M.F.  which  sets  it  up,  and 
this  will  enable  us  to  explain  what  is  meant  by  the 
sine  curve  or  sine  wave,  terms  frequently  used  in  speak- 
ing of  alternating  currents. 

The  reader  is,  of  course,  aware  that  when  a  simple 
coil  of  wire  is  rotated  in  a  magnetic  field,  it  has  alter- 


48  ALTERNATING-CURRENT  CIRCUIT.          [§  15. 

nating  E.M.Fs.  induced  in  it.*  A  simple  two-pole  field 
and  coil  is  shown  in  Fig.  27,  and  we  will  consider  what 
happens  to  the  top  half, p,  of  the  coil,  abed,  when  the 
latter  is  evenly  rotated  in  the  direction  shown  by  the 
curved  arrow. 

Now  p  will,  if  viewed  sideways  from  one  of  the  pole 
faces,  N  or  S,  have  an  up-and-down  motion;  and  its 
apparent  velocity  will  be  variable  during  any  one  com- 


FIG.  27. 


plete  revolution  of  the  coil ;  but  the  changes  that  take 
place  will  be  repeated  over  and  over  again  at  regular 
intervals.  This  will  be  more  clearly  understood  from 
Figs.  28  and  29.  Fig.  28  represents  the  circular  path 
traversed  by  p  when  the  coil  is  looked  at  from  the  front 
end,  only  one  pole,  N,  being  shown  for  simplicity's  sake  ; 
and,  as  we  suppose  that  the  coil  is  being  turned  with 
uniform  velocity,  the  actual  rate  of  progress  of  p  round 

*  See  the  Author's  Electric  Lighting  and  Power  Distribution,  Third 
Edition,  vol.  i.  §  153. 


15.] 


VARIATION  OF  MOTION  AND 


its  circular  path  will  also  be  uniform.  But  if  we  look 
at  p  from  one  of  the  sides  of  the  coil  ,  it  will  appear  to 
travel  up  and  down  in  a  straight  line,  a  b  (Fig.  29),  and 
its  rate  of  motion  in  an  actual  up  or  down  direction 
will  not  be  uniform.  When  p  has  travelled  round  10° 
from  its  topmost  position,  i.e.  from  p  to  pl  (Fig.  28), 
its  actual  progress  in  a  downward  direction  will  be  repre- 
sented by  the  distance  p  pl  in  Fig.  29,  which  is  relatively 


FIG.  28.   (View  from  end  of  coil.)          FIG.  29.    (View  from  side  of  coil.) 

much  less  than  the  circumferential  distance  ppl  in  the 
first  figure.  Another  10°  travel  is  from  p^  to  p2  (Fig.  28), 
from  p2  to  p3,  from  p3  to  p4,  and  so  on ;  and,  as  these 
distances  are  traversed  in  equal  times,  the  apparent 
velocity  of  p,  as  viewed  in  Fig.  29,  will  at  first  be  very 
slow,  and  will  gradually  increase  until  it  reaches  the 
90°  position,  pb.  From  p5  to  p6  its  apparent  velocity 
will  gradually  decrease.  The  same  thing  will  be 


50  ALTERNATING-CURRENT   CIRCUIT.  [§  15. 

observed  when  the  coil  is  making  its  second  half-turn — 
i.e.  when  p  is  travelling  from  _p6  back  again  to  its  top- 
most position.  Now,  the  E.M.F.  induced  at  p  depends 
upon  the  rate  at  which  it  cuts  the  lines  of  the  field,  and, 
supposing  the  field  to  be  uniform,  this  depends  upon  its 
rate  of  motion  in  an  actual  up  or  down  direction,  as 
viewed  in  Fig.  29.  It  therefore  follows  that  the  E.M.F. 
in  p  will  vary  just  as  the  rate  of  its  travel  along  the 
path  a  b  (Fig.  29)  varies,  and  it  will  change  from  zero  to 
a  maximum  during  the  first  quarter-turn  of  the  coil ; 
from  maximum  to  zero  during  the  second  quarter-turn  ; 
from  zero  to  maximum,  in  the  reverse  direction,  during 
the  third  quarter-turn ;  and  from  maximum  to  zero 
during  the  last  quarter-turn :  by  which  time  it  will 
have  made  one  complete  revolution.  The  other  half, 
c  d,  of  the  coil  (Fig.  27)  will  be  acted  upon  in  a  precisely 
similar  manner. 

Motion  of  the  kind  described  in  connection  with 
Fig.  29  is  called  harmonic,  and  obeys  a  simple  law 
called  the  sine  law.  This  can  be  explained  by  the  aid  of 
Figs.  30  and  31,  which  are  closely  related  to  the  two 
preceding  figures. 

Looking  at  the  coil  from  the  collector  or  front  end 
(Fig.  27),  the  path  described  by  the  point  p  (Fig.  30)  will 
be  a  circle,  having  its  centre  at  0,  p  0  being  its  zero  or 
starting  position,  and  1,  2,  3,  4,  5,  &c.,  successive  points 
on  its  journey  during  one  revolution  of  the  coil. 

The  sine  curve  or  curve  of  E.M.F.  is  plotted  as 
follows.  Take  a  horizontal  line,  such  as  that  marked 
TIME  BASE  (Fig.  31)  :  since  the  point  p  moves  with 


15.] 


THE  SINE   CURVE. 


51 


uniform  velocity 
round  its  circular 
path,  distances  mea- 
sured along  the  time 
base  may  be  taken 
to  represent  either 
'time  from  the  be- 
ginning of  measure- 
ment,' or  '  distance 
moved  by  p  round 
its  circular  path.'  p 
is  connected  to  its 
'  centre  of  travel '  or 
axis,  0,  by  the 
radius  R  of  the 
circle  in  which  it 
moves,  and  this  is 
clearly  the  greatest 
height  to  which  it 
can  rise,  as  in  posi- 
tion 0  4  (Fig.  30)  : 
we  therefore  take 
this  height  as  the 
maximum  height  for 
our  sine  curve  (Fig.  > 
31),  which  represents  CrJ  \ 
the  rise,  fall,  and 
reversal  of  E.M.F. 
The  radius  R  will 
make  an  angle  with 





E   2 


52  ALTERNATING-CURRENT  CIRCUIT.  [§  15. 

the  horizontal  diameter  of  the  circle,  which  will  begin  at 
zero  when  p  is  in  the  position  p  0,  and  will  increase  as  p 
travels  round  the  circle,  until,  at  position  4,  the  radius 
is  90°  from  its  original  position.  To  draw  the  E.M.F. 
curve,  we  must  first  take  a  length  along  the  time  base, 
and  call  it  360° :  this  may  conveniently  be  made  equal 
to  half  the  length  of  the  circumference  of  the  circle  in 
which  p  moves.*  This  length  is  then  equally  divided 
up,  and  we  get  a  straight  line  with  subdivisions  repre- 
senting the  distances  moved  by  p  along  its  circular 
path,  or,  what  is  the  same  thing,  the  angles  made  by 
the  radius  with  its  first  position  in  its  revolution  round 
the  centre  0 ;  and  these  divisions,  as  before  pointed  out, 
may  also  be  taken  to  represent  time.| 

Suppose  p  has  reached  the  point  1,  we  take  a 
distance  along  the  time  base  equal  to  half  the  circum- 
ferential distance,  0 1,  and  at  that  point  erect  a 
perpendicular  :  where  this  cuts  a  horizontal  line  drawn 
through  point  1  on  the  circle,  we  get  one  point  on  the 
curve.  In  the  same  way  for  position  2,  we  take  half  the 
distance  along  the  circumference  0  2,  and  mark  this 
off  on  the  time  base,  then  erect  a  perpendicular,  and 

*  Distances  along  the  time  base  are  proportional  to  circumferential 
distances,  and  may  be  drawn  to  any  scale.  In  the  present  case  they  are 
equal  to  half  the  circumferential  distances  which  they  represent,  this 
being  a  convenient  scale. 

f  If  p  has  moved  from  its  zero  position  to  position  2  (Fig.  30),  the 
radius  will  have  travelled  round  45°.  When  p  reaches  the  position  4 
the  radius  will  have  travelled  or  have  described  an  angle  of  90°.  When 
p  has  made  one  half-turn,  i.e.  when  it  has  reached  the  position  8, 
the  radius  may  be  said  to  have  travelled  180°  from  its  zero  position. 
When  p  has  made  one  complete  revolution,  we  say  that  its  radius  has 
travelled  round  or  described  an  angle  of  360°. 


§15.] 


SIN. 


58 


where  the  latter  cuts  a  horizontal  line  drawn  through  2 
on  the  circle,  we  get  the  second  point  on  our  curve. 
This  operation  being  repeated  for  different  positions  of 
p  round  its  circular  path  (3,  4,  5,  6,  etc.),  a  series  of 
points  is  obtained,  which,  when  connected,  are  found  to 
lie  on  a  wavy  line  called  the  sine  curve  (Fig.  31). 

This  curve  depends  upon  the  relationship  that  the 
distance,  H,  of  each  position  of  p  (above  or  below  the 
horizontal  line)  bears  to  the  radius,  R.  For,  the  greater 
H  is,  that  is,  the  greater  the  distance  p  is  above  or 
below  the  base  line,  the  more  effectively  is  it  cutting 
the  magnetic  lines  of  the  field,  and  the  greater  is  the 
E.M.F.  H  is  a  maximum  at  the  positions  4  and  12,  and 
these  are  consequently  the  maximum  points  on  the 
curve.  The  connection  between  H  and  R  is  as 
follows : — 

H  =  R  sin  A, 

where  A  is  the  angle  which  the  radius,  R,  makes  with 
the  horizontal  line,  in  the  par- 
ticular position  taken. 

The  sine  of  the  angle  A 
(Fig.  32)  (written  sin  A  or  sine 
A)  is  the  number  obtained  by 
dividing  the  length  of  the  per- 
pendicular or  height  H  by  the 
length  of  the  hypotenuse  (side 
opposite  the  right  angle)  or  third  Fm-  32- 

side  R  ;  in  this  case  the  radius  of  the  circle — i.e. 


54  ALTERNATING-CURRENT  CIRCUIT.  [§  15. 

this  ratio  being  dependent  on  the  angle  itself,  not  on 
the  individual  length  of  either  of  its  sides.* 

The  curve  obtained  shows  the  variation  in  the  E.M.F. 
of  a  simple  alternator,  such  as  that  illustrated  in  Fig.  27, 
for  one  revolution  of  its  coil  or  armature.  The  E.M.F.  is 
at  zero  when  the  plane  of  the  coil  or  armature  is  at  right 
angles  to  the  lines  of  force  of  the  field,  and  gradually 
rises,  reaching  a  maximum  when  the  plane  of  the  coil 
is  parallel  with  the  direction  of  the  field.  The  field  in 
this  case  is  assumed  to  be  uniform  ;  if  it  is  not  so,  the 
simple  sine  law  no  longer  holds  good,  and  the  E.M.F. 
curve  will  be  more  or  less  altered  in  form.  In  prac- 
tical alternators,  owing  to  the  non-uniformity  of  the 
fields,  and  the  various  shapes  of  coils  used,  the  form  of 
the  E.M.F.  curve  may  vary  considerably  from  that  of 
the  true  sine  curve.  The  design  of  alternators  has  been 

*  Let  ABC  (Fig.  33)  be  any  angle,  a,  of  which  the  sine  value  is 
required.  Take  any  point,  D,  in  either  side,  say  in  A  B,  and  drop 
therefrom  a  perpendicular,  D  E,  to  the  other  side,  B  C,  cutting  it  at  E. 
Then  BDE  will  be  a  right-angled  triangle,  of  which  BD  is  the 
hypotenuse,  and  D  E  the  perpendicular. 

Now,  in  such,  the  ratio  P^^dJcfOar   [e 

hypotenuse 

-r\  TTI 

— -,   represents  the   sine   value  of    the 
JD  D 

angle  a.    If  the  angle  remains  the  same 
(in  the  present  case  it  is  30°),  no  matter 
P       QQ  how  long  the  sides  B  D  or  B  E  may  be, 

or  from  which  point  or  side  the  perpen- 
dicular is  dropped,  the  ratio  ^Pendicular  wm  aiways  be  the  same. 

hypotenuse 

In  the  present  case,  for  instance,  it  is  f ,  i.e.  sin  30°  =  -5.  Sine  values 
may  be  directly  obtained  from  tables.  (Vide  the  Author's  Electric 
Lighting  and  Power  Distribution,  Third  Edition,  vol.  i.  §  104.) 


