nh
materials ave amat,oositve Suscostibat
Homapncte fields. hoe naterials ale sliahtl
&macnete Jield ad do not
Redat
btracted by
he manete ppe
h n the external fteld ic remoVed. 'toeamapnetue Aaterialg
repertis olu due t» tha presence
Sbme paised le cdron s,
e r r o maaneko Hateriale a e a laie, potik ve suscaftubb
tena manetie feld, uy exibit a Sn
o
4Hracton lo mapntkc fields and als able to reeth her
mautie felds prpertis ater theexternas ield he bea. fero
tertoapnee
m adcrials have Some Unedured ee etrons So thelr
atoms a t e a
net
mapnztke momat
magne
tation
H Hjield vtensity
H
TAL
1.4 Sauration and
Por
e
peiut w a
2
ttysteres1
between
magneitc crcuct, thw relatsnship
flu densily B es las (b|) and la maneterin rce8
1 amperes per meter (Am) u give ba:
B HHH
Lwhe
47 10and P. u to rrlateu Perneab1lty
L ree spa
PMedium
fermeabi
ameakure
nagntgat.on tht a matertal
b b u io respdnge to an apeied mapnebe field.
.P
appcatan s an externall mapnae feld tundsto bringths
tho domato axa to lina to reinrpr he
field t s Hheur oun
magne
achan so that u y o be comes Nerr much mote
ce
wense
shan ll
t
ho, domains ayes,a n one p thoeeterna feld dx
matertal
Said
ee Ea tu.rat.
i.4.1. Hsteresis Loop
* Ferromagnatkc materiak are caracteried lon 2 B-H ehxcs
that u b oth non-naar and m ultialued. Thi s
to
al a
Generally rejerrn
Hatderesít Chec.
hoide
ot
k
Llluatrate ic ohenomenon, we e 4h Sezuene t
a
htereste loop f
tqure belocs <hosng the evolutron f
Unittterom.apnekc core.
Torot4 H
Fi
>H
O
Evo|nhion s
Hsteer,
3T4
5T/4
T4
da
6A
(IT)-
T4
m)
()
3T
3T4
o
31l4
t
Assume hat tw MMF, and hence H, is slowely Vagi So8oidal waveform wth period T, as Show t s Sae torm
arp
T h cvplion
f 4 B-H Hpsteredir lbop udis cussed ttff iteral
LnervalI: Bedweeu t-0 and Tla.
maonete field tensity His pasit«
and tntreasiuo. e
flux density mcreasina alo na R. ttd Curveoa)
tp fo th SaturaWan Value s . Increasing
Sevel daes uot
Lnterva
tes ult
v an inerease io B3.
: Betwean t Tl and Th Ho magnetie field utesly
positzve but dacreasina, I
alo
H be yond tho saturdho
flur denaty B8 u Dbserved to decreag
tho SiGment ab. Aote that ab
higher than 0a, dnd us
o
Ha Aane Valuefor B Br ad is dlt qnn 2ero esenthough H
Ls ero t at point To tine t-T/2. (( valae
of br u rejerred
toa ts restolual tield, re man ence, or reentvity. TA we leae
Hto Coil
Uneraine.d, 4h Core sill be magnakred
Lnterva
:
Be Husoen t= T ad 8T/4 4e Hu revesed and
Creasihq tn magn:tude.
uValue H
b de creages ts zero 2t poini C.
esults inreversala
2t skrch mapnetzat.
is tero D called he Coercive force (He).
Furtherdecrease a
H results i» reversal
B up t»
fo
d,
point
t T4
correspon din9
Linterval 1V: Betveen t 8T/ and T, t
Value H is
leaatve ut decreasen he B ts nesatve and ncreates
fDm t» e. The estdual field S ohserved e
itH=O.
Interval
V: Beeen t-Tand 5T/4, His increase.d trom O,
and the lux density bis negative but docreastn 9 up to t
ure
omaterial is demagnctgad. beiyond f ve frnd
hat b Tncreasts up to
aagain.
