Principle of Electrical Machine
72
MEE 20 / PRINCIPLE OF ELECTRICAL MACHINES / FEBRUARY 2008 SUL 1 1 To Be A World Class Maritime Academy To Be A World Class Maritime Academy Principle of Electrical Machines Energy and Power Systems
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Transcript of Principle of Electrical Machine
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 11
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Prin
cipl
e of
Ele
ctri
cal M
achi
nes
Ene
rgy
and
Pow
er S
yste
ms
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 22
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
�B
uild
stro
ng fo
unda
tion
in th
e ba
sic
prin
cipl
es
of m
agne
tic c
ircu
its a
nd e
lect
rom
agne
tic
ener
gy c
onve
rsio
n.
Intro
duct
ion
At t
he c
ompl
etio
n of
this
cha
pter
, you
sho
uld
be a
ble
to
�A
cqui
re a
nd a
pply
fund
amen
tal p
rinc
iple
s of
scie
nce
and
engi
neer
ing.
�Id
entif
y, fo
rmul
ate
and
mod
el p
robl
ems a
nd fi
nd
engi
neer
ing
solu
tions
bas
ed o
n a
syst
ems a
ppro
ach.
�W
ork
inde
pend
ently
as w
ell a
s with
oth
ers i
n a
team
.�
Com
mun
icat
e ef
fect
ivel
y.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 33
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Spec
ific
Topi
cs�
Pow
er g
ener
atio
n-Tr
ansf
orm
atio
n an
d D
istri
butio
n�
Pow
er sy
stem
on
land
and
ship
�
Intro
duct
ion
to m
achi
ne p
rinci
ple
�Pr
inci
ple
of e
lect
rom
agne
tic e
nerg
y co
nver
sion
�D
efin
ition
of m
agne
tic q
uant
ities
�In
duct
ion
�Lo
sses
�3
phas
e ci
rcui
t�
Act
ive
and
reac
tive
pow
er
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 44
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Pow
er g
ener
atio
n-Tr
ansf
orm
atio
n an
d D
istri
butio
n�
1. M
echa
nica
l ene
rgy
(mov
emen
t)�
2.R
otat
iona
l ene
rgy
(mov
emen
t abo
ut a
n ax
is-c
ircul
ar)
�3.
Elec
trica
l ene
rgy
(ele
ctro
n flo
w)
�4.
Wor
k (a
pplic
atio
n of
forc
e th
roug
h a
dist
ance
)�
5.Po
wer
(rat
e of
doi
ng w
ork)
�6.
Effic
ienc
y (o
utpu
t / in
put )
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 55
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Uni
t Con
vers
ions
& D
imen
sion
al A
naly
sis
*Po
wer
(Wat
ts)
*
A
ltern
ativ
e un
its th
at m
ay b
e us
edhp
(hor
sepo
wer
)=74
5.7
W*
Ele
ctri
c Po
tent
ial(V
olts
)*
Cur
rent
(Am
pere
s), A
*R
esis
tanc
e(O
hms)
*SI
uni
ts(V
olts
, Am
pere
s, O
hms,
Wat
ts)
*E
nglis
h un
its(in
ches
, fee
t, po
unds
etc
.).
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 66
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Intro
duct
ion
to M
achi
nery
Prin
cipl
esA
n el
ectr
ic m
achi
ne:
is a
dev
ice
that
can
con
vert
eith
er m
echa
nica
l ene
rgy
to
elec
tric
ene
rgy
or v
ice
vers
a.
A g
ener
ator
:co
nver
ts m
echa
nica
l ene
rgy
to e
lect
rica
l ene
rgy.
A m
otor
:co
nver
ts e
lect
rica
l ene
rgy
to m
echa
nica
l ene
rgy.
Alm
ost
all p
ract
ical
mot
ors
and
gene
rato
rs c
onve
rt e
nerg
y fr
om
one
form
to a
noth
er th
roug
h th
e ac
tion
of a
mag
netic
fiel
d.
Ano
ther
clo
sely
rel
ated
dev
ice
is th
e tr
ansf
orm
er.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 77
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
A tr
ansf
orm
er:
is a
dev
ice
that
con
vert
s ac
ele
ctri
cal
ener
gy a
t on
e vo
ltage
leve
l to
ac e
lect
rica
l ene
rgy
at a
noth
er v
olta
ge le
vel,
but
with
the
sam
e fr
eque
ncy.
In g
ener
al,
tran
sfor
mer
s op
erat
e on
the
sam
e pr
inci
ples
as
gene
rato
rs a
nd m
otor
s, a
nd a
re u
sual
ly s
tudi
ed t
oget
her
with
ge
nera
tors
and
mot
ors.
The
se 3
type
s of d
evic
es a
re in
evita
ble
in m
oder
n da
ily li
fe.
*Mot
ors f
ind
appl
icat
ions
in se
vera
l hom
e ap
plia
nces
.*I
n th
e w
ork
plac
e, m
otor
s pro
vide
the
mot
ive
pow
er fo
r
alm
ost a
ll to
ols.
*Gen
erat
ors a
re e
ssen
tial t
o su
pply
the
pow
er u
sed
by a
ll th
ese
mot
ors.
Intr
oduc
tion
to M
achi
nery
Prin
cipl
es .
. . .
.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 88
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Mec
hani
cal E
nerg
y
Elec
trica
l Ene
rgy
Mot
orG
ener
ator
Tra
nsfo
rmer
Intr
oduc
tion
to M
achi
nery
Prin
cipl
es .
. . .
.
Elec
trica
l Ene
rgy
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 99
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
INTR
OD
UC
TIO
N
The
elec
trom
agne
tic s
yste
m is
a n
eces
sary
ele
men
t of a
ll ro
tatin
g an
d st
atic
ele
ctric
mac
hine
ry a
nd e
lect
rom
echa
nica
l dev
ices
.
The
role
of
elec
trom
agne
tic s
yste
m i
s to
est
ablis
h an
d co
ntro
l el
ectro
mag
netic
fiel
ds fo
r car
ryin
g ou
t con
vers
ion
of e
nerg
y, a
nd
trans
fer.
Prac
tical
ly a
ll m
otor
s an
d ge
nera
tors
, dep
end
upon
the
mag
netic
fie
ld a
s th
e co
uplin
g m
ediu
m a
llow
ing
inte
rcha
nge
of e
nerg
y in
ei
ther
dire
ctio
n be
twee
n el
ectri
cal a
nd m
echa
nica
l sys
tem
s.
A
trans
form
er
thou
gh
not
an
elec
trom
echa
nica
l co
nver
sion
de
vice
, pr
ovid
es
a m
eans
of
tra
nsfe
rrin
g el
ectri
cal
ener
gy
betw
een
two
elec
trica
l por
ts v
ia th
e m
ediu
m o
f a m
agne
tic fi
eld.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 1010
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
It is
, th
eref
ore,
se
en
that
al
l el
ectri
c m
achi
nes
incl
udin
g tra
nsfo
rmer
s us
e th
e m
ediu
m
of
mag
netic
fie
ld
for
ener
gy
conv
ersi
on a
nd tr
ansf
er.
INTR
OD
UC
TIO
N .
. . .
.
τ,ω
Blo
ck d
iagr
am r
epre
sent
atio
n of
ele
ctro
mec
hani
cal e
nerg
y co
nver
sion
e, I
Ele
ctri
cal
Syst
emC
oupl
ing
Syst
emM
echa
nica
lSy
stem
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 1111
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
PRIN
CIP
LE O
F EL
ECTR
O-M
ECH
AN
ICA
L EN
ERG
Y C
ON
VER
SIO
N
The
elec
trom
agne
tic s
yste
m is
a n
eces
sary
ele
men
t of a
ll ro
tatin
g an
d st
atic
ele
ctric
mac
hine
ry a
nd e
lect
rom
echa
nica
l dev
ices
.
The
role
of
elec
trom
agne
tic s
yste
m i
s to
est
ablis
h an
d co
ntro
l el
ectro
mag
netic
fie
lds
for
carr
ying
out
con
vers
ion
of e
nerg
y, i
ts
proc
essi
ng a
nd tr
ansf
er.