§  16.]   FREQUENCY,  ALTERNATION,  AND  PERIOD.    55 

brought  to  such  a  pitch  of  perfection  that  they  may  be 
made  to  give  a  true  sine  wave  of  E.M.F.,  or  one  which 
differs  in  respect  of  height,  breadth,  &c.,  according  to 
the  ideas  of  the  designer.  The  fact  of  thus  being  able 
to  obtain  variously  shaped  waves  of  E.M.F.  within 
certain  limits  is  of  importance ;  and  one  question  which 
naturally  arises  is,  what  is  the  most  efficient  form  of 
wave  for  a  given  circuit  ?  This  is  a  matter,  however, 
beyond  the  scope  of  this  book. 

16.  FREQUENCY. — The  E.M.F.  of  the  coil  shown  in 
Fig.  27  is  nothing  in  the  upright  position  there  depicted, 
but  gradually  increases  until  the  plane  of  the  coil  lies 
horizontal — i.e.  until  the  coil  has  moved  through 
90°  and  has  no  lines  through  it ;  it  then  gradually 
decreases,  reaching  zero  when  the  coil  has  made  one 
half-turn.  In  the  second  half-turn  the  E.M.F.  will  again 
gradually  rise  and  fall,  but  this  time  in  the  reverse 
direction.  This  rise,  fall,  and  reversal,  and  the  corre- 
sponding distance  travelled  by  the  coil,  are  shown  in 
Figs.  30  and  31. 

If  the  coil  is  connected  up  with  an  outer  circuit,  in 
one  revolution  the  induced  E.M.F.  and  resulting  current 
will  make  two  alternations,  or  one  complete  period  or 
cycle  ;  and  the  rate  of  double  alternations  per  second,  or 
number  of  complete  periods  or  cycles  per  second,  which  is 
termed  the  frequency  or  periodicity,  will  depend  upon  the 
number  of  revolutions  which  the  coil  makes  in  that 
time.  Thus,  supposing  it  revolves  600  times  in  one 
minute,  the  frequency  of  the  E.M.F.  and  of  the  current 
set  up  will  be  10, 


56  ALTEENATING-CUEEENT  CIRCUIT.  [§  17. 

Frequency  is  denoted  by  the  symbol  ^ ,  thus  70  ^ 
signifies  a  pressure  or  current  making  70  complete 
periods  per  second — i.e.  having  a  frequency  of  70. 

The  frequency  of  alternating  currents,  as  used  for 
ordinary  work  in  this  country,  varies  from  40  /•*  to 
130  ~,  the  present  tendency  in  central  station  work 
being  to  reduce  it  to  something  like  50  ^  or  60  /•*.  For 
special  purposes,  E.M.Fs.  of  very  much  higher  fre- 
quency are  sometimes  employed. 

The  rise  and  fall  of  the  current  in  one  direction 
should  be  called  an  alternation ;  but  this  term  is  some- 
times employed  to  indicate  a  complete  reversal — i.e.  a 
period  or  cycle,  a  disagreement  which  is  somewhat 
confusing.  Keferring  to  Fig.  31,  the  portion  of  the 
curve  from  0  to  180  is  really  an  alternation,  and  the 
portion  from  0  to  360  a  cycle  or  period,  and  the 
symbol  for  frequency  (^),  being  derived  from  the  shape 
of  the  curve,  should  assist  the  student  in  remembering 
this.  An  alternation  is,  as  its  name  indicates,  an  alterna- 
tive wave  or  alteration  in  direction.  Thus  a  frequency  of 
80  ^  means  80  periods,  or  160  alternations  per  second. 

17.  FREQUENCY  OF  ALTERNATORS. — In  the  case  of  a 
simple  coil  rotating  in  a  2-pole  field,  it  was  shown 
that  the  frequency  is  proportional  to  the  number  of 
revolutions  per  second  (§15).  Practical  alternators  are, 
with  few  exceptions,  constructed  with  multipolar  field- 
magnets,  as  well  as  a  number  of  coils  :  but  the  frequency 
is  got  by  simply  multiplying  together  the  revolutions 
per  second  and  the  number  of  pairs  of  poles,  a  con- 
sequent pole  counting  as  a  single  pole. 


§  18.]  VIRTUAL  E.M.F.  AND   CURRENT.  57 

Example.  —  An  alternator  has  12  pairs  of  poles 
(2V  and  S),  and  runs  at  300  revolutions  per  minute. 
Each  coil  will  pass  through  12  fields  in  one  revolution 
—  i.e.  there  will  be  12  complete  reversals  or  waves  of 
E.M.F.  (~)  in  each  revolution.  Consequently,  the  re- 
sulting frequency  will  be  :  — 


12  x          =  12  x  5  =  60  ~. 
oO 

18.  VIBTUAL  VOLTS  AND  AMPEKES.  —  The  E.M.F.  of 
a  practical  alternator  is  constantly  rising,  falling,  and 
reversing,  in  much  the  same  manner  as  described  in 
§  15  ;  and  the  current  in  the  circuit  must  rise,  fall,  and 
reverse  in  sympathy  though  not  necessarily  in  step  with 
the  E.M.F.  (§  19). 

It  is  clear  that  we  cannot  take  the  maximum  points 
of  the  pressure  or  current  wave  as  the  nominal  value,  for 
the  pressure  or  current  are  only  at  these  maxima  for 
comparatively  short  periods.  What  is  rightly  called  an 
alternating  E.M.F.  of,  say,  100  volts,  must  at  some 
times  be  considerably  above  100  volts,  and  at  other 
times  at  zero.  Similarly,  an  alternating  current  of, 
say,  10  amperes,  is  at  times  greater  than  10  amperes, 
and  at  others  less.  We  must  take  a  value,  called  the 
virtual  value,  which  is  equivalent  to  that  of  a  direct 
E.M.F.  or  current  which  would  produce  the  same  effect  : 
and  those  effects  of  the  E.M.F.  and  current  are  taken 
which  are  not  affected  by  rapid  changes  in  direction  and 
strength  ;  in  the  case  of  E.M.F.  or  pressure  —  the  read- 
ing on  an  electrostatic  voltmeter  ;  and  in  the  case  of 
current  —  the  heating  effect. 


THK 


58  ALTERNATING-CURRENT  CIRCUIT.  [§18. 

Thus,  a  virtual  E.M.F.  of  100  volts  is  one  that 
would  produce  the  same  deflection  on  an  electrostatic 
voltmeter  as  a  direct  E.M.F.  of  100  volts  ;  and  a  virtual 
current  of  5  amperes  is  that  current  which  would  pro- 
duce the  same  heating  effect  as  a  direct  current  of 
5  amperes — say,  on  a  'bank'  or  group  of  incandescent 
lamps  :  but  both  pressure  and  current  will  be  continually 
varying  above  and  below  these  values. 

Neglecting  the  effects  produced,  the  virtual  value  of 
an  alternating  E.M.F.  or  current  having  a  sine  curve 
form  (Fig.  31)  is  about  *707  of  its  maximum  value. 
For  example,  an  E.M.F.  which  alternates  between 
maximum  values  of  100  volts  in  one  direction,  and 
100  volts  in  the  other,  will  have  a  virtual  value  of  about 
70*7  volts.  Similarly,  a  current  which  alternates  between 
10  amperes  in  one  direction,  and  10  amperes  in  the  other, 
will  have  a  virtual  value  of  about  7*07  amperes.  The 
reciprocal*  of  *707  is  1*41,  so  that  if  any  virtual  value 
of  pressure  or  current  be  multiplied  by  this  number,  the 
product  will  give  the  approximate  maximum  value. 
Thus,  a  virtual  alternating  pressure  of  220  volts  alter- 
nates between  (220  x  1-41  =)  310  volts  in  one  direc- 
tion, and  310  volts  in  the  other ;  and  a  virtual  current  of 
50  amperes  alternates  between  maxima  of  (50  x  1*41  =), 
say,  70  amperes  in  one  direction,  and  70  amperes  in  the 
other  direction. 

*  The  reciprocal  of  any  number,  n,  is  obtained  by  dividing  it  into 
unity — i.e.  reciprocal  of  n   =  -.      Thus,  reciprocal  of  -707   =    -— 

=  1*41442 . . .,  or,  say,  1*41.     The  product  of  any  number  multiplied 
into  its  reciprocal  is  unity  :  thus,  -707  x  1-41442 ...  =  1. 


§§19,20.]  AMPLITUDE,  PHASE,  LAG,  AND    LEAD.    59 

A  given  virtual  alternating  pressure  throws  more 
strain  on  the  insulation  of  a  circuit  than  a  direct  pressure 
of  the  same  value  (§  12) ;  and  in  this  connection  it  should 
be  remembered  that,  as  we  have  just  pointed  out,  any 
given  virtual  pressure  fluctuates  between  values  nearly 
half  as  high  again  as  its  virtual  value.  If  the  wave  of 
pressure  differs  from  the  sine  curve  form — a  matter 
which  depends  on  the  design  of  the  alternator,  as 
mentioned  at  the  end  of  §  15 — the  maxima  may  be  as 
much  as  twice  the  virtual  values. 

The  difference  between  virtual  and  effective  values  of 
pressure  and  current  is  explained  in  §  29. 

19.  AMPLITUDE   AND    PHASE. — The  amplitude   of   an 
impressed  (virtual)  alternating  E.M.F.  or  current  is  the 
maximum   value  or   height   of  each   wave.      Thus,  in 
Fig.  31,  the  distances  a  a  represent  the  amplitude  of  the 
waves  of  E.M.F. 

Both  E.M.F.  and  current  suffer  periodic  changes  of 
strength,  that  is,  they  pass  through  different  phases  or 
states.  If  we  take  a  case  where  the  current  rises,  falls, 
and  reverses  exactly  at  the  same  time  as  the  E.M.F., 
the  current  would  then  be  said  to  be  in  phase  or  in  step 
with  the  E.M.F. ;  but,  as  already  explained,  this  is  not 
always  so,  the  current  wave  being  more  often  out  of 
phase  with  the  E.M.F.  wave,  owing  to  the  effects  of 
inductance  and  capacity.  The  frequency  of  the  current 
is,  however,  always  the  same  as  that  of  the  impressed 
E.M.F. 

20.  LAG  AND  LEAD. — It  was   explained  in  §  4  that 
the   effect  of  inductance  in  a  circuit  is   to  cause  the 


60  ALTERNATING-CURRENT   CIRCUIT.  [§20. 

current  to  take  time  to  '  grow,'  and  time  to  die  away. 
In  fact,  the  current  does  not  generally  start  till  after 
the  E.M.F.  has  been  impressed  on  the  circuit,  and  does 
not  stop  until  after  the  E.M.F.  has  been  stopped  or 
reversed.  Inductance  in  an  alternating-current  circuit 
consequently  causes  the  wave  of  current  to  lag  behind 
the  wave  of  E.M.F.  This  is  depicted  in  Fig.  34,  where 
the  dotted  curve,  P,  represents  the  E.M.F.  or  pressure 
wave ;  and  the  other  curve,  (7,  the  current  wave.  Start- 


FIG.  34. 

ing  from  the  left-hand  end  of  the  horizontal  line  or 
time  base,  it  will  be  seen  that  the  current  starts  after 
the  E.M.F.  starts,  and  reverses  after  the  E.M.F.  re- 
verses, and  so  on.  In  other  words,  the  current  lags  in 
phase  behind  the  E.M.F.,  although  its  frequency  is 
exactly  the  same. 

The  amount  of  the  lag  is  measured  in  degrees  as 
set  out  along  the  time  base  (Fig.  31).  Thus,  in  Fig.  84, 
the  lag  is  indicated  by  the  distance,  d,  between  the 
beginning  of  the  pressure  curve  and  the  beginning  of 
the  current  curve,  and  is  in  this  case  about  70°.  The 
lag  due  to  inductance  may  be  anything  up  to  90°  (a 
quarter  period),  but  cannot  exceed  this. 