H isdecreased and
retra Cethe &ame
pa th.
reversed
,
the
Cure does not
B C T )
bs
Resdual f u l d
remanen te br
nLnC
b - H CuFtef o
materal
hetentvita
S a t u r a ' o n
a g
r)
m
non-
H
(A)
HysleresisCure
Cufe
f o r c eH e -
oerd
FEHstresit loop jo 2herronapnahe
naiial.
r
ounce
Cper Umt v»lume)
T
-
v
anea nclo sed H h e wsteens Joop. Tisenelr
dcsi patd saaat loss.
e
ousel disti patid throuph teresis P. iuiven
R
(
Bm)"
u
teresha lou pesecnnd ts
equadand
to the pndut' bat tla op alea
ka- Consta3
m- mayimum flug dersiky
n: 1 s 3'S.
in thw Core material,
ltapes
indue
w
i
lux
hanpe
C
e
currents cirulating do tho Core. (he induced
ochoil Fegult
CuRetstend t4stablish a flax thad Opprses tha ona'a Chang
upoud l
At
Cun
te dourcowich buill toult "in pouwer odue+
4l Core materi this n dutad Cunta a Callad Jd
Core lou u ud 4o denstu tw Cobination eddy- Currat
md Htares power osses tn
tlo malena
t5. Manetio Ctrcuds
Ciruit ú tho flow of mapnelko lux nkuced
The ame mameue
y
th
ferom ah
Circutt.
Core Can he mads analogous to an elecbriod
maqnee
f u r
h e s
mean core
t
Cross Sectona
are A
manetc Core
permeali
S t n a i n
N
urns
From t
Pa Stmple manelco ckt
abbve u e hu magneke fiald Can be Vtsu allted sterms
Axnes tich sform clased lops terkinbed vit% ths udind9
AS
Ppd to tho mapnahe cet tu 8Dura q 4 mapnelít fteloTu
Core is
Anpere -Hurn pnduct N. Tn magnatke bH terminolos
Ni u emapatumottve pre
On
Ckt
(Vmt)°actng
hmpneto fux crosst
otw
hormall Cowponm
a uaa S
tlectrtt Ckt aaln
-
tru
B.da. o
R
mapniht
Fl:(
-BA
Reluctan.a, .
Cnmf
Mapnatke Ckt analaT
Surfe ce tep0
From Avperesud
HS
lt
HCAmpere turns qer meler)Ts knouwn ag
h
etjort requised to produ a moprake
fild.
acore 3uch auin
the
fqun,
B-PH
didm BA
PNiA
Uet
Crcuit equivale.d eguahaA
mapne ht
FBR Sinilar to V IR)
kaenest o R w mapl ckt analegY 4 Ailar o Concat
hegLs an C
Keluclan ta 13 4
meagure et mari esishna to t
manettc lux,
Series reluctanco Keg
Rt R+ 23..
peralle reluckna
e&eana
"a e
T»Vera
a
elucknce (Pp)
Pal:A
(P- PA R
PA
flo
EXamale. 11
300m-
loc
15tm
N- 200uns
30Un
SUn
Deph:
Degth-loc
30Cm
tertomaqnelo core s snoo aboe. Turee sides o this Core aw
uni orm tidt, \stile Hhe,
fpuudb stde is Somat +hiwner. Te depth b t
Core (iato u
pae) Ts Mocm, amd tha oherdimensions are shotdn tu fue.
4 aoo twn Coi
i
wrapped arund tu losjsida o th tore. Acsuning
relatve permeabilty *
2500, How. much lux wl ba podu cad
by A put Current
A
n
Soutn Tkree Stdesth
Core
hove th sama.
csa,
dlt area. hus the tore Can be dindad
Stde os
(1) tu Siuala iner Sida
c) the dther 8 sidesta ken toge
a
tslile t
t
qdh
2 regto
thor.
The Correspondeng maprubie. e
h a lengl
aa s
loio cm
=
sian4. is 45tm, and th tnie-secta
Aoo cm. Tharefore, the
reluctanca iato Fac
4+5 m
PAL
MMA
250o) (4T#|6) (o-blm
14,300 A. turns/
ean
So Cm
prth Jength rion 0 u 12ocm and the Csa 4IS2I00
.
t
reluctance n 4h se Lond region
4+3m
PA
MPA
Sbo4Tro)(o-btr)
27, 600 A. turns/
'. ( t o tal eluchnceT tw Core s
41300 A. turntol + 27,6ooA.
turn shob
A, g00 A. turns Jwb
Te tata mmf is
N i (200turna) GoA)= 200 A.turns
e
tot flux iw 4l core
gve q
=200 A qurns
R
4-1,900 turns/w
0004-3 NL
ELample 12 (Hu
A ferromapnelto core tose mea ath engt id 40cm. The is a arall
q4p 005m in te 3rachusa ato ohenoio tele Coo. e
coss
Secknal 0Aua -tko Cpre u 42, » elatve permeabitty og t
C o A oo0, and tho C
wie m thu Cota has 400 tur&
AssumD bat
}ringjg
Hu a l e
hcreaer t
efeckie Croy
&ctonal asa {le aib 9ap by 5 peiee. Gven
Mi, wfora,
hiud ()o tolad teuctamu to u path (
iro pus ai2R
amdC) tu Cu
a
hequired
ts
produee
{wx dntity
oST t a &P
N 400
Ans
eg a82,8oo A-tum
D:602 A
0-05Om
A120m
16 Produ con of
(13)
an
EMF
iu
EXap 18,; A cil wire th l0o turns orfpA arund
Qun iron core. he flux In the cofe is aenor t
-
0.05 stn 877t
W.