It is
, th
eref
ore,
se
en
that
al
l el
ectri
c m
achi
nes
incl
udin
g tra
nsfo
rmer
s us
e th
e m
ediu
m
of
mag
netic
fie
ld
for
ener
gy
conv
ersi
on a
nd tr
ansf
er.
Sinc
e el
ectro
mec
hani
cal e
nerg
y co
nver
sion
invo
lves
the
inte
rcha
nge
of
ener
gy b
etw
een
an e
lect
rical
and
a m
echa
nica
l sy
stem
, th
e pr
imar
y qu
antit
ies
invo
lved
in
the
mec
hani
cal
syst
em a
re t
orqu
e an
d sp
eed,
w
hile
the
anal
ogou
s qu
antit
ies
in th
e el
ectri
cal s
yste
m a
re c
urre
nt a
nd
volta
ge re
spec
tivel
y.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 1212
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
PRIN
CIP
LE O
F EL
ECTR
O-M
ECH
AN
ICA
L EN
ERG
Y C
ON
VER
SIO
N .
. .
τ,ω
Blo
ck d
iagr
am r
epre
sent
atio
n of
ele
ctro
mec
hani
cal e
nerg
y co
nver
sion
e, I
Ele
ctri
cal
Syst
emC
oupl
ing
Syst
emM
echa
nica
lSy
stem
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 1313
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
PRIN
CIP
LE O
F EL
ECTR
O-M
ECH
AN
ICA
L EN
ERG
Y C
ON
VER
SIO
N .
. .
Mot
or a
ctio
n re
sults
whe
n th
e el
ectri
cal s
yste
m c
ause
s a
curr
enti
to fl
ow th
roug
h co
nduc
tors
that
are
pla
ced
in a
mag
netic
fiel
d.
A m
agne
tic f
orce
is
prod
uced
on
each
con
duct
or s
o th
at i
f th
e co
nduc
tors
are
loc
ated
on
a st
ruct
ure
that
is
free
to
rota
te,
anel
ectro
mag
netic
torq
ue T
resu
lts.
Mor
eove
r, w
ith
the
deve
lopm
ent
of
this
m
otor
ac
tion
the
revo
lvin
g co
nduc
tors
cut
thro
ugh
the
mag
netic
fie
ld a
nd t
here
by
unde
rgo
an e
.m.f.
ew
hich
is re
ally
a re
actio
n vo
ltage
ana
logo
us to
th
e in
duce
d e.
m.f
in th
e pr
imar
y w
indi
ng o
f a tr
ansf
orm
er.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 1414
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
DEF
INIT
ION
OF
MAG
NET
IC Q
UA
NTI
TIES
Mag
net:
It is
a s
ubst
ance
hav
ing
the
prop
ertie
s of
attr
actin
g iro
n an
d its
allo
ys.
Mag
netic
fiel
d:
It is
def
ined
as
the
regi
on o
f sp
ace
arou
nd a
mag
net
in
whi
ch
ther
e ex
ists
a
mag
netic
fo
rce
(eith
er
forc
e of
at
tract
ion
or fo
rce
of re
puls
ion)
. M
agne
tic li
nes
of fo
rce:
Thes
e ar
e th
e im
agin
ary
lines
of
forc
e in
a m
agne
tic f
ield
w
hich
st
art
from
th
e N
orth
po
le,
pass
th
roug
h ai
r or
m
agne
tic m
ediu
m, a
nd e
nd a
t Sou
th p
ole.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 1515
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Mag
netic
circ
uit:
DEF
INIT
ION
OF
MAG
NET
IC Q
UA
NTI
TIES
. . .
. .
It is
a c
ontin
uous
pat
h oc
cupi
ed b
y m
agne
tic li
nes o
f for
ce.
Mag
netic
flux
: It
is th
e nu
mbe
r of l
ines
of m
agne
tic fo
rce
cros
sing
the
spac
e oc
cupi
ed b
y a
mag
netic
fiel
d(φ)
.M
agne
to-m
otiv
e-fo
rce
(m.m
.f):
It dr
ives
or
tries
to
driv
e m
agne
tic f
lux
to f
low
thr
ough
m
agne
tic c
ircui
t and
is th
e an
alog
y to
e.m
.f. in
ele
ctric
circ
uit.
It is
equ
al to
the
effe
ctiv
e cu
rren
t flo
w a
pplie
d to
the
core
(Ni).
Its
unit
is A
mpe
re-tu
rn.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 1616
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
DEF
INIT
ION
OF
MAG
NET
IC Q
UA
NTI
TIES
. . .
. .
Am
pere
-turn
(AT)
:It
is th
e un
it fo
r m
mf
and
is g
iven
by
the
prod
uct o
f nu
mbe
r of
tur
ns o
f m
agne
tic c
ircui
t an
d th
e cu
rren
t flo
win
g in
the
tu
rns (
Ni )
. R
eluc
tanc
e (S
or ℜ
) :Th
is is
the
term
giv
en to
the
mat
eria
l cha
ract
eris
tics
that
resi
st
the
gene
ratio
n of
mag
netic
flux
insi
de it
. It
is t
he o
ppos
ition
off
ered
by
a m
agne
tic p
ath
to t
he
esta
blis
hmen
t of
a m
agne
tic f
lux
and
is t
he a
nalo
gy t
o re
sist
ance
in e
lect
ric c
ircui
t. (A
T/W
b)
Rel
ucta
nce,
S =
Al µA
lAl
=
ρ
ρ1
Res
ista
nce
R =
()
0A
lAl
rµµ
µ×
==
ℜ
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 1717
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
DEF
INIT
ION
OF
MAG
NET
IC Q
UA
NTI
TIES
. . .
. .
Perm
eanc
e:
Perm
eanc
eis
reci
proc
al to
relu
ctan
ce. I
t is
also
the
anal
ogy
to
cond
ucta
nce
in e
lect
ric c
ircui
t.
Res
idua
l mag
netis
m:
It is
the
mag
netis
m w
hich
rem
ains
in
a m
ater
ial
even
the
ef
fect
ive
mag
netiz
ing
forc
e ha
s bee
n re
duce
d to
zer
o.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 1818
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
MAG
NET
IC F
IELD
The
mag
netic
fie
ld in
a m
achi
ne f
orm
s th
e en
ergy
link
bet
wee
n th
e el
ectri
cal a
nd m
echa
nica
l sys
tem
s.
It is
pro
duce
d by
cur
rent
s flo
win
g in
coi
ls o
f w
ire i
nsid
e th
e m
achi
ne, o
r by
a co
mbi
natio
n of
coi
ls a
nd p
erm
anen
t mag
nets
.
It is
def
ined
as
the
regi
on o
f sp
ace
arou
nd a
mag
net
in w
hich
th
ere
exis
ts a
mag
netic
forc
e (e
ither
forc
e of
attr
actio
n or
forc
e of
re
puls
ion)
.
1.A
cu
rren
t-car
ryin
g w
ire
(con
duct
or)
prod
uces
a
mag
netic
fiel
d in
the
area
aro
und
it.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 1919
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
3.A
cur
rent
-car
ryin
g w
ire in
the
pres
ence
of a
mag
netic
fie
ld h
as a
forc
e in
duce
d on
it. T
his i
s the
ba
sis
of
mot
or a
ctio
n.
4.A
mov
ing
wire
in th
e pr
esen
ce o
f a m
agne
tic fi
eld
has
a vo
ltage
indu
ced
in it
. Thi
s is t
he b
asis
of t
he g
ener
ator
ac
tion.
2.A
tim
e-ch
angi
ng m
agne
tic f
ield
ind
uces
a v
olta
ge i
n a
coil
of w
ire i
f it
pass
es t
hrou
gh t
hat
coil.
Thi
s is
the
ba
sis o
f tra
nsfo
rmer
act
ion.
MAG
NET
IC F
IELD
. . .
.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 2020
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
PRO
DU
CTI
ON
OF
MA
GN
ETIC
FIE
LD
Figu
re
show
s a
rect
angu
lar
core
with
a
win
ding
of
N t
urns
of
wire
w
rapp
ed
arou
nd
one
leg
of th
e co
re.