§  20.]  LEAD.  61 

The  effect  of  capacity  in  a  circuit  is  generally  said  to 
cause  the  current  to  lead  in  phase,  but  this  effect  is 
rather  difficult  to  conceive,  though  we  will  endeavour  to 
explain  it  by  means  of  a  mechanical  analogy,  such  as  has 
already  been  employed.  Apply  an  alternating  twisting 
force  to  the  top  of  the  wire,  T  (Fig.  35)  ; 
the  action  of  the  spring,  S,  being  taken  to 
represent  the  effect  of  capacity,  and  the 
movement  of  the  vane,  V,  movement  of 
electricity  or  current.  On  commencing 
the  experiment,  of  course  the  twisting 
force  (E.M.F.)  must  first  be  applied  before 
the  rotation  (current)  starts;  but  after  a 
time,  though  it  will  be  difficult  to  discern, 
the  resiliency  or  rebounding  effect  of  the 
spring  acts  so  as  to  cause  the  vane,  F,  to  move  in 
advance  of  the  twisting  force  (E.M.F.),  thus  representing 
the  current  leading  in  phase.  The  explanation  of  the 
effect  of  capacity,  as  given  in  §§  8  and  10,  will  also 
assist  the  reader  to  understand  what  is  meant  by  the 
term.  As  a  general  rule,  alternating  currents  lag  more 
or  less  in  phase,  as  the  inductance  usually  greatly  pre- 
ponderates over  the  capacity ;  but,  on  very  long  lines,  or 
by  purposely  introducing  capacity  into  a  circuit,  the  lag 
may  be  neutralised  or  even  exceeded  by  the  lead,  and 
the  current  will  then  be  either  in  phase  with  the  pressure, 
or  it  may  lead  in  phase. 

There  has  been  some  objection  to  the  terms  *  lead  of 
current '  or  '  lead  in  phase,'  principally  on  the  ground 
that  they  tend  to  convey  the  idea  that  the  effect  pre- 


62  ALTERNATING-CURRENT  CIRCUIT.  [§20. 

cedes  the  cause — i.e.  that  the  current  is  in  advance  of 
the  E.M.F.  causing  it.  The  latter  is  true  in  one  sense, 
but  untrue  in  another.  Of  course,  there  can  be  no  flow 
of  electricity  in  a  circuit  until  E.M.F.  has  been  applied; 
but  if  the  circuit  has  capacity,  and  supposing  firstly 
that  a  direct  E.M.F.  is  applied,  the  current  will  on 
starting  be  momentarily  greater  than  the  ultimate 
steady  current ;  and  it  will  again  be  momentarily 
greater  on  stopping  the  E.M.F. 

In  §  3,  we  likened  an  electrical  circuit  to  a  pipe  filled 
with  water  :  this  analogy  may  be  extended  by  supposing 
that  an  electrical  circuit  with  capacity  is  like  a  pipe 
circuit  only  partially  filled  with  water :  then  when 
watermotive  force  is  applied — for  instance,  when  the 
connection  of  the  circuit  with  a  cistern  or  reservoir  is 
established  by  opening  a  tap — there  will  be  a  rush  of 
water  (till  the  pipe  is  filled  up)  that  will  be  greater  than 
the  ultimate  steady  flow.  The  hydraulic  circuit,  how- 
ever, does  not  offer  a  good  analogy  for  the  electric  circuit 
when  capacity  is  taken  into  account.  The  capacity  of  a 
rigid  pipe  for  water  is  fixed,  whereas  the  capacity  of  a 
conductor  for  electricity  depends  upon  its  surroundings, 
and  on  the  E.M.F.  or  P.D.  applied.*  The  illustration 
just  put  forward  will  serve  to  give  the  reader  an  idea  as  to 
how  capacity  may  be  said  to  '  suck '  the  current  out  in 
advance  of  a  direct  E.M.F.,  but  does  not  afford  a  parallel 
for  the  discharge  flow,  or  for  the  action  with  an  alter- 
nating E.M.F. 

*  See  the  Author's  First  Book  of  Electricity  and  Magnetism,  Second 
Edition,  §  159. 


21.] 


REACTANCE. 


68 


When  the  direct  E.M.F.  is  cut  off,  the  direction  of 
the  capacity  or  condenser  current  of  discharge  may  be 
roughly  said  to  be  opposite  to  the  charging  current 
when  the  capacity  is  in  series  with  the  circuit  as  in 
Figs.  6,  7,  8,  etc.,  but  in  either  or  both  directions  when 
the  capacity  is  in  parallel  with  the  circuit,  as  in  Figs.  11, 
16,  &c. 

The  lead  of  current  due  to  capacity  in  an  alternating- 
current  circuit  is  best  illustrated  by  the  mechanical 
analogies  given  at  the  beginning  of  this  paragraph  and 


FIG.  36. 

in  §  10,  while  a  partial  explanation  is  given  in  §§  6,  8, 
and  28.  Fig.  36  represents  the  current  curve,  C  C  C, 
leading  in  advance  of  the  pressure  curve,  P  P  P. 

Lag  and  lead  are  further  dealt  with  in  §§  31  and  32. 

21.  EEACTANCB. — The  resistance  offered  by  a  con- 
ductor to  a  steady  flow  of  electricity  is  expressed  in 
ohms  ;  and  this  value  is  the  same  whether  the  conductor 
is  coiled  up  or  stretched  out,  and  is  unaffected  by  the 
presence  of  neighbouring  conductors.  With  a  constantly 
changing  current,  such  as  an  alternating  one,  the  ap- 
parent resistance  offered  to  its  flow  is  greater  if  the 
circuit  conductor  be  coiled  up  than  if  it  is  straight,  is 
affected  by  the  presence  of  neighbouring  conductors, 


64  ALTERNATING-CURRENT  CIRCUIT.  [§22. 

and  also  depends  upon  the  frequency.     In  short,  Ohm's 
simple  law  cannot  be  applied  to  alternating-current  work. 

The  cause  of  this  apparent  additional  resistance  in 
the  circuit  is  the  combined  effect  of  inductance  and 
mutual  induction,  and  is  called  reactance.*  The  in- 
ductance is  increased  by  the  presence  of  electro-magnets 
or  coils  of  wire  in  the  circuit,  but  is  decreased  by 
capacity ;  while  the  mutual  induction  depends  upon  the 
presence  of  neighbouring  conductors  ;  and  their  com- 
bined effect  —  i.e.  the  reactance — increases  with  the 
frequency.  Keactance  constitutes  a  kind  of  spurious 
resistance,  over  and  above  the  ordinary  or  ohmic  resist- 
ance (§  30). 

22.  KEACTANCE  AND  IMPEDANCE. — Impedance  is  the 
'  virtual '  or  '  effective '  resistance  offered  to  the  flow  of 
an  alternating  current ;  and,  from  what  was  said  in  the 
preceding  paragraph,  is  clearly  the  combined  effect  of 
the  ohmic  and  spurious  resistance  in  a  circuit;  or  in 
other  words  : — 

-p      ,  inductance,  mutual  induction,  and  frequency. 

capacity. 

And:- 

Impedance  oc  resistance  and  reactance.f 

The  two  terms  reactance  and  impedance  must  not  be 
confused. 

*  The  term  inductance  was  originally  introduced  to  take  the  place 
of  self-induction.  Some  few  writers  extend  its  meaning,  and  make  it 
include  mutual  induction  also  ;  the  terms  self -inductance  and  mutual- 
inductance  are  consequently  sometimes  employed. 

f  The  sign  oc  signifies  '  is  proportional  to.' 


§23.]  REACTANCE  AND  IMPEDANi 

It  should  be  easy  to  remember  that  reactancexQfews 
only  to  the  reactive  effects  in  the  circuit,  or  what  is 
otherwise  called  the  '  spurious  resistance' — i.e.  an  extra 
resistance  brought  about  when  the  flow  of  electricity  is 
not  steady  ;  whereas  impedance  implies  the  virtual  or 
effective  or  total  resistance  which  impedes  the  flow  of 
an  alternating  current  of  electricity. 

The  connection  between  resistance,  reactance,  and 
impedance  is  further  explained  in  §  30. 

23.  DIFFERENT  ACTION  OF  RESISTANCE  AND  REACT- 
ANCE ON  CURRENT.  CHOKING  COILS. — There  is  a  very 
important  difference  in  the  obstruction  offered  to  an 
alternating  current  by  ordinary  resistance  and  by  re- 
actance, as  the  reader  will  have  observed  in  performing 
the  experiments  mentioned  in  §§11  and  14.  Resistance 
obstructs  the  current  by  dissipating  its  energy,  which  is 
converted  into  heat.  Reactance,  on  the  other  hand, 
obstructs  the  current  by  setting  up  an  alternating 
E.M.F.  in  opposition  to  the  impressed  E.M.F.,  and 
so  reduces  the  effective  current  in  the  circuit  ivithout 
wasting  much  energy,  except  by  hysteresis  in  any  iron 
magnetised.* 

This  may  be  regarded  as  one  of  the  advantages  of 
alternating  over  direct  currents,  for,  by  introducing 
reactance  into  a  circuit,  we  can  cut  the  current  down 
with  comparatively  little  loss  of  energy.  This  is  generally 
done  by  increasing  the  inductance  in  a  circuit,  and  con- 
sequently also  its  reactance  and  impedance,  by  means  of 

*  See  the  Author's  Electric  Lighting  and  Power  Distribution,  Third 
Edition,  vol.  i.  §  100. 

F 


66 


ALTERNATING-CURRENT  CIRCUIT. 


'23. 


a  device  called  variously  a  reactance  coil,  impedance  coil, 

choking  coil,  or  *  choker.' 

Figures  37,  38,  and  39  illustrate  the  principle  of 
choking  coils.  In  Fig.  37,  C 
is  a  coil  of  thick  wire  provided 
with  a  laminated  iron  core,  I  C, 
which  may  be  either  fixed  or 
movable.  In  the  first  case, 
the  inductance,  and  therefore, 
also  the  reactance  of  the  coil, 
is  invariable,  with  a  given  fre- 
quency :  in  the  second  case, 
the  inductance  and  consequent 
FieK  37-  reactance  may  be  respectively 

increased  or  diminished  by  inserting  the  core  farther 

within  the  coil  or  by  withdrawing  it. 

In  Fig.  38,  C  is  a  coil  of  thick  wire  with  a  fixed 

laminated  iron  core,  1C,  and  a  movable  thick  copper 


FIG.  38. 


sheath  or  sleeve,  C  S.    When  C  S  is  apart  from  C,  the 
latter    will    have    its    maximum    inductance— i.e.    its 


§23.] 


CHOKING  COILS. 


greatest  choking  effect :  but  this  will  decrease  as  C  S  is 
slipped  more  or  less  on  to  C.  When  C  S  is  placed  over  Ct 
mutual  induction  takes  place  between  C  and  C  S,  the 
latter  forming  a  closed  secondary  circuit.  The  E.M.F. 
due  to  the  inductance  of  the  coil  C  will  then  expend  more 
or  less  of  its  energy  in  setting  up  currents  in  C  S, 
instead  of  in  weakening  the  current  in  the  main  circuit. 
The  sheath,  C  S,  however,  also  tends  to  absorb  some  of 
the  energy  of  the  current  flowing  through  C ;  hence  a 
choking  coil  on  the  first-described  principle  (Fig.  37)  is 
more  generally  used. 