Alhc VolHape is pmdued at t
Polarity á o
egualLbn
teminals ott
Ceil
VolHpe duing e tme n
tla
cneag Cna t t referena drs Shosn in ta s u ,
4L
eros
Sola:h anihuds
leu f
Cid Nd44
- loo turns)( o«as Stn 34t)
t
885tos
lss5 sin (9nt +6)V
Attemat ve
.7
Produdkon of tnduced. Fore Dh 2wire.
Examelo: 14
A
amagi.c fils
bead.
t
ire u Casn a Cuko the PReRnce a
mgntiup den
oiei
dn fon trp to bottom
rce
33att
O25T, direckd vto h
Im Jona and cauus D5A
Cud u
et Hl baard, uId ase te mog e
Dnoucadon tho Lbie?
Ans: D-1sn, direct to t rad
EEEB344 Electromechanical Devices
Chapter
I1. Eddy Current Loss
.
2.
3.
4.
.
A time-changing flux induces voltage within a ferromagnetic core.
These voltages cause swirls of current to flow within the core eddy currents.
Energy is dissipated (in the form of heat) because these eddy currents are flowing in a resistive
material (iron)
The amount of energy lost to eddy currents is proportional to the size of the paths they follow
within the core.
To reduce energy loss, ferromagnetic core should be broken up into small strips, or laminations,
and build the core up out of these strips. An insulating oxide or resin is used between the strips, so
that the current paths for eddy currents are limited to small areas.
Path of eddy
current
perpendicular
to
Solid
Laminated
Core
core
Conclusion
Core loss is extremely important in practice, since it greatly affects operating temperatures, efficiencies,
and ratings of magnetic devices.
3. How Magnetic Ficld canaffectitssurroundings
3.1 FARADAY'S LAW-Induced Voltage from a Time-Changing Magnetic Field
Before, we looked at the production of a magnetic field and on its properties. Now, we will look at the
various ways in which an existing magnetic field can affect its surroundings.
1
Faraday's Law:
fa flux passes through a turn of a coil of wire, voltage will be induced in the turn of the wire that is
directly proportional to the rate of change in the flux with respect oftime'
eind
dh
dt
If there is N number of turns in the coil with the same amount of flux flowing through it, hence:
ind-N-
dt
where:
N - number of turns of wire in coil.
Note the negative sign at the equation above which is in accordance to Lenz' Law which states:
The direction ofthe build-up voltage in the coil is as such that ifthe coils were short circuited it would
produce current that would cause a flur opposing the original fux change.
16
3)
EEEB344 Electromechanical Devices
Chapter 1
Examine the figure below:
D.rection of i required
id
tums
Direction of
opposing flux
increasing
(6)
(a)
if the fHux-shown is increasing in strength, then the voltage built up in the coil will tend to
establish a flux that will oppose the increase.
" A curTent flowing as shown in the figure would produce a flux opposing the increase.
So, the voltage on the coil must be built up with the polarity required to drive the current through
the external circuit. So, -eind
"NOTE: In Chapman, the minus sign is often left out because the polarity of the resulting voltage
can be determined from physical considerations.
Equation ed-d o/dt assumes that exactly the same flux is present in each turn of the
coil. This is not true, since there is leakage flux. This equation will give valid answer if the
windings are tightly coupled, so that the vast majority of the flux passing thru one turn of the coil
does indeed pass through all of them.
2
3.
Now consider the induced voltage in the ith turn of the coil,
do
dt
Since there is N number of turns,
ind
i=l
The equation above may be rewritten into,
ind
d
dt
where a (flux linkage) is def+ned as:
(weber-turns)
17
EEEB344 Electromechanical Devices
Chapter 1
4.
Faraday's law is the fundamental property of magnetic fields involved in transformer operation.
5.
Lenz's Law in transformers is used to predict the polarity of the voltages induced in transformer
windings.
3.2 Production of Induced Force on a Wire.
1.