If t
he c
ore
is c
ompo
sed
of i
ron
or c
erta
in o
ther
sim
ilar
met
als,
esse
ntia
lly a
ll th
e m
agne
tic f
ield
pro
duce
d by
the
cur
rent
will
rem
ain
insi
de th
e co
re
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 2121
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Acc
ordi
ng t
o A
mpe
re�s
Law
for
the
pro
duct
ion
of a
mag
netic
fie
ld b
y a
curr
ent:
∫=
•ne
tI
dIH
Whe
re H
is
the
mag
netic
fiel
d in
tens
ity p
rodu
ced
by th
e cu
rren
t I ne
t.
PRO
DU
CTI
ON
OF
MA
GN
ETIC
FIE
LD .
. . .
So t
he p
ath
of i
nteg
ratio
n in
Am
pere
�s l
aw i
s th
e m
ean
path
le
ngth
of t
he c
ore
l c.
The
curr
ent p
assi
ng w
ithin
the
path
of
inte
grat
ion
I net
is th
en N
i, si
nce
the
coil
of w
ire c
uts
the
path
of
inte
grat
ion
N ti
mes
whi
le
carr
ying
cur
rent
i.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 2222
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
PRO
DU
CTI
ON
OF
MA
GN
ETIC
FIE
LD .
. . .
Am
pere
�s la
w th
us b
ecom
es
Ther
efor
e, t
he m
agni
tude
of
the
mag
netic
fie
ld i
nten
sity
in
the
core
due
to th
e ap
plie
d cu
rren
t is
clNi
H=
The
stre
ngth
of
the
mag
netic
fie
ld f
lux
prod
uced
in th
e co
re(H
) al
so d
epen
ds o
n th
e m
ater
ial o
f the
cor
e.
HB
µ=Th
e re
latio
nshi
p be
twee
n th
e m
agne
tic fi
eld
inte
nsity
(H) a
nd th
e re
sulti
ng m
agne
tic f
lux
dens
ity B
pro
duce
d w
ithin
a m
ater
ial
is
give
n by
µ=
mag
netic
per
mea
bilit
yof
mat
eria
lB
= re
sulti
ng m
agne
tic fl
ux d
ensi
ty p
rodu
ced.
iN
lH
c×
=×
rµµ
µ×
=0
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 2323
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
PRO
DU
CTI
ON
OF
MA
GN
ETIC
FIE
LD .
. . .
The
perm
eabi
lity
of fr
ee sp
ace
is c
alle
d µ
0, an
d is
giv
en b
y
H/m
10
47
0−
×=
πµ
Rel
ativ
e pe
rmea
bilit
y (µ
r)is
a c
onve
nien
t w
ay t
o co
mpa
re t
he
perm
eabi
lity
of m
ater
ials
.
The
mag
nitu
de o
f the
flux
den
sity
is g
iven
by
clNi
HB
µµ
==
The
equa
tion
for t
otal
flux
isBA
=φ
Thus
, the
tota
l flu
x in
the
core
due
to th
e cu
rren
t iin
the
win
ding
is
whe
re A
is th
e cr
oss-
sect
iona
l are
a of
the
core
.clNiA
BAµ
φ=
=
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 2424
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
MA
GN
ET
IC C
IRC
UIT
S
It is
a c
ontin
uous
pat
h oc
cupi
ed b
y m
agne
tic li
nes o
f for
ce.
Mag
netic
circ
uit
is a
nalo
gous
to a
n el
ectri
c ci
rcui
t.
In a
sim
ple
elec
tric
circ
uit,
the
volta
ge s
ourc
e V
driv
es a
cur
rent
I
acro
ss a
resi
stan
ce R
. In
the
ele
ctric
circ
uit,
it is
the
vol
tage
or
elec
trom
otiv
e fo
rce
that
dr
ives
the
curr
ent f
low
.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 2525
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
By
anal
ogy,
the
corr
espo
ndin
g qu
antit
y in
the
mag
netic
circ
uit i
s ca
lled
the
mag
neto
mot
ive
forc
e (m
mf).
MA
GN
ET
IC C
IRC
UIT
S . .
. . .
The
pola
rity
of th
e m
mff
rom
a c
oil o
f wire
can
be
dete
rmin
ed a
s fo
llow
s:
If
the
finge
rs
of
the
right
han
d cu
rl in
the
di
rect
ion
of t
he c
urre
nt
flow
in
a co
il of
wire
, th
en t
hum
b w
ill p
oint
in
the
dire
ctio
n of
the
po
sitiv
e m
mf.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 2626
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
The
mm
fof t
he m
agne
tic c
ircui
t is
equa
l to
the
effe
ctiv
e cu
rren
t flo
w a
pplie
d to
the
core
, or
MA
GN
ET
IC C
IRC
UIT
S . .
. . .
F= φ
ℜ
Whe
re F
= m
agne
tom
otiv
efo
rce
of th
e ci
rcui
tφ=
flux
of t
he c
ircui
tℜ
= re
luct
ance
of t
he c
ircui
t
We
see
that
the
relu
ctan
ce o
f the
cor
e is
Al c µ=
ℜ
Rel
ucta
nces
in
a
mag
netic
ci
rcui
t ob
ey
the
sam
e ru
les
as
resi
stan
ces i
n an
ele
ctric
circ
uit.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 2727
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
...5
43
21
+ℜ
+ℜ
+ℜ
+ℜ
+ℜ
=ℜ
eq
MA
GN
ET
IC C
IRC
UIT
S . .
. . .
The
equi
vale
nt re
luct
ance
of a
num
ber o
f rel
ucta
nces
in
serie
s is
ju
st th
e su
m o
f the
indi
vidu
al re
luct
ance
s:
Sim
ilarly
, re
luct
ance
s in
pa
ralle
l co
mbi
ne
acco
rdin
g to
th
e eq
uatio
n
...1
11
11
1
54
32
1ℜ
+ℜ
+ℜ
+ℜ
+ℜ
=ℜ
eq
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 2828
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
LE
AK
AG
EFL
UX
In a
ll pr
actic
al m
agne
tic c
ircui
ts, m
ost o
f th
e flu
x is
con
fined
to
the
inte
nded
pat
h by
use
of m
agne
tic c
ores
but
a s
mal
l am
ount
of
flux
alw
ays l
eaks
thro
ugh
the
surr
ound
ing
air.
This
stra
y flu
x is
cal
led
the
leak
age
flux.
i P(t
)
v P(t
)+ -
+ -
v S(t
)~
Lea
kag
e fl
ux,
φL
PM
utu
al f
lux,
φM
Lea
kag
e fl
ux,
φL
S
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 2929
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Leak
age
is c
hara
cter
istic
of a
ll m
agne
tic c
ircui
ts a
nd c
an n
ever
be
fully
elim
inat
ed.
Cal
cula
tions
con
cern
ing
the
mai
n m
agne
tic c
ircui
t ar
e us
ually
ca
rrie
d ou
t w
ith t
he e
ffec
t of
lea
kage
flu
x ei
ther
ign
ored
or
empi
rical
ly a
ccou
nted
for.
Spec
ial
stud
ies
of l
eaka
ge m
ust
be m
ade
for
ac m
achi
nes
and
trans
form
ers s
ince
thei
r per
form
ance
is a
ffec
ted
by it
.
LE
AK
AG
EFL
UX
. . .
.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 3030
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
FRIN
GIN
G
At
an a
ir-ga
p in
a m
agne
tic c
ore,
the
flu
x fr
inge
s ou
t in
to
neig
hbor
ing
air p
aths
as s
how
n in
Fig
ure
belo
w.
The
resu
lt is
non
-uni
form
flu
x de
nsity
in
the
air-
gap
(dec
reas
ing
outw
ard)
, enl
arge
men
t of
the
effe
ctiv
e ai
r-ga
p ar
ea a
nd a
dec
reas
e in
th
e av
erag
e ga
p flu
x de
nsity
.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 3131
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
The
effe
ct o
f frin
ging
incr
ease
s with
the
air-
gap
leng
th.
FRIN
GIN
G .