The  choking  coil  depicted  in  Fig.  38  is  virtually  a 
small  transformer,  of  which  C  is  the  primary  coil,  and 
C  S  the  secondary  coil.  Now  the  copper  sheath,  C  S,  has 
very  little  resistance,  and  the  currents  set  circulating  in 
it — which  represent  energy  transferred  from  the  primary 
circuit,  C — are  comparatively  large.  If  we  could  in- 


FlG. 


crease  or  diminish  the  resistance  of  C  S  at  will,  instead 
of  slipping  it  on  or  off  C,  we  should  be  equally  well  able 
to  regulate  the  choking  effect  of  the  apparatus  as  a  whole. 
This  is  sometimes  done  in  practice,  as  diagrammatically 
represented  in  Fig.  39,  where  I C  is  a  laminated  iron  core, 

F2 


68  ALTERNATING-CURRENT   CIRCUIT.          [§24. 

on  which  are  wound  the  fixed  primary  and  secondary 
coils  P  and  S.  P  is  in  the  main  circuit,  and  joined  up 
to  S  is  an  adjustable  resistance,  R,  and  some  kind 
of  sliding  contact,  C,  by  which  the  amount  of  R 
may  be  increased  or  diminished.  The  iron  core,  1C, 
may  or  may  not  be  movable.  Supposing,  first  of  all, 
that  it  is  fixed.  The  greater  the  resistance  of  the 
secondary  circuit,  SEC,  the  smaller  will  be  the  currents 
induced  therein,  and  the  less  the  energy  of  inductance 
absorbed  from  the  primary  circuit,  P :  consequently, 
when  R  is  small,  the  least  choking  effect  will  be  exercised, 
but  as  R  is  increased  the  choking  effect  will  increase. 
If  I  C  is  movable,  the  choking  effect  may  be  further 
diminished  or  increased  by  respectively  withdrawing  or 
inserting  it. 

24.  PRACTICAL  FORMS  OF  CHOKING  COILS. — All  the 
'  chokers  '  described  here  belong  to  the  class  depicted  in 
Fig.  37 — i-e.  they  consist  of  one  winding  with  a  movable 
or  fixed  core.  Choking  coils  acting  on  the  principle 
shown  in  Fig.  38  are  used  in  America,  but  besides 
being  less  efficient,  as  pointed  out  in  the  last  paragraph, 
they  are  also  more  expensive  in  construction.*  For  the 
same  reasons,  the  principle  shown  in  Fig.  39  is  not 
altogether  satisfactory  when  the  choking  coil  is  in 
circuit  for  hours  at  a  time. 

Fig.  40  shows  a  choking  coil  for  heavy  work,  as 
made  by  Messrs.  Johnson  &  Phillips.  The  coil  consists 
of  one  winding  in  two  sections,  the  bobbin  being  divided 

*  The  sliding  brass  tube  regulator  used  in  some  medical  coils  is  of 
this  type,  and  works  on  the  principle  enunciated. 


§24.]  CHOKING   COILS. 

midway     by    an     insulating 
'  cheek.'      A    guide-tube    of 
'  presspahn '  *  is  fixed  to  the 
top   of  the  bobbin,    and    in 
this    slides    the    core.      The 
latter  is  made  of  a  bundle  of 
fine  iron  wires  securely  bound 
together ;  it  is  hung  at  one 
end   of  a   steel  cord,   which 
makes    a    couple    of    turns 
round   a  pulley,   and   termi- 
nates in  a  counterweight ;  the 
cord  being  fixed  at  one  point 
to  the  pulley,  so  that  it  cannot 
slip  thereon.     A  sensitive  ad- 
justment is  secured,  the  hand- 
wheel  operating  a  worm  which 
gears  into  a  spur-wheel  fixed 
alongside    the   pulley.      The 
latter  may  be  locked  in  any 
required  position  by  means  of 
the  small  bolt  at  the  right- 
hand  end  of  its  spindle.    The 
terminals  of  the  coil  are  at  the 
back  of  the  wooden  stand,  the 
switch  at  the  top  being 
so  connected  as  to  short- 
circuit    it    if    required. 

*  Presspahn  is  a  material 
made  of  wood  fibre.  It  is 
cheaper  than  vulcanite,  and 
more  durable  than  pasteboard. 

FIG.  40. 


70 


ALTERNATING-CURRENT   CIRCUIT. 


[§24. 


The  height  of  the  stand  is  4  ft.  10  in.,  the  particular 
size  shown  carrying  a  maximum  current  of  15  amperes, 
and  choking  the  P.D.  of  the  circuit  in  which  it  is  fixed 
down  from  1,400  to  200  volts. 

Fig.  41  gives  an  outline  of  a  choking  coil  made  by 
Messrs.  Crompton  &  Co.,  the  main  difference  between 
this  and  the  one  just  described  being  in  the  method  of 


H 


Fm.  41. 


adjustment,  and  the  absence  of  a  counterweight.  The 
coil  C>  and  iron  core  I C,  are  both  great  in  diameter,  as 
compared  with  length,  the  core  thus  having  to  pass 
through  only  a  relatively  small  distance  to  secure  a 
large  difference  of  effect.  The  core,  I  C>  is  made  up  of 


§24.]  CHOKING  COILS.  71 

fine  soft  iron  wires  bound  together,  and  is  fastened  to 
one  end  of  a  steel  band  which  passes  over  the  pulley  P : 
the  other  end  of  this  steel  band  is  secured  to  the  cross- 
piece  C  P,  which  travels  along  the  two  guide  rods,  GR. 
The  horizontal  lines  on  I C  represent  the  binding  round 
the  iron  wires,  the  latter  running  of  course  in  a  per- 
pendicular direction.  The  handwheel  H,  on  the  front 
of  the  switchboard,  turns  the  screwed  spindle  S  S, 
which  is  tapped  into  C  P ;  and  according  to  the 
direction  in  which  H  is  turned,  so  C  P  moves  either  to 
the  right  or  to  the  left,  and  I C  is  withdrawn  from  or 
dropped  further  into  the  coil.  Of  course  the  design  of 
the  coil  and  the  method  of  adjustment  may  be  altered  to 
suit  different  circumstances. 

A  *  choker '  for  use  on  arc-lamp  circuits,  also  made  by 
this  firm,  consists  of  a  bobbin  about  9^  in.  long,  wound 
with  a  single  coil  of  wire,  and  provided  with  a  movable 


FIG.  42. 

rectangular  core  made  up  of  thin  flat  strips  of  soft  iron. 
It  thus  resembles  Fig.  37. 

Fig.  42  shows  a  choking  coil,  or  rather  a  collection 
of  choking  coils,  mounted  in  a  cast-iron  case.  Each 
separate  '  choker  '  has  two  coils  mounted  on  a  laminated 
core,  as  illustrated  in  Fig.  43.  These  cores  cannot  be 


72 


ALTERNATING-CUEEENT  CIRCUIT. 


[§24 


seen  in   Fig.    42,    as   strips   of  vulcanised    fibre    are 
placed  between  their  ends  and  the  holding-down  bolts. 

The  coils  are  rela- 
tively small,  as  the 
case  which  holds  them 
is  only  2  feet  long. 
This  apparatus  was 
made  by  the  Electric 
Construction  Corpora- 
tion for  the  system  of 
street  lighting  adopted 
at  Lagos,  W.  Africa, 
where  50-c.p.  incandescent  lamps  are  run  in  series 
circuits  off  constant  potential  mains.  A  sketch  of  the 
connections  is  given  in  Fig.  44,  where  it  will  be  seen 
that  the  choker  (or  rather  chokers)  are  joined  up  with 
a  multiple-contact  regulating  switch.  Each  lamp  is 
provided  with  an  automatic  short-circuiting  cut-out,  and 


FIG.  43. 


CHOKING  COIL 


MAIN  SWITCH 


LAMPS 

-X — X  —  X-  — 


FIG.  44. 


should  one,  two,  or  more  of  them  fail,  a  corresponding 
number  of  sections  of  the  choking  apparatus  is  put  in 


24.] 


CHOKING  COILS. 


circuit  to  take  the  place  of  the  broken  lamp  or  lamps, 
and  thus  keep  the  current  constant.  It  must  not  be 
supposed  that  this  ar- 
rangement of  lamps, 
&c.,  is  a  general  one ; 
it  being  adopted  to  suit 
certain  special  condi- 
tions. The  matter  is 
cited  as  illustrating  an 
application  of  choking 
coils. 

Another  type  of 
choking  coil,  made  by 
Messrs.  Miller  &  Woods 
for  very  light  work, 
consists  of  a  fixed  core 
and  coil,  the  turns  of 
the  latter  in  circuit 
being  varied  by  means 
of  a  sliding  contact.  A 
diagram  of  this  ar- 
rangement is  given  in 
Fig.  45,  and  an  exterior 
view  in  Fig.  46.  IE 
is  a  laminated  iron 
ring  built  up  of  soft 
iron  ribbon :  on  this,  'hut  well  insulated  therefrom,  is 
a  coil  of  thickly  covered  copper  wire,  one  end  of  which 
E  is  free,  i.e.  unconnected  with  anything,  while  the 
other  is  joined  to  terminal  T.  I  R  is  closely  wound 


SUPPLY  MAINS 


FIG.  45. 


74 


ALTERNATING-CURRENT   CIRCUIT. 


[§24, 


with  the  wire — i.e.  the  turns  lie  close  side  by  side,  not  as 
shown  in  the  figure.  Pivoted  in  the  centre  of  the  ring 
and  operated  by  a  handle  H,  is  a  brass  arm  A,  the  end 
of  which  bends  over  and  makes  contact  with  the  turns 
of  wire  on  the  outer  edge  of  the  ring,  the  insulating 
covering  being  scraped  off  for  this  purpose,  after  the 
wire  is  wound  on  the  ring,  without,  however,  short- 


FIG.  46. 

circuiting  the  neighbouring  turns.  A  is  connected  with 
terminal  T't  and  the  figure  shows  the  connection  of  the 
choking  coil  to  the  mains,  with  one  lamp  L  in  circuit. 
If  the  arm  A  is  in  the  dotted  position  A'  no  current  will 
flow.  By  turning  H  one  way  or  the  other,  more  or  less 
of  the  turns  of  wire  will  be  put  in  circuit  with  the  lamp, 
and  the  latter  will  give  less  or  more  light.  It  may 


§25.]  USE  OF  CHOKING  COILS.  75 

appear  that  the  actual  resistance  of  the  turns  put  in 
circuit  has  something  to  do  with  the  cutting  down  of  the 
current ;  but  if  the  apparatus  is  well  designed,  the  resist- 
ance of  the  whole  of  the  coil  shoulfl^ 
be  such  that  if  it  were  all  put  in 
circuit  with  the  lamp  without  its  iron 
core,  i.e.  without  appreciable  react- 
ance, there  should  be  very  little  effect 
on  the  brightness  of  the  lamp.  For  if 
the  coil  has  much  resistance  as  well  as 
reactance,  energy  will  be  absorbed  in 
heating  the  coil,  and  the  current  will 
not  be  cut  down  without  material 
waste,  the  primary  object  of  a  choking 
coil.  An  exterior  view  (about  half- 
size)  is  given  in  Fig.  46,  where  will  be 
seen  the  handle  and  the  terminals  (H 
and  TTf  in  Fig.  45). 

These  choking  coils  are  suitable  for 
regulating  a  single  or  even  two  or 
three  lamps,  but  cannot  be  used  for 
large  currents.  Fig.  47  shows  this 
apparatus  or  regulating  sivitch  as  it  is 
sometimes  called  (for  it  acts  both  as 
switch  and  regulator),  fixed  in  con- 
junction with  a  glow  lamp. 

25.  USE  OF  CHOKING  COILS. — It  has  been  shown  that 
choking  or  impedance  coils  are  made  in  many  different 
forms ;  and  their  use  is  to  cut  down,  '  choke,'  or 
'  throttle '  the  current  in  a  circuit  or  portion  of  a 


76  ALTERNATING-CURRENT   CIRCUIT.          [§26. 

circuit :  the  principle  of  their  action  being  illustrated 
by  the  experiment  mentioned  in  §  11.  In  electric-light 
work  for  instance,  a  glow  lamp  or  group  of  lamps  may 
be  '  turned  down  '  or  '  dimmed '  to  any  desired  extent 
by  operating  a  choking  coil  in  its  or  their  circuit.  In 
theatres,  music-halls,  churches,  etc.,  where  there  are  a 
large  number  of  inaccessible  lamps  to  be  simultaneously 
raised  or  lowered  in  brilliancy,  the  use  of  choking  coils 
is  of  great  advantage.  In  ordinary  house  work,  an 
alteration  in  the  light  is  generally  effected  by  simply 
turning  lamps  on  or  off,  though  a  choking  coil  such  as 
is  shown  in  Fig.  47,  is  useful  in  some  cases — e.g.  in  bed- 
rooms, etc.  Of  course  this  turning  down  or  lowering  of 
the  lights  could  be  effected  by  simply  inserting  ordinary 
resistance  in  the  circuit,  but  here,  as  previously  ex- 
plained, much  of  the  energy  taken  from  the  lamps  would 
be  expended  in  heating  the  resistance,  whereas  by  using 
impedance  coils  the  current  is  cut  down  with  very  little 
waste. 