A current carrying conductor present in a uniform magnetic field of flux density B, would produce
a force to the conductor/wire. Dependent upon the direction of the surrounding magnetic field, the
force induced is given by:
F=i(lx B)
where:
i -represents the current flow in the conductor
with direction of l defined to be in the direction of current flow
B magnetic field
density
-length of wire,
The direction of the force is given by the right-hand rule. Direction of the force depends on the
direction of current flow and the direction of the surrounding magnetic field. A rule of thumb to
determine the direction can be found using the right-hand rule as shown below:
Thumb
(resultant force)
Index Finger
(current direction)
Middle
Finger
(Magnetic Flux Direction)
3
Right Hand rule
The induced force formula shown earlier is true if the current
carry ing conductor is perpendicular
to the direction of the magnetic field. If the current carrying conductor is position at an angle to the
magnetic field, the formula is modified to be as follows:
F = ilB sin 6
Where:
4.
angle between the conductor and the direction of the magnetic field.
In summary, this phenomenon is the basis of an electric motor where
torque or rotational force of
the motor is the effect of the stator field current and the
magnetic field of the rotor.
Example 12
X
B
The figure shows a wire carry ing a current in the
presence of a
magnetic field. The magnetic flux density is 0.25T, directed into the
page. If the wire is Im long and carries 0.5A of current in the
direction from the top of the page to the bottom, what are the
magnitude and direction of the force induced on the wire?
X
X
X
18
length of the wire in the magnetic field
(vx B) I
(v x B)l cos6
vxB
X
X
30
X
in.d
T
X
X
X
+
types of generators.
X
X
X B
19
to the right in a magnetic field. The flux density is 0.5T, out
ofthe page, and the wire is Im in length. What are the
magnitude and polarity of the resulting induced voltage?
Figure shows a conductor moving with a velocity of 10m/s
the resulting induced voltage?
The figure shows a conductor moving with a velocity of
Sm/s to the right in the presence of a magnetic field. The
flux density is 0.5T into the page, and the wire is Im length,
oriented as shown. What are the magnitude and polarity of
The induction of voltages in a wire moving in a magnetic field is fundamental to the operation of all
where:
0 angle between the conductor and the direction of (vx B)
emd
magnetic field. Hence a more complete formula will be as follows:
Example 1.9
X
eind
Note: The value of I (length) is dependent upon the angle at which the wire cuts through the
-
V-velocity of the wire
B-magnetic field density
where:
If a conductor moves or 'cuts' through a magnetic field, voltage will be induced between the
terminals of the conductor at which the magnitude of the induced voltage is dependent upon the
velocity of the wire assuming that the magnetic field is constant. This can be summarised in terms
of formulation as shown:
Example 1.8
3
2.
1.
3.3 Induced Voltage on a Conductor Moving in a Magnetic Field
Chapter 1
EEEB344 Electromechanical Devices
EEEB344 Electromechanical Devices
Chapter 4
1. Asimple loop in a uniform magneticfield
The figure below shows a simple rotating loop in a uniform magnetic field. (a) is the front view and (b) is
the view of the coil. The rotating part is called the rotor, and the stationary part is called the stator.
N
S
Cdc
ha
Vcd
B is a uniform magnetic
field, aligned as shown.
+0-
(a)
(b)
This case in not representative
is not constant in either
torque on the loop are the
of real ac machines (flux in real ac machines
and
magnitude or direction). However, the factors that control the voltage
the factors that control the voltage and torque in real ac machines.
same as
The voltageinduced inasimple rotatingloop
If the rotor (loop) is rotated, a voltage will be induced in the wire loop.
To determine the magnitude and
shape, examine the phasors below:
ab
(a)
(b)
(c)
induced etot On the loop, examine each segment of the loop separately and
To determine the total voltage
The voltage on each segment is given by equation
sum all the resulting voltages.
Cind(VXB).1
(remember that these ideas all
revert back to the linear DC machine concepts in Chapter 1).
1. Segment ab
to the
the path of rotation, while the magnetic field B points
The velocity of the wire is tangential to
the
ab.
Thus,
into the page, which is the same direction as segment
right. The quantity vxB points
induced voltage on this segment is:
2
ecb0
segment
segment. Thus,
into the page
Thus,
Segment be
In the first half of this
segment, the quantity vx B points into the page, and in the second half of this
segment, the quantity vxB points out of the page. Since the length
l is in the plane of the page, v x B
is perpendicular to / for both
portions of the
vBl sin Oab
-
=
ot
=
2r1,
2r @BI sin ot
eindABo sin ot
Cind
Thus,
density lines, so
endPmay
Sin wt
max
PmaxAB
Finally, since maximum flux through the loop occurs when the loop is perpendicular to the magnetic flux
since area, A
loop. Hence,
where r is the radius from axis of rotation out to the edge of the loop and o is the angular velocity of the
also, the tangential velocity v of the edges of the loop is:
0
Ifthe loop is rotating at a constant angular velocity a, then the angle 0 of the loop will increase linearly
with time.