. . .
g
gg
Al 0µ=
ℜTh
e ga
p re
luct
ance
wou
ld n
ow b
e gi
ven
by
A g>
A
Frin
ging
incr
ease
s the
cro
ss se
ctio
nal a
rea
of th
e ai
r gap
.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 3232
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
IND
UC
TIO
N
Self
indu
ctio
nSe
lf-in
duct
ion
is th
e ph
enom
enon
whe
reby
a c
hang
e in
the
curr
ent
in a
con
duct
or in
duce
s an
emfi
n th
e co
nduc
tor i
tsel
f.i.e
., w
hen
a co
nduc
tor
is g
iven
a c
urre
nt, f
lux
will
be
prod
uced
, an
d if
the
curr
ent i
s cha
nged
the
flux
also
cha
nges
.
Va
ria
ble
ac
sup
ply
~
Cu
rre
nt,
iS
Ma
gne
tic li
ne
s o
f fo
rce
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 3333
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
SEL
F IN
DU
CT
ION
. . .
.
As
per F
arad
ay�s
Law
whe
n th
ere
is a
cha
nge
of fl
ux, a
n em
fwill
be
indu
ced.
This
is c
alle
d se
lf in
duct
ion.
The
indu
ced
emf
will
be
alw
ays
oppo
site
in
dire
ctio
n to
the
ap
plie
d em
f.
The
oppo
sing
em
fthu
s pr
oduc
ed is
cal
led
the
coun
ter e
mfo
f sel
f in
duct
ion.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 3434
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
It is
the
elec
trom
agne
tic in
duct
ion
prod
uced
by
one
circ
uit i
n th
e ne
arby
circ
uit
due
to t
he v
aria
ble
flux
of th
e fir
st c
ircui
t cu
tting
th
e co
nduc
tor o
f the
nea
rby
circ
uit.
MU
TU
AL
IN
DU
CT
ION
Whe
n tw
o co
ils a
re k
ept n
ear
to e
ach
othe
r an
d if
a va
riabl
e cu
rren
t is
give
n to
one
coi
l, th
e flu
x pr
oduc
ed d
ue t
o th
at v
aria
ble
curr
ent
cuts
bo
th th
e co
ils, a
nd a
n em
fwill
be
indu
ced
in b
oth
the
coils
.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 3535
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
The
phen
omen
on b
y w
hich
em
fis
indu
ced
in th
e se
cond
coi
l due
to
the
varia
tion
of c
urre
nt in
firs
t coi
l is c
alle
d m
utua
l ind
uctio
n.
Bot
h th
e ci
rcui
ts h
ave
no e
lect
rical
con
tact
s w
ith e
ach
othe
r bu
t m
agne
tic li
nkag
es.
This
prin
cipl
e is
app
lied
in tr
ansf
orm
ers,
indu
ctio
n co
ils, e
tc.
MU
TU
AL
IN
DU
CT
ION
. . .
. .
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 3636
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
MA
GN
ET
IC C
IRC
UIT
CA
LC
UL
AT
ION
S
Nor
mal
ly m
agne
tic c
ircui
t ca
lcul
atio
ns i
nvol
ve t
wo
type
s of
pr
oble
ms.
In t
he f
irst
type
of
prob
lem
it
is r
equi
red
to d
eter
min
e th
e ex
cita
tion
(mm
f) n
eede
d to
est
ablis
h a
desi
red
flux
orflu
x de
nsity
at
a g
iven
poi
nt in
a m
agne
tic c
ircui
t.
In th
e se
cond
cat
egor
y th
e flu
x (o
r flu
x de
nsity
) is
unk
now
n an
dis
requ
ired
to b
e de
term
ined
for a
giv
en g
eom
etry
of t
he m
agne
tic
circ
uit a
nd sp
ecifi
ed m
mf.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 3737
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
MA
GN
ET
IC C
IRC
UIT
CA
LC
UL
AT
ION
S . .
. . .
CO
RE
WIT
H A
IR-G
AP
Tran
sfor
mer
s are
wou
nd o
n cl
osed
cor
es.
Rot
atin
g m
achi
nes
have
a m
ovin
g el
emen
t an
d m
ust
ther
efor
e ha
ve a
ir-ga
ps in
the
core
s out
of n
eces
sity
. A
typ
ical
mag
netic
circ
uit
with
an
air-
gap
is s
how
n in
Fig
ure
belo
w.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 3838
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
It is
ass
umed
that
the
air-
gap
is n
arro
w a
nd th
e flu
x co
min
g ou
t of
the
core
pas
ses s
traig
ht d
own
the
air-
gap
such
that
the
flux
dens
ity
in th
e ai
r-ga
p is
the
sam
e as
in th
e co
re.
Act
ually
the
flux
in th
e ga
p fr
inge
s out
so th
at th
e ga
p flu
x de
nsity
is
som
ewha
t les
s tha
n th
at o
f the
cor
e.
MA
GN
ET
IC C
IRC
UIT
CA
LC
UL
AT
ION
S . .
. . .
CO
RE
WIT
H A
IR-G
AP.
. . .
.
The
mm
fNi i
s now
con
sum
ed in
the
core
plu
s the
air-
gap.
Fr
om th
e Fi
gure
, N
i = H
cl c+
Hgl g
gg
ccc
lB
lB
Ni
0µµ
+=
Ass
umin
g th
at a
ll th
e co
re fl
ux p
asse
s stra
ight
dow
n th
e ai
r-ga
p , i
.e.,
no
frin
ging
eff
ect,
Bg
= B
cA
BA
Bg
c=
=φ
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 3939
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
gg
ccc
il
Bl
BN
0µµ
+=
MA
GN
ET
IC C
IRC
UIT
CA
LC
UL
AT
ION
S . .
. . .
CO
RE
WIT
H A
IR-G
AP.
. . .
.
Bg
= B
c
AB
AB
gc
==
φ
+
=A
lA
lN
g
cci
0µφ
µφ
()
eqg
cℜ
=ℜ
+ℜ
=φ
φF
==
ℜA
l cc
0µco
re re
luct
ance
==
ℜA
l gg
0µai
r-ga
p re
luct
ance
Whi
ch m
eans
tha
t in
a m
agne
tic c
ircui
t w
ith a
ir-ga
p, c
ore
relu
ctan
ce
may
be
negl
ecte
d w
ith n
o si
gnifi
cant
loss
of a
ccur
acy.
This
ass
umpt
ion
will
be
gene
rally
mad
e in
mod
elin
g ro
tatin
g m
achi
nes.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 4040
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
MA
GN
ET
IC B
EH
AV
IOR
OF
FER
RO
MA
GN
ET
IC M
AT
ER
IAL
S
To
illus
trate
th
e be
havi
or
of
mag
netic
pe
rmea
bilit
y in
a
ferr
omag
netic
mat
eria
l, ap
ply
a di
rect
cur
rent
to th
e co
re, s
tarti
ng
with
0 A
and
slo
wly
wor
king
up
to t
he m
axim
um p
erm
issi
ble
curr
ent.
Whe
n th
e flu
x pr
oduc
ed i
n th
e co
re i
s pl
otte
d ve
rsus
the
mm
fpr
oduc
ing
it, th
e re
sulti
ng p
lot l
ooks
like
figu
re b
elow
.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 4141
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
This
type
of
plot
is c
alle
d a
satu
ratio
n cu
rve
or a
mag
netiz
atio
n cu
rve.
MA
GN
ET
IC B
EH
AV
IOR
OF
FER
RO
MA
GN
ET
IC M
AT
ER
IAL
S. .
. .
At f
irst,
a sm
all i
ncre
ase
in th
e m
mf
prod
uces
a h
uge
incr
ease
in
the
resu
lting
flux
. A
fter
a ce
rtain
poi
nt,
thou
gh,
the
mm
fin
crea
ses
ther
e is
a
rela
tivel
y sm
alle
r inc
reas
e in
the
flux.
Fina
lly, a
n in
crea
se in
the
mag
neto
mot
ive
forc
e pr
oduc
es a
lmos
t no
cha
nge
at a
ll.