Choking  coils  are  also  used  for  regulating  purposes  in 
central  station  work. 

26.  *  SKIN  EESISTANCE  '  OB  CONDUCTOR  IMPEDANCE. — 
When  a  direct  current  begins  to  traverse  a  conductor,  it 
commences  to  flow  first  at  the  surface,  and  then  at  last 
penetrates  to  the  interior :  on  stopping  it,  it  leaves  off 
first  at  the  surface  and  lastly  in  the  interior.  This 
effect  is  due  to  the  inductance  of  the  conductor,  and  may 
be  explained  as  follows.  Imagine  the  conductor  to  con- 
sist of  a  number  of  separate  small  insulated  wires 
packed  closely  together  side  by  side  (Fig.  48) ;  now,  when 


§26.]  CONDUCTOE  IMPEDANCE.  77 

a  current  is  started  along  these  separate  wires,  mutual 
induction  will  take  place  between  them,  and  momentary 
reverse  E.M.Fs.  will  be  set  up  therein ;  but  clearly 
those  wires  which  are  nearer  the  centre  and  conse- 
quently completely  surrounded  by  neighbouring  wires 
will  have  stronger  reverse  E.M.Fs.  set  up  in  them  than 
those  on  or  near  the  outer  surface,  so  that  a  direct 
current  will  find  less  momentary  opposition  to  it  near 
the  surface  than  in  the  interior  of  the  conductor,  and 
thus  a  direct  current  may  be  said  to  flow  first  at  the 
surface  and  lastly  evenly  through  the  whole  section  of 


OOOQOnnrXXXXXXXX)QO 
OOOOOOOOuuuOOOOOOOO 


FIG.  48. 

the  conductor  ;  the  time  occupied  in  settling  down  being 
of  course  only  a  fraction  or  so  of  a  second.  When  the 
steadily  flowing  direct  current  is  suddenly  stopped,  again 
imagining  our  conductor  as  subdivided  (Fig.  48),  mutual 
induction  will  take  place,  and  momentary  '  direct ' 
E.M.Fs.  (i.e.  in  the  same  direction  as  the  current  is 
flowing)  will  be  set  up  in  the  separate  wires  and  tend  to 
prolong  the  current,  and  these  induced  E.M.Fs.  will  be 
greater  in  those  wires  in  the  centre  than  in  those  on  the 
surface,  hence  the  current  will  leave  off  first  at  the 
surface  and  lastly  at  the  interior.  If  we  suppose  for 
argument  that  the  conductor  is  subdivided  into  separate 


78  ALTERNATING-CURRENT  CIRCUIT.         [§27. 

conductors  or  wires  we  may  put  down  the  effect  to 
mutual  induction  :  but  if  we  think  of  the  conductor  as  a 
whole,  the  effect  may  be  attributed  to  self  induction 
(inductance),  which  is  perhaps  after  all  the  real  cause. 
This  phenomenon  has  been  given  the  name  of  skin 
resistance,  a  bad  and  misleading  term,  as  after  all  it  is 
plainly  an  effect  of  induction  and  has  nothing  to  do 
with  ohmic  resistance  :  for  if  we  take  two  conductors  of 
equal  length,  resistance,  and  material ;  one  being  of 
circular  section  and  the  other  in  the  shape  of  a  ribbon 
(Fig.  48),  it  will  be  found  that  the  so-called  '  skin 
resistance '  of  the  former  is  greater  than  that  of  the 
latter,  for  the  reason  that  the  latter  conductor  is  more 
spread  out — i.e.  the  imaginary  separate  wires  composing 
it  are  not  so  much  under  each  other's  inductive  in- 
fluence. The  term  *  skin  resistance  '  would  lead  one  to 
think  that  of  two  or  more  conductors  of  equal  cross- 
section,  the  phenomenon  in  question  would  be  more 
marked  in  the  one  with  the  greater  surface,  whereas  the 
reverse  is  the  case.  The  Author  suggests  that  conductor 
impedance  would  be  a  better  and  more  expressive  name 
for  this  effect. 

It  was  stated  in  §  4  that  inductance  is  not  very 
noticeable  in  straight  conductors,  but  there  is  no  doubt 
that  it  exists  therein,  and  is  the  greater  the  greater  the 
mass  and  the  more  compact  the  shape  of  the  conductor  : 
but  with  direct  currents  its  effects  (in  uncoiled  con- 
ductors) may  be  disregarded. 

27.  CONDUCTORS  FOE  ALTERNATING  CURRENTS. — As  is 
the  case  with  circuits  in  general  (§  4),  the  inductance  of 


§27.]    ALTERNATING-CURRENT   CONDUCTORS.         79 

a  straight  or  thick  conductor  (i.e.  its  so-called  '  skin 
resistance  ')  exercises  a  continual  effect  on  the  flow  of  an 
alternating  current,  which  effect  increases  with  the 
frequency  and  strength  of  the  current.  When  an  alter- 
nating current  commences  to  flow  in  a  conductor,  it 
starts  first  at  the  outer  surface  and  then  penetrates 
more  or  less  to  the  interior ;  but,  unless  the  frequency 
be  very  low,  and  the  conductor  thin  (presuming  it  to  be 
of  circular  solid  section),  it  may  happen  that  very  little 
or  no  current  flows  through  the  centre.  Let  us  try  to 
picture  the  probable  cause  of  this.  When  the  current 
starts  in  one  direction,  the  reverse  E.M.F.  due  to  induct- 
ance is,  as  was  shown  in  the  preceding  paragraph, 
greatest  at  the  centre ;  supposing  the  current  to  stop, 
the  induced  E.M.F.  is  again  greatest  at  the  centre,  and 
is  at  the  moment  '  direct ' — i.e.  in  the  same  direction 
as  the  current,  but  the  latter  at  this  moment  reverses, 
and  this  direct  E.M.F.  acts  in  opposition  to  it,  and  so 
on.  Thus  the  induced  E.M.Fs.,  which  exist  principally 
in  the  centre,  alternate  as  the  current  alternates,  but 
are  constantly  opposite  in  direction.  Though  the  effect 
of  inductance  in  straight  conductors  is  practically 
nothing  as  compared  with  that  in  coiled  ones  (as  in 
electromagnetic  apparatus),  still  it  does  exist,  and  Lord 
Kelvin  has  shown  that  in  the  case  of  a  current  at  a 
frequency  of  150  /%  the  current  only  penetrates  the 
copper  conductor  to  a  depth  of  about  three  millimetres* — 
i.e.  a  little  over  one-tenth  of  an  inch.  At  the  ordinary 
frequency  of  100  ~,  it  has  been  calculated  that  the 

*  A  millimetre  =  -03937  in.— i.e.  between  ^  in.  and  i  in. 


80  ALTERNATING-CURRENT   CIRCUIT.         [§28. 

current  in  a  copper  conductor  at  a  depth  of  12  milli- 
metres (nearly  *5  in.)  from  the  surface  is  only  about  one- 
seventh  of  its  value  at  the  surface.  Thus  the  largest 
useful  size  of  cable  in  alternating  current  work  at  ordi- 
nary frequencies  is  about  19 /14s — i.e.  a  strand  made  up 
of  19  wires  each  of  No.  14  size,  the  total  diameter  being 
about  *4  in. :  larger  sizes  are  less  efficient  than  the 
cross-section  of  copper  would  seem  to  indicate. 

Some  writers  aver  that  the  effect  just  dealt  with  is 
due  to  capacity  as  well  as  inductance,  and  such  is  pro- 
bably the  case,  though  if  capacity  be  taken  into  account 
the  explanation  becomes  less  simple.  However  the 
effect  be  explained,  it  seems  certain  that  the  conducting 
power  of  a  conductor  for  alternating  currents  depends 
not  so  much  upon  its  mass  as  upon  its  surface,  so  that 
a  hollow  tube  may  conduct  nearly  as  well  as  a  solid  rod 
of  the  same  diameter  :  and  with  the  same  area  of  cross- 
section,  a  ribbon-shaped  or  tubular  conductor  is  prefer- 
able to  a  circular  stranded  or  solid  one.  As  was  men- 
tioned at  the  beginning  of  this  paragraph,  the  '  skin 
resistance  '  or  conductor  impedance  (§  26)  increases  as 
the  frequency  increases  ;  but  unless  either  the  frequency 
of  the  current  or  the  thickness  of  the  conductor  be  very 
great,  it  may  be  disregarded  in  practice  for  the  sizes 
most  commonly  in  use.  Low-tension  alternating  cables 
for  large  currents  have  however  to  be  designed  with  this 
point  in  view. 

28.  ELECTEICAL  EESONANCE.  —  The  mains  of  the 
London  Electric  Supply  Corporation  extend  from  their 
central  station  at  Deptford  to  various  distributing 


§  28.]  RISE   OF  PRESSURE  EFFECT.  81 

centres  in  the  western  and  southern  districts  of  London. 
Each  '  go '  and  '  return '  main  has  up  to  the  present 
consisted  of  concentric  copper  tubes  insulated  from  each 
other  with  tightly  compressed  paper,  and  owing  both  to 
their  shape  and  great  length  they  possess  considerable 
capacity.*  Soon  after  the  supply  was  started,  the  fall 
of  pressure  along  the  mains  was  found  to  be  much  less 
than  anticipated,  in  other  words  the  pressure  at  the  dis- 
tributing ends  was  greater  than  could  be  then  accounted 
for.  After  a  time  it  was  seen  that  this  effect  was  owing 
to  the  great  length  and  consequent  capacity  of  the  mains, 
and  the  rise  of  pressure  due  to  this  cause  has  been  given 
the  name  electrical  resonance,  though  it  is  more  popularly 
known  as  '  rise  of  pressure  effect,'  '  capacity  effect,'  or 
'  condenser  effect.' 

In  his  Koyal  Institution  Lectures  (§  9),  Professor 
Forbes  presented  a  mechanical  analogy  for  this  so-called 
'  electrical  resonance.'  A  long  spiral  spring  was  suspended 
from  the  ceiling,  the  free  end  being  held  in  the  hand  : 
the  end  of  the  spring  was  pulled  down  and  allowed  to  rise 
again  at  regular  intervals  and  with  small  force.  After  a 
time,  the  spring  accumulated  energy  by  reason  of  its  re- 
silience, and  its  movements  up  and  down  showed  greater 
amplitude  than  that  which  the  operator  gave  it  at  each 
downward  pull — that  is  to  say,  the  spring  'jumped '  up  and 
down  of  its  own  accord  beyond  the  range  of  the  hand  at 
the  other  end. 

To  construe  this  effect  into  electrical  language,  we 

*  The  London  Electric  Supply  Corporation  are  now  (1897)  replacing 
these  tubular  conductors  by  ordinary  cables, 

G 


82  ALTERNATING-CURRENT  CIRCUIT.         [§28. 

must  first  of  all  assume  that  the  direction  of  the  axis 
passing  down  the  centre  of  the  spring  is  the  direction  of 
the  circuit  conductor  (or  rather  part  of  it)  ;  that  the 
elongation  and  shortening  of  the  spring  represent 
currents  first  in  one  direction  and  then  in  the  other ; 
and  that  the  downward  pulls  represent  the  impressed 
E.M.Fs.  in  one  direction,  there  being  no  analogue  for 
E.M.Fs.  in  the  opposite  direction,  as  the  spring  is  not 
pushed  up,  but  contracts  of  its  own  accord. 