2 vBL sin6
vBl sin 0ab t vBI sin Ocd
0d and sin 0 sin (180° - 0)
Alternative way to express eind
since 0ab=180°
out of the page
Cbat ecb t ede t ead
Cda0
segment bc, v x B is perpendicular to l. Thus,
Total induced voltage on the loop Cind
same as
4. Segment da
vBI sin Gcd
ecd(x B).l
3. Segment cd
The velocity of the wire is
tangential to the path of rotation, while B points to the right. The quantity
VXB points into the
page, which is the same direction as
ca.
2.
=
Cba(vx B).l
EEEB344 Electromechanical Devices
Chapter 4
EEEB344 Electromechanical Devices
Chapter 4
From here we may conclude that the
induced voltage is
dependent upon:
Flux level (the B
component)
beed of Rotation (the v
component)
Machine Constants (the I
component and machine
materials)
The Torque Induced ina Current-Carrying Loop
Assume that the rotor
loop is at some arbitrary angle 0 wrt the magnetic field, and that current is
in the loop.
flowing
a
B
(a)
To determine the
(b)
magnitude and direction of the torque, examine the phasors below:
l into page
.
r, F
into page
The0
(a)
b)
r, F out of page
I out of page
Tda=0
(c)
(d)
The force on each segment of the loop is given by:
F=i (lxB)
T=rFsin
B
EEEB344 Electromechanical Devices
Chapter 4
Torque on that segment,
. Segment ab
The direction of the current is into the page, while the magnetic field B points to the right.
points down. Thus,
(x B)
F=i (lx B)
= ilB
Resulting torque,
down
ab (Frsin 0)
= rilB sin ,
clockwise
ab
2.
Segment be
The direction of the current is in the plane of the page, while the magnetic field B points to the right.
x
B) points into the page. Thus,
F=i (lx BB)
=
ilB into the page
Resulting torque is zero, since vector r and I are parallel and the angle 8be is 0.
The(Fr sin, )
=
0
3. Segment cd
The direction of the current is out of the page, while the magnetic field B points to the right.
(x B)
points up. Thus,
F=i (lx B)
= ilB up
Resulting torque,
Tcd(FXrsin 9)
= rilB sin Gd
clockwise
4. Segment da
The direction of the current is in the plane of the page, while the magnetic field B points to the right.
(x B) points out of the page. Thus,
F=i(lx B)
ilB out of the page
Resulting torque is zero, since vector r and I are parallel and the angle Oa is 0.
The=(Frsin )
= 0
The total induced torque on the loop:
Tindabhe+Ted +Tda
=
rilB sin +rilB sin 6
arilB sin 0
EEEB344 Electromechanical Devices
Chapter 4
Note: the torque is maximum when the plane of the loop is parallel to the magnetic ficld, and the torque
is zero when the plane of the loop is perpendicular to the magnetic field.
An altemative way to express the torque equation can be done which clearly relates the behaviour of the
single loop to the behaviour of larger ac machines.
Examine the phasors below:
f the current inthe loop is as shown, that current will generate a magnetic flux density Bony with the
direction shown. The magnitude of Bloop is:
Bioop
Where G is a factor that
G
depends on the geometry of the loop.
The area ofthe loop A is 2rl and substituting these two equations into the torque equation earlier yields:
ind
AG B
sin6
= kBanB sin
Where k=AG/u is a factor depending on the construction of the machine, Bs is used for the stator
magnetic field to distinguish it from the magnetic field generated by the rotor, and 0 is the angle between
Bloop and Bs.
Thus,
TindkBloonB
From here, we may conclude that torque is dependent upon:
Strength of rotor magnetic field
Strength of stator magnetic field
Angle between the 2 fields
Machine constants
2. The Rotating Magnetic Field
Before we have looked at how if two magnetic fields are present in a machine, then a torque will be
created which will tend to line up the two magnetic fields. If one magnetic field is produced by the stator
of an ac machine and the other by the rotor, then a torque will be induced in the rotor which will cause
the rotor to turn and align itself with the stator magnetic field.
If there were some way to make the stator magnetic field rotate, then the induced torque in the rotor
would cause it to 'chase' the stator magnetic field.
How do we make the stator magnetic field to rotate?
Fundamental principle -a 3-phase set ofcurrents, each ofequal magnitude and differing in phase by
120° Aows in a 3-phase winding, then it will produce a rolating magnetic field ofconstant magnitude.
The rotating magnetic field concept is illustrated below empty stator containing 3 coils 120 apart. It is
a 2-pole winding (one north and one south).
6