The
regi
on in
whi
ch th
e cu
rve
flatte
ns o
ut is
cal
led
the
satu
ratio
n re
gion
, and
the
core
is sa
id to
be
satu
rate
d.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 4242
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
In c
ontra
st,
the
regi
on w
here
the
flu
x ch
ange
s ve
ry r
apid
ly i
s ca
lled
the
unsa
tura
ted
regi
onof
the
curv
e, a
nd th
e co
re is
sai
d to
be
uns
atur
ated
. Th
e tra
nsiti
on r
egio
n be
twee
n th
e un
satu
rate
d re
gion
and
the
sa
tura
ted
regi
on is
som
etim
es c
alle
d th
e kn
eeof
the
curv
e.
Not
e th
at th
e flu
x pr
oduc
ed i
n th
e co
re i
s lin
early
rel
ated
to
the
appl
ied
mm
fin
the
unsa
tura
ted
regi
on.
MA
GN
ET
IC B
EH
AV
IOR
OF
FER
RO
MA
GN
ET
IC M
AT
ER
IAL
S. .
. .
From
the
figur
e, it
is s
een
that
the
rela
tions
hip
betw
een
B a
nd H
ha
s the
sam
e sh
ape
as th
e re
latio
nshi
p be
twee
n flu
x an
d m
mf.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 4343
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
EN
ER
GY
LO
SSE
S IN
A F
ER
RO
MA
GN
ET
IC C
OR
EM
AG
NE
TIC
BE
HA
VIO
R O
F FE
RR
OM
AG
NE
TIC
MA
TE
RIA
LS.
.
Inst
ead
of a
pply
ing
a di
rect
cur
rent
to
the
core
, le
ts a
pply
an
alte
rnat
ing
curr
ent.
Ass
ume
that
the
flux
in th
e co
re is
initi
ally
zer
o.
As
the
curr
ent
incr
ease
s fo
r th
e fir
st t
ime,
the
flu
x in
the
cor
e tra
ces o
ut p
ath
abin
figu
re.
This
is b
asic
ally
the
satu
ratio
n cu
rve
expl
aine
d ab
ove.
H
owev
er,
whe
n th
e cu
rren
t fa
lls a
gain
, th
e flu
x tr
aces
out
a
diffe
rent
pat
h fr
om t
he o
ne i
t fo
llow
ed w
hen
the
curr
ent
was
in
crea
sed.
As
the
curr
ent d
ecre
ases
, the
flux
in th
e co
re tr
aces
out
pat
hbc
d,
and
late
r whe
n th
e cu
rren
t inc
reas
es a
gain
, the
flux
trac
es o
ut p
ath
deb.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 4444
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
EN
ER
GY
LO
SSE
S IN
A F
ER
RO
MA
GN
ET
IC C
OR
E .
. . .
MA
GN
ET
IC B
EH
AV
IOR
OF
FE
RR
OM
AG
NE
TIC
MA
TE
RIA
LS.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 4545
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Not
ice
that
the
am
ount
of
flux
pres
ent
in t
he c
ore
depe
nds
not
only
on
the
amou
nt o
f cur
rent
app
lied
to th
e w
indi
ngs
of th
e co
re,
but a
lso
on th
e pr
evio
us h
isto
ry o
f the
flux
in th
e co
re.
This
dep
ende
nce
on t
he p
rece
ding
flu
x hi
stor
y an
d th
e re
sulti
ng
failu
re to
retra
ce fl
ux p
ath
is c
alle
d hy
ster
isis
.
Path
bcd
ebtra
ced
out i
n th
e fig
ure
is c
alle
d a
hyst
eris
islo
op.
EN
ER
GY
LO
SSE
S IN
A F
ER
RO
MA
GN
ET
IC C
OR
E .
. . .
MA
GN
ET
IC B
EH
AV
IOR
OF
FER
RO
MA
GN
ET
IC M
AT
ER
IAL
S. .
. .
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 4646
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
HY
STE
RIS
IS L
OSS
Ene
rgy
loss
es in
a fe
rrom
agne
tic c
ore
. . .
.M
agne
tic b
ehav
ior
of F
erro
mag
netic
mat
eria
ls. .
.
Whe
n th
e m
agne
to-m
otiv
e fo
rce
is r
emov
ed, t
he f
lux
in t
he c
ore
does
not
go
to z
ero.
In
stea
d, so
me
mag
netic
flux
is le
ft in
the
core
.
This
mag
netic
flux
is c
alle
d th
e re
sidu
al fl
uxin
the
core
.
It is
pre
cise
ly th
e m
anne
r tha
t per
man
ent m
agne
ts a
re p
rodu
ced.
To fo
rce
the
flux
to z
ero,
an
amou
nt o
f mm
fkno
wn
as th
e C
oerc
ive
mag
neto
-mot
ive
forc
em
ust b
e ap
plie
d to
the
core
in th
e op
posi
te
dire
ctio
n.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 4747
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
HY
STE
RIS
IS L
OSS
. . .
.E
nerg
y lo
sses
in a
ferr
omag
netic
cor
e . .
. .
Mag
netic
beh
avio
r of
Fer
rom
agne
tic m
ater
ials
. . .
To u
nder
stan
d th
e be
havi
our
of f
erro
mag
netic
mat
eria
ls,
it is
ne
cess
ary
to k
now
abo
ut th
eir s
truct
ure.
With
in th
e m
etal
, the
re a
re m
any
smal
l reg
ions
cal
led
dom
ains
.
In e
ach
dom
ain,
all
the
atom
s ar
e al
igne
d w
ith t
heir
mag
netic
fie
lds
poin
ting
in th
e sa
me
dire
ctio
n, s
o ea
ch d
omai
n w
ithin
the
mat
eria
l act
s as a
smal
l per
man
ent m
agne
t.
The
reas
on th
at a
who
le b
lock
of
iron
can
appe
ar t
o ha
ve n
o flu
xis
th
at t
hese
num
erou
s tin
y do
mai
ns a
re o
rient
ed r
ando
mly
with
in t
he
mat
eria
l as s
how
n in
figu
re.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 4848
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
HY
STE
RIS
IS L
OSS
. . .
.E
nerg
y lo
sses
in a
ferr
omag
netic
cor
e . .
. .
Mag
netic
beh
avio
r of
Fer
rom
agne
tic m
ater
ials
. . .
Fig.
b
Fig.
a :
Mag
netic
dom
ains
orie
nted
rand
omly
.Fi
g. b
: M
agne
tic d
omai
ns li
ned
up in
the
pres
ence
of a
n ex
tern
al
mag
netic
fiel
d.
Fig.
a
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 4949
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Whe
n an
ext
erna
l mag
netic
fiel
d is
app
lied
to th
is b
lock
of i
ron,
it
caus
es d
omai
ns te
nd to
alig
n th
emse
lves
in a
n or
derly
fas
hion
inth
e di
rect
ion
of a
pplie
d m
agne
tic fi
eld.
As
the
stre
ngth
of
the
exte
rnal
mag
netic
fie
ld c
ontin
ues
to
incr
ease
, who
le d
omai
ns t
hat
are
alig
ned
in t
he w
rong
dire
ctio
n re
-orie
nt th
emse
lves
as a
uni
t to
line
up w
ith th
e ex
tern
al fi
eld.
HY
STE
RIS
IS L
OSS
. . .
.E
nerg
y lo
sses
in a
ferr
omag
netic
cor
e . .
. .
Mag
netic
beh
avio
r of
Fer
rom
agne
tic m
ater
ials
. . .
The
key
to h
yste
resi
sis
that
whe
n th
e ex
tern
al m
agne
tic f
ield
is
rem
oved
, the
dom
ains
do
not c
ompl
etel
y ra
ndom
ize
agai
n.W
hy d
o th
e do
mai
ns re
mai
n lin
ed u
p?B
ecau
se tu
rnin
g th
e at
oms i
n th
em re
quire
s ene
rgy.