If  both  the  E.M.F.  and  frequency  be  very  low  indeed, 
i.e.  if  the  end  of  the  spring  be  pulled  down  slowly  and 
at  long  intervals,  the  'jumping  effect '  (capacity  current) 
will  be  absent :  but  as  the  '  frequency '  of  the  downward 
pulls  increases,  or  the  E.M.F. ,  as  represented  by  the 
sharpness  with  which  the  spring  is  pulled,  so  also  will  the 
jumping  effect,  until  at  last  the  movement  of  the  spring 
(current)  will  refuse  to  be  governed  by  and  will  be  out 
of  step  with  the  successive  pulls  (E.M.F.  impulses)  given 
by  the  observer.  The  above  explanation  is  doubtless 
somewhat  crude  and  weak,  but  it  will  serve  to  give  the 
reader  an  inkling  of  the  cause  of  electrical  resonance, 
the  effect  being,  as  already  stated,  to  lessen  the  fall  of 
potential  along  the  conductor  or  circuit  in  which  it 
exists. 

It  is  hardly  necessary  to  point  out  that  what  is  here 
termed  '  electrical  resonance '  is  merely  the  ordinary 
effect  of  capacity  on  an  alternating  current  in  circuits 
(or  portions  thereof)  which  happen  to  have  considerable 
capacity  and  very  little  inductance,  the  result  being  a 
lead  in  phase  of  the  current. 


§29.]         EFFECTIVE  E.M.F.  AND   CURRENT.  83 

29.  EFFECTIVE  VOLTS  AND  AMPERES. — In  §  18  an  ex- 
planation was  given  of  the  meaning  of  the  terms 
virtual  pressure,  virtual  current,  &c.  A  virtual 
E.M.F.  is  about  '707  of  the  maximum  values  reached 
by  the  tops  of  the  curve  if  the  latter  is  of  the  sine 
shape  (Fig.  31),  and  varies  slightly  as  the  form  of  the 
curve  varies.  When  we  speak  of  the  E.M.F.  impresssed 
on  the  circuit,  we  mean  the  virtual  E.M.F.  In  most 
circuits  the  impressed  or  virtual  E.M.F.  meets  with  an 
opposing  E.M.F.  of  reactance,  and  the  effective  E.M.F. 
is  something  less  than  the  virtual  E.M.F.,  it  being  that 
pressure  which  is  ultimately  available  for  driving  elec- 
tricity round  the  circuit,  or  for  doing  work. 

For  illustration,  let  us  imagine  a  given  non-inductive 
circuit,  without  appreciable  capacity,  containing  a  short- 
circuited  choking  coil;  and  suppose  that  a  constant 
virtual  or  impressed  E.M.F.  is  maintained  at  its  ends : 
while  the  choking  coil  is  short-circuited,  there  being  no 
opposing  E.M.F.  in  the  circuit,  the  whole  of  the  im- 
pressed E.M.F.  will  be  effective  in  driving  electricity 
round— i.e.  the  virtual  and  effective  E.M.Fs.  will  be 
equal.  If  the  choking  coil  is  thrown  in  circuit,  the 
reactive  E.M.F.  due  to  its  inductance  will  oppose  the 
virtual  E.M.F.,  and  the  effective  E.M.F.  and  con- 
sequent current  will  be  proportionately  reduced,  and  will 
be  still  further  reduced  as  the  reactance  of  the  coil  is 
increased,  the  virtual  or  impressed  E.M.F.  remaining 
constant  the  whole  time. 

Eeferring  to  what  was  said  in  §  18,  if  an  electrostatic 
voltmeter  be  applied  to  the  ends  of  a  circuit,  the 

G   2 


84  ALTERNATING-CURRENT   CIRCUIT.        [§30. 

reading  will  give  the  virtual  volts  under  all  circum- 
stances, and  if  there  be  no  reactance  present,  this  reading 
will  also  represent  the  effective  volts. 

Current  necessarily  implies  the  flow  of  electricity, 
and  a  virtual  current  is  that  indicated  when  a  reliable 
ammeter  is  put  in  circuit.  If  the  current  happens  to  be 
in  phase  with  the  pressure,  this  reading  will  also  give 
what  may  be  called  the  effective  current.  It  has  been 
shown  (§§  19  and  20)  that  the  current  is  not  always  in 
phase  or  step  with  the  pressure ;  it  frequently  lagging 
or  leading  in  phase— generally  the  former.  The  amount 
of  this  lag  or  lead  is  called  the  phase  difference  or  angle 
of  lag  or  lead,  as  the  case  may  be,  and  the  greater  this  is 
the  less  is  the  power  of  a  given  virtual  current  to  do 
useful  work.  That  proportion  of  the  current  which  can 
do  useful  work  may  be  called  the  effective  current.  When 
there  is  no  phase  difference,  the  effective  current  is  the 
same  as  the  virtual  current ;  but  as  the  angle  of  lag  or 
lead  increases,  so  does  the  value  of  the  effective  as  com- 
pared with  the  virtual  current  diminish. 

The  difference  between  virtual  and  effective  current 
is  further  referred  to  in  §  31. 

It  is  necessary  to  point  out  that  the  terms  virtual 
and  effective  are  employed  indiscriminately  by  some 
writers,  while  others  use  only  the  one  term  or  the  other. 
The  necessity  for  both,  and  the  distinction  between 
them,  should  be  clear  from  what  has  been  said  above 
and  in  §  18. 

30.  CONNECTION  BETWEEN  INDUCTANCE,  EEACTANCE, 
IMPEDANCE,  IMPRESSED  VOLTS,  AND  VIRTUAL  CURRENT. 


§  30.]        REACTANCE  AND  REACTIVE  DROP.  85 

— We  have  seen  (§§  21  and  22)  that  the  reactance  in  an 
alternating-current  circuit  depends  directly  upon  the 
inductance  and  the  frequency,  and  inversely  upon  the 
capacity.  In  a  circuit  with  negligible  capacity,  if  L  be 
the  inductance,  and  n  the  frequency,  the  reactance  will 
be  2 TT n L* 

Eeactance  or  spurious  resistance  is,  like  ohmic 
resistance,  independent  of  the  current ;  but  the  current 
must  be  taken  into  account  when  we  wish  to  find  the 
volts  necessary  to  overcome  these  resistances.  Thus  if 
C  be  any  virtual  current,  R  C  denotes  the  volts  neces- 
sary to  force  it  through  an  ohmic  resistance  E  :  simi- 
larly %7rnLC  will  be  the  volts  necessary  to  force 
the  same  current  through  an  inductive  or  spurious 
resistance  ZirnL.  This  quantity  ZirnLC  represents, 
in  fact,  the  counter  E.M.F.  of  reactance,  or  the  reactive 
drop  or  loss  of  volts ;  just  as  R  C  represents  the  ohmic 
drop. 

Thus  for  example,  if  C  =  60  amperes,  n  —  80  ^ 
per  sec.,  and  L  =  -005  henry,1  the  E.M.F.  of  reactance 
will  be  2  x  3-1416  x  80  x  -005  x  60  =  150  volts. 

If,  as  already  stated,  the  reactive  drop  in  an  alter- 
nating-current circuit  carrying  a  virtual  current  C  be 
2  irn L  C  volts,  it  seems  to  follow  that  the  total  volts 
necessary  to  be  impressed  on  the  circuit  would  be  equal  to 

*  This  formula  requires  an  application  of  the  differential  calculus 
for  its  proof :  so  we  will  therefore  take  it  for  granted.  TT  (Greek  pi) 
stands  for  the  ratio  of  the  circumference  of  any  circle  to  its  diameter, 
i.e.  3-1416  (approximately). 

f  The  henry  is  the  unit  of  inductance.  See  the  Author's  Electric 
Lighting  and  Power  Distribution,  Third  Edition,  vol.  i.  §  67. 


86  ALTERNATING-CURRENT  CIRCUIT.        [§  30. 

the  sum  of  the  volts  required  to  send  the  given  current 
through  the  ohmic  resistance  and  the  volts  equal  to 
opposed  volts  of  reactance  (i.e.  R  C  +  2  TT  n  L  C).  This, 
however,  is  not  the  case,  owing  to  the  fact  that  the 
E.M.F.  of  reactance  is  not  in  phase  with  the  impressed 
E.M.F. ;  that  is  to  say,  the  wave  of  alternating  E.M.F. 
of  reactance  does  not  reach  its  maximum  values  at  the 
same  time  as  the  wave  of  impressed  E.M.F.,  but  after- 
wards :  in  other  words,  the  E.M.F.  of  reactance  lags 
behind  the  impressed  E.M.F. 

The  impressed  or  virtual  volts  necessary  to  set  up  a 
current  of  C  (virtual)  amperes  in  a  circuit  of  known 
ohmic  resistance  R,  and  reactance  2  TT  n  L,  is  found  as 
follows : — 

Draw  a  horizontal  line  A  B  (Fig.  49)  proportional 
in  length  to  the  volts  (R  C)  required  to  send  the  current 
P  through  the  ohmic 

resistance  R  of  the 
circuit:  and  from 

^^^^  \/  A    draw  A  C  Per' 

:^^£s  pendicular  to  A  B, 

^C^v.  and     proportional 


UJ 


OHMIC   DROP  B 

J  active  drop,  i.e.  to 

FlG-  49'  -si         rm, 

/  C.      Then 


join  B  C.     The  length  of  B  C  will  represent  the   re- 
quired value  of  impressed  E.M.F. 

Now  CAB  is  a  right-angled  triangle,  of  which  C  B 
is  the  hypotenuse— i.e.  the  side  opposite  the  right 
angle.  In  such,  the  square  of  the  hypotenuse  is  equal 


§  30.]  LAW  OF   THE   CIECUIT.  87 

to  the  sum  of  the  squares  of  the  other  two  sides  (by 
Euclid,  I.  47.)- 
Thus  :— 


and 


That  is  : 


Impressed  or  virtual  volts  (JET)  =  V  (R  <?)2  +  (2  TT  n  L  C)' 


=  V^2  x  (E  +  2 


=  C  V  E2  +  (2  TT  w  L)2 
Now  as  in  Ohm's  simple  law  :— 


and  :  — 


We  may  write  :  — 

(Impressed  or  virtual)  E 

(Virtual)C=      V^TH^Xp— 

and  this  may  be  termed  the  Ohm's  law  for  alternating 
currents  :  V  -R2  +  (2  TTH  L)2   being  in  fact   the   imped- 
ance or  '  virtual  resistance  '  in  the  circuit. 
In  words  the  above  may  be  written  thus  :  — 

Virtual  current  =  ^pressed  E.M.F. 
Impedance. 

If,  in  a  steady-current  circuit  we  multiply  together 


ALTERNATINO-CURRENT   CIRCUIT. 


[§30. 


the  current  and  the  resistance,  the  product  will  give  the 
E.M.F.  in  the  circuit :  or 

C  x  R  =  E 

The  same  result  follows  in  an  alternating-current 
circuit,  for  multiplying  the  current  (virtual)  by  the 
virtual  resistance  (impedance)  will  give  us  the  impressed 
E.M.F.  i.e.  :— 


C  x  V  R2  +  (2 


=  E  (impressed) 


This  being  merely  the  foregoing  equation  transposed. 

It  will  be  seen  (Fig.  49)  that  in  each  of  the  three 
quantities  —  impressed  volts,  reactive  drop,  and  ohmic 
drop,  the  quantity  C  (virtual  current)  occurs.  Ob- 

viously C,  being 
a  common  factor, 
may  be  eliminated 
in  each  case,  and 
the  quantities  will 
then  respectively 
represent  impe- 
dance, reactance, 
and  resistance,  as 


RESISTANCE 

FIG.  50. 


"B 


shown  in  Fig.  50. 
Thus  :— 


i.e. : — 


Impedance2  =  resistance2  +  reactance2 
Impedance  =  Vresistance2  +  reactance2. 


In  Figs  49  and  50  it  will  be  noticed  that  the  length 
of  A  C,  and  therefore  also  the  angle  ABC,  depends 


§  31.]  POWER.  89 

upon  the  reactive  drop,  or,  with  a  given  current,  upon 
the  reactance.  A  B  C  is,  in  fact,  the  angle  of  phase. 

31.  POWER  IN  ALTERNATING-CURRENT  CIRCUITS. — 
The  power  in  a  direct-current  circuit  is  obtained  by 
simply  multiplying  together  the  pressure  and  the 
current ;  or — presuming  the  circuit  has  no  back  E.M.F. 
in  it — the  square  of  the  current  and  the  resistance,  the 
product  in  either  case  representing  watts* 

It  might  be  thought  that  by  taking  the  product  of 
the  virtual  volts  and  virtual  amperes  in  a  circuit  we 
should  obtain  the  actual  power  developed.  Such  would 
be  true  in  a  sense,  but  the  product  would  only  represent 
useful  power  when  the  current  was  in  phase  with  the 
E.M.F.,  the  product  in  question  being  in  all  cases  the 
apparent  power  or  apparent  ivatts.  The  phase  differ- 
ence or  angle  of  lag  or  lead  (§  29)  has  to  be  taken  into 
account,  and  the  greater  this  is  the  less  is  the  power 
actually  being  developed  in  a  circuit  with  a  given  virtual 
pressure  and  current.  In  fact,  if  the  phase  difference 
is  very  great — i.e.  if  there  is  a  large  amount  of  either 
inductance  or  capacity  in  a  circuit  of  comparatively 
low  resistance — we  may  have  what  is  known  as  a  nearly 
wattless  current,  the  true  power  or  effective  watts  being 
far  less  than  the  apparent  power  or  zvatts. 

The  idea  of  a  '  wattless  current '  is  difficult  to  grasp. 
If  there  be  any  current  at  all,  it  is  not  easy  to  under- 
stand why  it  cannot  do  work.  But  when  it  is  re- 

*  The  watt  is  a  unit  of  power  or  rate  of  doing  work.  See  the 
Author's  Electric  Lighting  and  Power  Distribution,  Third  Edition* 
vol.  i.  §  33. 


90  ALTEBNATING-CURBENT  CIRCUIT.  [§31. 

membered  that  a  flow  of  electricity — as  of  water — must 
have  pressure  behind  it  to  enable  it  to  do  work,  and 
when  we  are  dealing  with  alternating  pressure  and  flow, 
and  can  conceive  that  they  may  be  more  or  less  out  of 
step  with  each  other,  comprehension  becomes  fairly 
simple. 

The  following  analogy  affords  a  rough  but  useful 
explanation.  Let  P  P  (Fig.  51)  represent  a  pipe  filled 
with  water  (or  a  conductor  forming  a  closed  circuit), 
and  W,  Wl}  W2,  and  W3  water  wheels  to  which  an 
alternating  movement  may  be  given  by  means  of  the 
handles  h,  h,  h,  h ;  these  may  consequently  be  looked 
upon  as  alternators.  Let  W  W  be  a  fifth  waterwheel, 
to  which  a  reciprocating  motion  is  imparted  by  the  to- 
and-fro  movement  of  the  water  in  the  pipe.  The  motion 
of  water  in  the  lower  part  of  the  pipe-circuit  may  be 
considered  as  analogous  to  the  effective  current,  and  the 
consequent  movement  given  to  W  W  the  effective 
power. 

We  will  first  consider  a  case  where  there  is  no  phase 
difference — that  is,  when  the  watermotive-force  of  W 
acts  directly  in  line  with  the  circuit,  as  indicated  by  the 
dotted  line :  here  the  virtual  E.M.F.  of  W  may  be  said 
to  also  represent  the  effective  E.M.F. ,  and  the  virtual 
current  or  motion  given  to  the  water  by  W  to  equal  the 
effective  current  operating  W  W.  Then  the  apparent 
or  virtual  watts  (i.e.  virtual  E.M.F.  x  virtual  current) 
will  also  represent  the  true  or  effective  watts  (power 
given  to  W  W). 

To  illustrate  the  effect  of  a  small  phase  difference, 


31.] 


EFFECT  OF  PHASE  DIFFERENCE. 


91 


we  will  next  consider  the  waterwheel  as  placed  slightly 
skew  with  the  circuit,  as  at  W} ;  the  angle  of  lag  (or  lead) 
being  denoted  by  the 
angle  a  between  the  two 
lines.  Supposing  the 
frequency  and  virtual 
E.M.F.  of  W}  to  be  the 
same  as  in  the  first  case 
(W),  the  virtual  current 
or  actual  movement  of 
electricity  (water)  imme- 
diately about  Wl  will 
also  be  the  same ;  but,  as 
part  of  the  pressure  will 
be  uselessly  employed 
in  driving  the  water  FlG-  51- 

against  the  sides  of  the  pipe,  the  repelling  effect 
of  which  may  be  looked  upon  as  analogous  to  the 
counter  E.M.F.  of  reactance,  the  effective  pressure  and 
also  the  effective  current  about  W  W  (the  product  of 
which  is  the  effective  or  true  power)  will  be  less.  The 
backwash  from  the  sides  of  the  pipe  is  thus  analogous  to 
wattless  current. 

If  there  be  a  still  greater  phase-difference,  as  at  W^ 
though  the  virtual  E.M.F.,  virtual  current,  and  there- 
fore also  the  virtual  power,  be  the  same  is  in  the  first  case 
(W),  the  effective  power  will  be  still  further  lessened. 

If  the  phase  difference  be  90°,  as  at  W3,  where  the 
waterwheel  is  placed  at  right  angles  with  its  most 
effective  position  ;  we  may  suppose  that  the  current  will 


92  ALTERNATING-CURRENT   CIRCUIT.  [§31. 

be  perfectly  wattless — i.e.  that  there  will  be  no  useful 
power  developed,  and  consequently  no  movement  of 
WW. 

The  above  explanation  is  rather  weak,  but  is 
perhaps  better  than  none  at  all ;  for  at  all  costs  the 
reader  must  get  some  idea — however  vague — that  a 
more  or  less  '  wattless  current '  can  exist. 

A  virtual  current  is  the  actual  flow  of  electricity  as 
observed  at  the  terminals  of  a  circuit,  and  depending 
on  the  conditions  of  the  latter  a  certain  proportion  will  be 
available  for  useful  work  (effective  current),  while  the 
remainder  represents  the  '  wattless  current.'  It  must  be 
borne  in  mind  that  a  '  wattless  current '  is  only  wattless 
so  far  as  its  power  of  doing  useful  work  is  concerned.  A 
current,  whether  *  wattless '  or  not,  will  develop  a  propor- 
tionate amount  of  heat  in — or  a  proportionate  magnetic 
field  around — a  given  conductor. 

The  '  wattless  current '  is  objectionable  in  central- station  work 
for  two  reasons  :  firstly,  because  it  loads  up  the  armatures  of  the 
alternators,  causing  heating  and  reducing  the  useful  load  that  may 
be  put  upon  each;  because  the  power  developed  with  a  given 
E.M.F.  is  limited  by  the  maximum  current  the  conductors  in  the 
armature  coils  can  carry.  Secondly,  owing  to  the  watts  lost  in 
the  cables  and  conductors  through  which  the  '  wattless  current ' 
flows.  Although  the  current  is  wattless  so  far  as  any  use  that  may 
be  made  of  it  is  concerned,  there  still  remains  the  drop  of  power 
due  to  ohrnic  resistance,  the  fall  of  pressure  in  a  cable  carrying 
a  given  current  depending  solely  on  the  resistance,  taking  into 
account,  of  course,  the  reduction  in  useful  area  due  to  any  so- 
called  *  skin  resistance  '  (§  26). 

Thus  :— 

Pressure  drop  =  resistance  x  current 
and  watts  lost  in  conductors  =  pressure  drop  x  current, 


§  32.]  POWER.  93 

the  *  current  '  being  that  measured  on  an  ammeter  connected  to 
the  cable  in  question,  and  therefore  the  virtual  current. 

In  addition  to  the  foregoing  matters,  a  lag  (or  lead)  in  phase  of 
the  current  sets  up  a  troublesome  reaction  between  the  field- 
magnets  and  the  armature  coils,  tending  to  weaken  the  former,  and 
this  necessitates  an  increase  in  the  exciting  current  if  the  impressed 
virtual  E.M.F.  is  to  be  maintained. 

32.  POWER  IN  ALTERNATING-CURRENT  CIRCUITS  (CONT.) 
—The  formula  for  the  power  in  an  alternating-current 
circuit  is  as  follows  : 

(True  or  effective)  Pw  =  Er  Cv  cos  X 

where  Pw  stands  for  ]  power  (in  watts),  Ev  for  virtual 
electromotive  force,  Cv  for  virtual  current,  and  \  for  the 
angle  of  lag  or  lead.* 

On  reference  to  any  table  of  cosines  it  will  be  seen 
that  for  an  angle  of  no  degrees  the  cosine  value  is  unity, 
or  1.  Thus,  in  the  formula  above,  if  the  current  and 
pressure  are  in  phase  —  i.e.  if  there  is  no  phase  difference 
(angle  of  lag  or  lead)  —  the  true  or  effective  watts  may  be 
obtained  by  simply  multiplying  together  the  virtual 
volts  and  virtual  amperes,  as  mentioned  in  the  preceding 
paragraph. 

Thus  :— 

True  or  effective  watts  =  Ev  x  Cv  x  cos  0° 

=  Ev  x  Cv  x  1 
=  Ev  x  Cv 

*  x  =  Greek  1  (lambda),  used  for  lag  or  lead  values.  Cos  \  signifies 
the  cosine  of  any  angle  A.  The  cosine  of  angle  a  (Fig.  33),  for  instance, 


is  the  ratio  of  the  adjacent  side  B  E  to  the  hypotenuse  B  D,  or  —  —  - 

JB  D 


94  ALTERNATING-CURRENT   CIRCUIT.  [§32. 

As  the  phase-difference  increases,  the  cosine  values 
decrease  below  unity,  thus  cos  10°  =  *985,  cos  30°  = 
•866,  cos  80°  =  -173,  cos  90°  =  -000;  and  the  true 
watts  become  proportionately  less  than  the  apparent 
watts.  Thus,  supposing  the  phase  difference  X  =  60°, 
the  true  watts  will  only  be  half  the  apparent  watts,  for 
cos  60°  =  *5.  It  will  thus  be  seen  how  important  it  is 
to  keep  the  phase  difference  as  low  as  possible. 

In  practice,  X  cannot  be  directly  calculated  with  any 
degree  of  accuracy,  for  it  varies  with  every  variation 
in  the  conditions  of  the  circuit,  and  also  with  the 
frequency.  It  can,  of  course,  be  calculated  for  any 
given  case,  but  it  is  not  a  fixed  or  constant  quantity. 
The  true  watts  may  be  directly  ascertained  by  means 
of  a  non-inductive  wattmeter.* 

The  apparent  or  virtual  watts  put  into  a  circuit 
feeding  arc  lamps  or  motors  through  transformers,  as 
calculated  from  the  indications  of  a  voltmeter  and  am- 
meter at  the  station  end,  may  be  far  in  excess  of  the 
actual  power  conveyed  to  the  lamps,  motors,  or  other 
inductive  apparatus — i.e.  may  give  the  idea  that  a  far 
larger  number  of  consuming  devices  are  in  circuit  than 
is  actually  the  case,  owing  to  the  excessive  reactance  in 
the  transformer  circuit. 

The  following  record  of  actual  observations  furnishes 

The  cosine  value  of  any  angle  is  simply  dependent  on  the  angle  itself, 
and  may  be  obtained  from  tables.  See  the  Author's  Electric  Lighting 
and  Power  Distribution,  Third  Edition,  vol.  i.  §  104. 

*  See  the  Author's  Electric  Lighting  and  Power  Distribution, 
Third  Edition,  vol.  i.  §  147. 


§  32.]  POWER  FACTOR.  95 

an  instructive  example.  The  virtual  current  passing  in 
and  out  from  an  alternator  was  44  amperes,  and  the 
pressure  2,050  volts.  The  exciting  current  for  a  corre- 
sponding non-inductive  load  would  have  been  between  50 
and  55  amperes,  at  about  80  volts ;  but  this  had  to  be 
increased  to  from  75  to  80  amperes  in  order  to  maintain 
the  2,050  volts  pressure  at  the  alternator  terminals. 
By  tests  made  with  a  wattmeter,  which  measures  true 
power,  it  was  found  that  the  latter  was  only  56,000  watts. 