Orig
inal
ly,
ener
gy w
as p
rovi
ded
by t
he e
xter
nal
mag
netic
fie
ld t
o ac
com
plis
h th
e al
ignm
ent;
whe
n th
e fie
ld i
s re
mov
ed,
ther
e is
no
sour
ce o
f ene
rgy
to c
ause
all
the
dom
ains
to ro
tate
bac
k.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 5050
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Hys
tere
sisL
oss:
The
hyst
eres
islo
ss i
n an
iro
n co
re i
s th
e en
ergy
req
uire
d to
ac
com
plis
h th
e re
-orie
ntat
ion
of d
omai
ns d
urin
g ea
ch c
ycle
of t
he
ac a
pplie
d to
the
core
.
HY
STE
RIS
IS L
OSS
. . .
.E
nerg
y lo
sses
in a
ferr
omag
netic
cor
e . .
. .
Mag
netic
beh
avio
r of
Fer
rom
agne
tic m
ater
ials
. . .
The
piec
e of
iron
is n
ow a
per
man
ent m
agne
t.O
nce
the
dom
ains
are
alig
ned,
som
e of
them
will
rem
ain
alig
ned
until
a so
urce
of e
xter
nal e
nerg
y is
supp
lied
to c
hang
e th
em.
Exam
ples
of
exte
rnal
sou
rces
of
ener
gy t
hat
can
chan
ge t
he
alig
nmen
t are
: a la
rge
mec
hani
cal s
hock
, hea
ting,
etc
.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 5151
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
HY
STE
RIS
IS L
OSS
. . .
.E
nerg
y lo
sses
in a
ferr
omag
netic
cor
e . .
. .
Mag
netic
beh
avio
r of
Fer
rom
agne
tic m
ater
ials
. . .
If th
e m
agne
tizin
g fo
rce
appl
ied
to a
mag
netic
mat
eria
l is
carr
ied
thro
ugh
a co
mpl
ete
cycl
e fr
om +
Hm
axto
�H
max
and
back
to
+Hm
ax, t
he B
-H c
hara
cter
istic
is d
escr
ibed
by
a hy
ster
isis
loop
as
show
n in
figu
re.
The
area
of
the
loop
rep
rese
nts
the
ener
gy lo
ss in
a u
nit c
ube
of
the
core
mat
eria
l dur
ing
one
cycl
e.Th
e hy
ster
isis
loss
is e
xpre
ssed
em
piric
ally
as
Whe
re t
he v
alue
of
n de
pend
s on
the
nat
ure
of
the
core
mat
eria
l. n
may
var
y be
twee
n 1.
5 an
d 2.
5 fo
r di
ffer
ent
mat
eria
ls.
How
ever
, th
e hy
ster
isis
loss
var
ies
dire
ctly
as
the
freq
uenc
y fo
r a g
iven
Bm
ax.
nh
Bf
Pm
ax∝
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 5252
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
ED
DY
CU
RR
EN
T L
OSS
Ene
rgy
loss
es in
a fe
rrom
agne
tic c
ore
. . .
.M
agne
tic b
ehav
ior
of F
erro
mag
netic
mat
eria
ls. .
.
A t
ime-
chan
ging
flu
x in
duce
s vo
ltage
with
in a
fer
rom
agne
tic
core
in
just
the
sam
e m
anne
r as
it
wou
ld i
n a
wire
wra
pped
ar
ound
that
cor
e.
Thes
e vo
ltage
s cau
se sw
irls o
f cur
rent
to fl
ow w
ithin
the
core
It is
the
sha
pe o
f th
ese
curr
ents
tha
t gi
ves
rise
to t
he n
ame
eddy
cu
rren
ts.
Thes
e ed
dy c
urre
nts
are
flow
ing
in a
res
istiv
e m
ater
ial (
the
iron
of
the
core
), so
ene
rgy
is d
issi
pate
d by
them
.
The
lost
ene
rgy
goes
into
hea
ting
the
iron
core
.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 5353
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
The
amou
nt o
f ene
rgy
lost
due
to e
ddy
curr
ents
(edd
y cu
rren
t los
s)is
pro
porti
onal
to th
e si
ze o
f the
pat
hs th
ey fo
llow
with
in th
e co
re.
ED
DY
CU
RR
EN
T L
OSS
. . .
. .
Ene
rgy
loss
es in
a fe
rrom
agne
tic c
ore
. . .
.M
agne
tic b
ehav
ior
of F
erro
mag
netic
mat
eria
ls. .
.
For t
his
reas
on, i
t is
cust
omar
y to
bre
ak u
p an
y fe
rrom
agne
tic c
ore
into
man
y sm
all s
trips
or laminations,a
nd to
bui
ld th
e co
re u
p ou
t of
thes
e st
rips.
An
insu
latin
g re
sin
is u
sed
betw
een
the
strip
s, so
tha
t th
e cu
rren
t pa
ths f
or e
ddy
curr
ents
are
lim
ited
to v
ery
smal
l are
as.
Bec
ause
the
insu
latin
g la
yers
are
ext
rem
ely
thin
, thi
s act
ion
redu
ces
eddy
cur
rent
los
ses
with
ver
y lit
tle e
ffec
t on
the
cor
e�s
mag
netic
pr
oper
ties.
The
iron
in th
e m
agne
tic c
ircui
ts is
lam
inat
ed to
pre
vent
exc
essi
ve
eddy
cur
rent
s.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 5454
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
The
eddy
cur
rent
loss
can
als
o be
exp
ress
ed in
term
s of
the
appl
ied
volta
ge a
nd th
e nu
mbe
r of
turn
s in
the
exci
ting
win
ding
whe
n th
ere
sist
ance
of t
he w
indi
ng is
neg
ligib
le.
ED
DY
CU
RR
EN
T L
OSS
. . .
. .
Ene
rgy
loss
es in
a fe
rrom
agne
tic c
ore
. . .
.M
agne
tic b
ehav
ior
of F
erro
mag
netic
mat
eria
ls. .
.
The
expr
essi
on is
2 max
2B
fP e
∝
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 5555
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
THR
EE �
PHA
SE C
IRC
UIT
SA
t the
com
plet
ion
of th
is c
hapt
er, y
ou s
houl
d be
abl
e to
: D
emon
stra
te in
dep
th u
nder
stan
ding
of t
he c
once
pts
of b
asic
thre
e-ph
ase
circ
uits
.
Dis
cuss
the
diffe
renc
es b
etw
een
thre
e-ph
ase
and
sing
le-p
hase
vo
ltage
s.D
iscu
ss th
e ch
arac
teris
tics
of d
elta
and
wye
conn
ectio
ns.
Com
pute
the
volta
ge a
nd c
urre
nt v
alue
s fo
r del
ta a
nd w
yeci
rcui
ts.
Con
nect
del
ta a
nd w
yeci
rcui
ts a
nd m
ake
mea
sure
men
ts w
ith
mea
surin
g in
stru
men
ts.
Acq
uire
and
app
ly fu
ndam
enta
l prin
cipl
es o
f sci
ence
and
eng
inee
ring.
Com
mun
icat
e ef
fect
ivel
y.A
cqui
re te
chni
cal c
ompe
tenc
e in
spe
cial
ized
are
a (th
ree-
phas
e ci
rcui
ts) o
f eng
inee
ring.
Wor
k in
depe
nden
tly a
s w
ell a
s w
ith o
ther
s in
a te
am
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 5656
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
THR
EE �
PHA
SE C
IRC
UIT
SIn
trod
ucti
onTh
ree-
phas
e sy
stem
s di
ffer
from
sin
gle-
phas
e sy
stem
s in
th
at th
ey u
se a
set
of t
hree
vol
tage
sin
stea
d of
one
.
In g
ener
al, a
thre
e-ph
ase
syst
em is
mer
ely
a co
mbi
natio
n of
thre
e si
ngle
pha
se s
yste
ms
of w
hich
the
thre
e vo
ltage
s di
ffer
in p
hase
by
120
elec
tric
al d
egre
esfr
om e
ach
othe
r in
a p
artic
ular
seq
uenc
e.
Ther
e ar
e tw
o ba
sic
thre
e-ph
ase
conn
ectio
ns,
the
wye
(sta
r) a
nd th
e de
ltaco
nnec
tions
.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 5757
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Arm
atur
e w
ith c
oils
Fig.
2.1
Sch
emat
ic d
iagr
am o
f a th
ree-
phas
e ge
nera
tor
Φ
ω
Rot
or w
ith
field
win
ding
Coi
l A
Coi
l BC
oil C
Phas
e se
quen
ceis
the
ord
er
in w
hich
the
var
ious
pha
ses
reac
h th
e m
axim
um v
alue
.A
BC
AC
B
EMFs
indu
ced
in
the
thre
e id
entic
al w
indi
ngs
A, B
and
Car
e el
ectr
ical
ly d
istr
ibut
ed b
y 12
0oan
d m
ay b
e ex
pres
sed
as b
elow
:
e B(t)
=EM
sin
(ωt -
120
o )
e C(t)
=EM
sin (ωt-
240o )
e A(t
)=E
M s
in (ωt -
0o ),
Thre
e-ph
ase
syst
ems
may
be
bala
nced
or u
nbal
ance
d.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 5858
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Bal
ance
d an
d un
bala
nced
syst
ems
A
thre
eA
th
ree --
phas
e po
wer
sy
stem
in
w
hich
th
e th
ree
phas
e po
wer
sy
stem
in
w
hich
th
e th
ree
gene
rato
rs
have
ge
nera
tors
ha
ve
volta
ges
volta
ges
that
ar
e ex
actly
th
at
are
exac
tly
equa
leq
ual
in
in
mag
nitu
de a
ndm
agni
tude
and
120
deg
120
deg
diffe
rent
in p
hase
, and
in w
hich
di
ffere
nt in
pha
se, a
nd in
whi
ch
all
all
thre
e lo
ads
are
iden
tical
thre
e lo
ads
are
iden
tical
is c
alle
d a
bala
nced
thr
eeis
cal
led
a ba
lanc
ed t
hree
--ph
ase
syst
em.
phas
e sy
stem
.
Bal
ance
d sy
stem
Un-
bala
nced
sys
tem
A
thre
eA
th
ree --
phas
e po
wer
sy
stem
in
w
hich
th
e th
ree
phas
e po
wer
sy
stem
in
w
hich
th
e th
ree
gene
rato
rs h
ave
gene
rato
rs h
ave
volta
ges
volta
ges
that
are
th
at a
re u
nequ
alun
equa
l in in
mag
nitu
dem
agni
tude
and/
or
and/
or d
o no
t diff
er b
y 12
0 de
g in
pha
sedo
not
diff
er b
y 12
0 de
g in
pha
se, a
nd/o
r in
whi
ch
, and
/or i
n w
hich
al
l al
l th
ree
load
s ar
e no
t id
entic
alth
ree
load
s ar
e no
t id
entic
alis
cal
led
a ba
lanc
ed
is c
alle
d a
bala
nced
th
ree
thre
e --ph
ase
syst
em.
phas
e sy
stem
.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 5959
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Req
uire
men
ts o
f a b
alan
ced
syst
em
!!A
ll th
ree
varia
bles
hav
e th
e A
ll th
ree
varia
bles
hav
e th
e sa
me
ampl
itude
.sa
me
ampl
itude
.
!!A
ll th
ree
varia
bles
hav
e th
e A
ll th
ree
varia
bles
hav
e th
e sa
me
freq
uenc
y.sa
me
freq
uenc
y.
!!A
ll th
ree
varia
bles
are
A
ll th
ree
varia
bles
are
120
deg
. in
phas
e.12
0 de
g. in
pha
se.
120
240
0
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 6060
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
The
pow
erde
liver
ed b
y a
sing
le-p
hase
syst
em p
ulsa
tes.
Ang
le
+ -0
1
20
2
40
36
0A
ngle
13
2
+ -
In 3
-pha
se sy
stem
, the
ave
rage
pow
erde
liver
ed to
load
is
con
stan
tra
ther
than
pul
satin
g
Adv
anta
ges o
f thr
ee-p
hase
circ
uits
In a
bal
ance
d 3-
phas
e sy
stem
, con
duct
ors
need
be
only
abo
ut 7
5%th
e si
ze o
f co
nduc
tors
for
a s
ingl
e-ph
ase
syst
em o
f th
e sa
me
kVA
ratin
g.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 6161
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Ana
lysi
s of t
hree
-pha
se b
alan
ced
circ
uits
In t
his
conn
ectio
n, s
imila
r en
ds (
star
t or
fin
ish)
of e
ach
of t
he t
hree
-ph
ase
win
ding
s ar
e jo
ined
toge
ther
.
s f
s f
s f
I PH
I PH
I PH
I L
I L
I L
V ab
V bc
V ca
N
The
com
mon
term
inal
is re
ferr
ed to
as
the
neut
ral p
oint
(N).
a cb
The
othe
r th
ree
term
inal
s ar
e co
nnec
ted
to 3
term
inal
s a,
b, a
ndc
(line
te
rmin
als)
.
Star
-con
nect
ed c
ircu
its
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 6262
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
The
sour
ce is
sai
d to
be
bala
nced
if th
e vo
ltage
s V a
n, V b
nan
d V c
n(c
alle
d ph
ase
volta
ges)
, hav
e th
e sa
me
mag
nitu
de a
nd s
um to
zer
o, i.
e.,
Line
and
pha
se v
olta
ges
are
diffe
rent
in
a w
ye(s
tar)
co
nnec
tion.
Line
and
pha
se c
urre
nts
are
the
sam
ein
a
wye
(sta
r)
conn
ectio
n.
I PH
I PH
I PHI L
I L I L
Na cb
Vab
Vbc
Vca
Ana
lysi
s of 3
-pha
se st
ar c
onne
cted
cir
cuits
. . .
.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 6363
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Ana
lysi
s of 3
-pha
se st
ar c
onne
cted
cir
cuits
. . .
.
Neu
tral
vol
tage
Con
side
r the
thre
e-ph
ase
bala
nced
wye
conn
ecte
d sy
stem
sho
wn
in F
ig.
N
Z L
Z L
Z L
I A
I B I C
E∠0o E∠
-120
oE∠
-240
o
Z G
Z G
Z G
A B C
N
LG
NA
ZZ
VE
I+−
∠=
o 0
LG
NB
ZZ
VE
I+
−−
∠=
o12
0
LG
NC
ZZ
VE
I+
−−
∠=
o24
0
Acc
ordi
ng t
o K
ircho
ff�s
curr
ent
law
,I A
+ I B
+ I C
= 0
024
012
00
=+
−−
∠+
+−
−∠
++−
∠⇒
LG
N
LG
N
LG
N
ZZ
VE
ZZ
VE
ZZ
VE
oo
o
LG
N
LG
ZZ
VZ
ZE
EE
+=
+−
∠+
−∠
+∠
⇒3
240
120
0o
oo
Sinc
e,
,024
012
00
=−
∠+
−∠
+∠
oo
oE
EE
V N=
0.
The
curr
ent i
n th
e th
ree
lines
are
:
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 6464
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Sing
le P
hase
Equ
ival
ent o
f a B
alan
ced
Thre
e Ph
ase
Syst
em
The
exp
ress
ion
for A
-pha
se c
urre
nt is
I A =
LZ
Z+
∠ G
No
V - 0
E=
LG
ZZ
+∠o 0
E.
The
circ
uit
diag
ram
th
at
repr
esen
ts
the
sing
le-p
hase
eq
uiva
lent
of
a
bala
nced
thre
e-ph
ase
syst
em is
Fig.
Sin
gle-
phas
e eq
uiva
lent
of a
thre
e-ph
ase
bala
nced
syst
em
I A
Z LV A
N
Z G
E∠0o
This
vi
ew
sim
plifi
es
com
puta
tion
of
a ba
lanc
ed
thre
e-ph
ase
pow
er s
yste
m.
The
resu
lts fo
r one
pha
se
can
be a
pplie
d to
all
the
thre
e ph
ases
.
Ana
lysi
s of 3
-pha
se st
ar c
onne
cted
cir
cuits
. . .
.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 6565
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Ana
lysi
s of 3
-pha
se st
ar c
onne
cted
cir
cuits
. . .