Thus  :— 

Apparent  watts  =  2,050    x    44 

=   90,200 

and  True  watts          =   56,000 

Consequently   the  ratio  between  the  true  and  apparent 
watts,  which  is  termed  the  poiver  factor,  was  in  this  case 
56,000 
90,200  = 

To  keep  this  power  factor  as  near  unity  as  possible 
is  thus  one  of  the  chief  problems  in  alternating-current 
distribution. 

When  transformers  are  feeding  non-inductive  circuits, 
such  as  glow  lamps,  the  reactance  due  to  the  former  is 
less  than  in  the  cases  cited  above,  and  is  diminished  as 
the  non-inductive  load  is  increased. 

It  is  here  that  the  question  of  the  introduction  of 
capacity  (in  the  shape  of  condensers)  to  reduce  the  lag 
crops  up ;  but,  though  most  engineers  are  familiar  with 
the  theory  of  their  application,  it  is  as  yet  a  moot  point 
as  to  whether  the  cost  and  upkeep  of  the  condensers,  as 
well  as  the  power  lost  in  them,  would  not  exceed  in  value 


96  ALTERNATING-CURRENT   CIRCUIT.         [$  33. 

the  saving  of  power  effected  by  their  use  (§  6) .  The  loss 
of  power  in  a  condenser  is  principally  due  to  a  pheno- 
menon known  as  dielectric  hysteresis,  which  is  somewhat 
analogous  to  magnetic  hysteresis.  The  rapidly  alter- 
nating charges  in  a  condenser  connected  up  in  an 
alternating-current  circuit  may  be  said  to  cause 
alternating  polarisation  of  the  dielectric,  and  consequent 
heating  and  loss  of  energy. 

On  the  other  hand,  if  the  phase  difference  is  due  to 
excessive  capacity  in  the  circuit,  the  introduction  of 
inductance  would  neutralise  it. 

33.  CONCLUSION. — The  kind  of  alternating  current 
dealt  with  in  this  little  book  is  that  known  as  the  single- 
phase  or  monophase  current.  There  are  other  kinds  of 
alternating  current  called  polyphase  currents,  which  may 
be  roughly  compared  with  two  (diphase),  three  (triphase), 
or  more  simple  alternating  currents  set  up  in  distinct 
circuits,  and  lagging  one  behind  the  other.  When  such 
are  properly  applied  to  a  specially  wound  gramme  ring, 
or  to  the  coils  of  a  multipolar  field  magnet,  they  pro- 
duce what  is  known  as  a  rotatory  or  rotating  magnetic 
field ;  and  within  such  a  field,  a  specially  wound  arma- 
ture without  commutator  or  collector,  i.e.  without  any 
external  electrical  connection,  will  revolve.*  Motors  so 
constructed  are  called  polyphase  asynchronous  motors, 
the  term  asynchronous  signifying  that  the  movement  of 
the  motor  armature  is  not  in  unison  with  the  frequency 
of  the  working  currents ;  and  polyphase  currents  thus 

*  See  the    Author's  Electric   Lighting  and   Power  Distribution, 
Third  Edition,  vol.  i.  §  100. 


OF    '  ; 

UNIV: 

§  33.]  POLYPHASE   CURRENTS.  \^0f  c         97 

possess  great  advantages  over  simple  alternating  currents 
for  motive-power  transmission  as  well  as  other  work  :  for 
though  a  commutatorless  motor  can  be  made  to  work 
with  a  monophase  or  ordinary  alternating  current 
(monophase  asynchronous  motor),  the  field  produced  is 
oscillatory  not  rotatory,  and  the  apparatus  is  less 
efficient.  As  in  such  motors  it  is  difficult  to  say  which 
part  should  be  called  the  armature  and  which  the  field 
magnet,  ambiguity  is  avoided  by  referring  to  the  fixed  por- 
tion as  the  stator,  and  to  the  moving  portion  as  the  rotor. 

The  conception  of  polyphase  currents  is  a  more 
difficult  matter  than  is  the  case  with  monophase 
currents,  and  their  study  requires  more  mathematical 
knowledge. 

The  whole  subject  of  alternating  currents  is  very 
great,  and  leads  up  to  vast  possibilities  of  electrical 
development.  This  introductory  book  will  have  fulfilled 
its  object  if  it  helps  the  reader  over  the  initial  diffi- 
culties, and  leads  him  to  take  up  further  and  more 
serious  study  of  the  matter. 


INDEX. 


The  figures  refer  to  the  numbered  paragraphs. 


ALTERNATING  CURRENT.  2, 3, 5, 6, 

circuit.     10. 

.     Power  in,  31,  32. 

.     Conductors  for,  27. 

.     Effects  of,  on  insulation 

of  circuit,  12,  18. 
— .     Graphical    representa-  j 

tion  of,  15. 
—  currents.    Ohm' slaw  applied 

to,  30. 

Alternation.     3,  16. 
Alternator.    Action  of,  1 
— .     Design  of,  15. 
— .     Frequency  of,  17. 
Ampere.    3. 
Amperes.  Effective,  29.  Virtual, 

18,  29. 

Amplitude.     19. 
Angle  of  lag  or  lead.     29. 
Angles  (footnote).     15. 
Apparent  resistance.    21. 


BATTERY.    ACTION  OF,  1. 
Bell  telephones.    Action  of,  5. 


CABLES.    CAPACITY  OF,  7. 


Capacity.     5,  6,  7,  8,  9,  10,  28. 

—  effect.    28. 
— .    Effect  of,  8. 

—  of  underground  mains.     7. 
Choking  coils.    23. 

.     Practical  forms  of,  24. 

.     Uses  of,  25.  . 

Circuit.  Alternating- current,  10. 
— .     Condensers  in,  6. 
— .     Direct -current,  9. 
Condenser  current.    8. 

—  effect.    28. 
Condensers  in  circuit.     6. 
— .     Practical  use  of,  6,  32. 
Conductor  impedance.    26. 
Conductors  for  alternating  cur- 
rents.   27. 

Cosine  or  cos.     32. 

Crompton  &  Co.'s  choking  coil. 

24. 

Current.     2. 

— .     Alternating,  2,  3,  5,  6. 
.     Effects  of,  on  insulation 

of  circuit,  12,  18. 
.    Graphical  representation 

of,  15 

— .     Condenser,  8. 
— .     Diphase,  33. 


100 


ALTERNATING-CURRENT   CIRCUIT. 


Current.     Direct,  3,  6. 

— .    Effective,  29. 

—  lag  and  lead.     20. 

— .   Periodic,  harmonic,  or  wave,  ' 

6. 

— .     Polyphase,  33. 
— .    Single -phase  or  monophase, 

33. 

— .    Triphase,  33. 
— .     Virtual,  18,  29. 
— .    Wattless,  31,  32. 
Curve.     2. 
Cycle.     16. 


DEPTFORD  MAINS.    12,  28. 
Dielectric  hysteresis.     32. 
Difference  of  pressure  or  poten- 
tial.    1. 

Diphase  current.     33. 
Direct  current.     3,  6. 

circuit.     9. 

Dynamo.    Action  of,  1. 

E.M.F.  =  ELECTROMOTIVE  FORCE, 

Edlund's  experiment  on  in- 
ductance. 14. 

Effective  resistance.     22. 

—  volts  and  amperes.     29. 

Electric  Construction  Corpora- 
tion's choking  coil.  24. 

—  inertia.     9. 
Electrical  resonance.     28. 
Electricity.     Theory  of,  1. 
Electrification.     1. 

—  of  conductor  dielectric.     13. 
Electromagnetic  inertia.     9. 
Electromotive  force.     3,  4. 
.     Effective,  29. 

.     Impressed,  2,  29. 

.     Virtual,  18, 29. 

waves.    15. 


Electrostatic  induction  (see  In- 
fluence) . 
Extra  current.     12. 


FREQUENCY.     3,  16. 

—  of  alternator.     17. 

Forbes'  (Prof.)  lectures  at  the 
Eoyal  Institution.  9,  28. 

GENERATOR.    5. 

HARMONIC  CURRENT.    5. 

—  motion.     15. 
Henry.     30. 

IMPEDANCE.     22,  30. 

—  coil.    23. 

Impressed  E.M.F.     2,  29. 
Inductance.     4,  9,  10, 11, 12,  21, 

30. 
— .     Effects  of,  on  insulation  of 

circuit,  12 
— .     Experiments  on,  11,  14. 

—  of  transformer.     8. 
— .     Unit  of,  30. 
Inductive  resistance.     8. 
Inertia  (footnote).     9. 

— .   Electric  or  electromagnetic, 

9. 

Influence.     6,  13. 
Insulation  of  circuit.    12,  18. 

JOHNSON  &  PHILLIPS'S  CHOKING 
COIL.  24 

LAG  AND  LEAD.    20. 

Lambda  (X).     32. 

London  Electric  Supply  Corpo- 
ration's mains  at  Deptford. 
12,  28. 


INDEX. 


101 


MAGNETO -MACHINE     AMD    BELL. 

ACTION  OF,  5. 
Mains.     Capacity  of,  7. 

—  of  London  Electric    Supply 
Corporation.     12,  28. 

Maxwell's    experiment    on   in- 
ductance.    14. 

Mechanical  analogies.    9, 10, 12, 
20. 

Miller  &Wood's  choking  coil.  24. 

Momentum  (footnote).     9. 

Monophase  asynchronous  motor. 

33. 
-  current.     33. 

Motors.     33. 

Mutual    inductance    (footnote). 
21. 

NIAGARA  POWER  CIRCUITS.     12. 
Non-inductive  resistance.     8. 

OHMIC  DROP.     30. 

—  resistance.     21. 

Ohm's  law  applied  to  alternat- 
ing currents.     30. 
Oscillation.     3. 

P.D.  =  POTENTIAL  DIFFERENCE. 
Period.     16. 
Periodic  current.     5. 
Periodicity.     16. 

—  of  alternator.     17. 
Phase.     19. 

—  difference.     29. 

— .     Lag  and  lead  in,  20. 

Pi  (TT).     30. 

Polyphase  current.     33. 

—  asynchronous  motors.     33. 
Potential  difference.     1 
Power.    Apparent  and  true,  31, 

32. 

—  factor.     32. 


Presspahn.     24. 
Pressure  drop.     31. 
Pressure.  Effective,  29.  Virtual, 
18,  29. 


KEACTANCE.  21,  22,  23,  30. 
—  coil.  23. 

Eeactive  drop  or  loss  of  volts.  30. 
Eeciprocal  (footnote).     18. 
Regulating  switch.     24. 
Resistance.     23 
— .     Apparent,  21. 
— .     Effective  or  virtual,  22. 
— .     Inductive   and  non-induc- 
tive, 8. 

— ,     Ohmic,  21. 
— .     Spurious,  21. 
Reversing  switch.     2. 
Ringer.     5. 

Rise  of  pressure  effect.  28. 
Rotary  magnetic  field.  33. 
Rotor.  33. 


SELF  INDUCTANCE  (footnote).  21. 

—  induction  (see  Inductance).  4. 
Sine  or  sin.     15. 

—  curve  or  wave.     15. 

—  law.     15. 

Single-phase  current.     33. 
'  Skin  resistance.'     26. 

'  Soaking-in  '  action.     13. 

Sparking  at  switches.     12. 

Spark- wear.     12. 

Spurious  resistance.     21. 

Stator.     33. 

Surplus  and  deficit  theory  of 
electricity.  1. 

Switch.     Regulating,  24. 

— .     Reversing,  2. 

Switches  for  circuits  with  in- 
ductance. 12. 


102 


ALTERNATING-CURRENT   CIRCUIT. 


TELEPHONES.    ACTION  OF  BELL,  ,   Volts.     Effective,  29. 

5.  !    —  Virtual,  18,  29. 

Theory  of  electricity.     1. 
Time  base.     15. 


Transformer.      Inductance    of, 

8. 
Triphase  current.    83. 


VIRTUAL  RESISTANCE.    22. 


WATTLESS  CURRENT.     31,  32. 


True 


Appa: 
,  and  Vi 


irtual.     31,  32. 


WAVE  CURRENT.     5. 


volts  and  amperes.     18,29.         Waves  of  E.M.F.     15. 

The  figures  refer  to  the  numbered  paragraphs. 


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