.V
olta
ges a
nd c
urre
nts
By
Kirc
hoff�
svo
ltage
law
, lin
e vo
ltage
(VA
B=
V L) i
s gi
ven
by
BA
BNAN
NB
ANL
VV
VV
VV
V−
=−
=+
=N
egle
ctin
g th
e so
urce
impe
danc
e (Z
G),
N
Z L
Z L
Z L
I A
I B I C
E∠0o E∠
-120
oE∠
-240
o
Z G
Z G
Z G
A B C
N
oo
120
0−
∠−
∠=
EE
V L[
])12
0si
n()
120
cos(
jE
E−
−=
Ej
Ej
EE
2323
2321
+=
−−
−=
()
oo
o30
330
sin
30co
s3
2123
3∠
=+
=
+=
Ej
Ej
E
Ther
efor
e, t
he r
elat
ions
hip
betw
een
the
mag
nitu
des
of t
he l
ine
volta
ge a
nd
phas
e vo
ltage
in a
wye
conn
ecte
d ge
nera
tor o
r loa
d is
o
303
∠=
PhL
VV Th
us, t
he li
ne-to
-line
vol
tage
has
a m
agni
tude
equ
al to
√3
times
the
mag
nitu
de
of th
e ph
ase
volta
gean
d it
lead
s th
e co
rres
pond
ing
phas
e vo
ltage
by
30°.
It ca
n be
obs
erve
d th
at th
e lin
e cu
rren
ts a
re id
entic
al to
the
corr
espo
ndin
g ph
ase
curr
ents
. Ph
LI
I=
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 6666
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Ana
lysi
s of 3
-pha
se d
elta
con
nect
ed c
ircu
itsIn
thi
s co
nnec
tion,
diff
eren
t en
ds (
star
t &
fin
ish)
of t
he t
hree
-pha
ses
are
join
ed to
geth
er. T
here
is n
o ne
utra
l poi
nt (N
)in
delta
con
nect
ion.
s f
s f
s f
I ph
I ph
I ph
BA C
a
bc
I L
I L I L
Line
and
pha
se c
urre
nts
are
diffe
rent
in a
de
lta c
onne
ctio
n.
Line
and
pha
se v
olta
ges
are
the
sam
ein
a
delta
con
nect
ion.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 6767
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Ana
lysi
s of 3
-pha
se d
elta
con
nect
ed c
ircu
its .
. . .
.V
olta
ges a
nd c
urre
nts
Con
side
r the
thre
e-ph
ase
bala
nced
del
ta s
yste
m s
how
n in
Fig
.
R LR L
R L
I CA
I AB
I BC
V AB
V CA
B
A C
A
BC
V BC
I A
I B I C
G
ener
ator
Load
o 0∠
=Ph
ABV
Vo
120
−∠
=Ph
BCV
Vo
240
−∠
=Ph
CA
VV
Bec
ause
the
load
is re
sist
ive,
the
phas
e cu
rren
tsar
e gi
ven
by:
o 0∠
=Ph
ABI
Io
120
−∠
=Ph
BCI
Io
240
−∠
=Ph
CA
II
In a
del
ta c
onne
ctio
n, th
e lin
e-to
-line
vol
tage
betw
een
any
two
lines
will
be
the
sam
eas
th
e vo
ltage
in
th
e co
rres
pond
ing
phas
e.
App
lyin
g K
CL
to n
ode
A y
ield
s,C
AAB
AI
II
−=
oo
240
0−
∠−
∠=
PhPh
II
[])
240
sin(
)24
0co
s(j
II
PhPh
−−
=
PhPh
PhPh
Ij
Ij
II
2323
2321
−=
+−
−=
()
oo
o30
330
sin
30co
s3
2123
3−
∠=
−=
−=
PhPh
PhI
jI
jI
o30
3−
∠=
PhL
IITh
e ph
ase
volta
ges
in th
e ge
nera
tor a
re g
iven
by
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 6868
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
I Ale
ads I
AB
by
30ο
VA
Bla
gs V
Aby
30
οA
CB
pha
se
sequ
ence
I Ala
gs I A
Bby
30
οV
AB
lead
s V
Aby
30
οA
BC
pha
se
sequ
ence
Cur
rent
m
agni
tude
s
Vol
tage
m
agni
tude
s
Del
ta c
onne
ctio
nW
yeco
nnec
tion
PhL
II
3=
PhL
II
=
PhLL
VV
=Ph
LLV
V3
=
Ana
lysi
s of t
hree
-pha
se b
alan
ced
circ
uits
. . .
.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 6969
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
DE
LT
A T
O S
TA
R T
RA
NSF
OR
MA
TIO
N
z CZ B
z A
CB
AA
CB
Fig.
Del
ta to
Sta
r tr
ansf
orm
atio
n.
The
val
ue o
f im
peda
nce
betw
een
any
two
ends
of
the
star
or
delta
mus
t be
eq
ual w
hen
the
two
circ
uits
are
equ
al.
Z A +
ZB
=
CA
BC
AB
CA
BC
AB
)(
ZZ
ZZ
ZZ
++
+(1
)Z B
+ Z
C =
C
AB
CA
B
AB
CA
BC
)(
ZZ
ZZ
ZZ
++
+(2
)
Z C +
ZA
=
CA
BC
AB
BC
AB
CA
)(
ZZ
ZZ
ZZ
++
+(3
)Eq
uatio
ns (1
) + (3
) -(2
)
z AB
z CA
z BC
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 7070
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Z A =
C
AB
CA
B
CA
AB
ZZ
ZZ.
Z+
+
z Cz B
z A
CB
A
z AB
zCA
z BC
Z B =
C
AB
CA
B
AB
BC
ZZ
ZZ.
Z+
+
Z C =
C
AB
CA
B
BC
CA
ZZ
ZZ.
Z+
+
Del
ta to
sta
r tra
nsfo
rmat
ion
. . .
.
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 7171
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
STA
R T
O D
EL
TA
TR
AN
SFO
RM
AT
ION
CB
AA
CB
Fig.
Sta
r to
Del
ta tr
ansf
orm
atio
n.
The
val
ue o
f adm
ittan
cebe
twee
n an
y tw
o en
ds o
f the
star
or
delta
mus
t be
equa
l w
hen
the
two
circ
uits
are
equ
al.
Y AB
= C
BA
BA
YY
YY.
Y+
+Y B
C=
CB
A
CB
YY
YY.
Y+
+Y C
A=
CB
A
AC
YY
YY.
Y+
+
Y CY B
Y AY A
BY C
A
Y BC
MEE
20
/ PR
INCI
PLE
OF
ELEC
TRIC
AL M
ACH
INES
/ FE
BR
UAR
Y 20
08SU
L 7272
To
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
yTo
Be A
Wo
rld
Cla
ss M
ari
tim
e A
cad
em
y
Act
ive
pow
er a
nd re
activ
e po
wer
cal
cula
tions
The
tota
l ave
rage
pow
er a
bsor
bed
by a
thre
e ph
ase
bala
nced
load
del
iver
ed
by a
thre
e-ph
ase
gene
rato
r, is
equ
alto
the
sum
of t
he p
ower
s in
eac
h ph
ase.
Th
e vo
ltage
and
cur
rent
in e
ach
phas
e ar
e eq
ual.
The
tota
l 3-p
hase
real
pow
eris
,θ
θco
s3
cos
33
××
×=
××
==
LLL
PhPh
PhT
IV
IV
PP
whe
re, θ
is th
e ph
ase
angl
e be
twee
n th
e vo
ltage
and
the
curr
ent o
r,
PhPhPh
PhT
RVR
IP
22
33
=×
=
Sim
ilarly
, the
tota
l rea
ctiv
e po
wer
is,
θθ
sin
3si
n3
3×
××
=×
×=
=L
LLPh
PhPh
TI
VI
VQ
Q
or
PhPhPh
PhT
XVX
IQ
22
33
=×
=
or,
Tota
l app
aren
t pow
eris
,
PhPh
LLL
PhPh
TT
T
ZI
IV
IV
jQP
S
××
=
××
=
×=
±=
2
*
*
3
3
3
PhPhT
ZVS
2
3=
P 1 P 2
P 1 P 3
P 2
PT
= P
1+
P2
+ P
3P
T=
P1
+ P
2
