1. Magnetic Circuits

8/12/2019 1. Magnetic Circuits

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Energy Conversion Flow Chart

Magnetic Circuits [1]

Transformers Electric Machines

AC DC

1-phase [] !-phase [!]

"n#uction Motor $ynchronous Motor [%] &en'

!-phase [(] 1-phase [)] &en' [*] Motor [+]

1


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Electric ,ower $ystem

Mechanical

Power InputSynchronous

Generator

6.6, 13.8, 24 KV

Stepup

!rans"or#er66,132,22$,4$$,

%$$,&%$ KV

Pri#ary

!rans#ission

'ine

Step(own

!rans"or#er12, 6.6 KV

Very 'ar)e 'oa(s

3phase In(uction

3phase Synch.

Step(own

!rans"or#er

4.16 KV'ar)e 'oa(s

3phase In(uction

Step(own

!rans"or#er22$ V

''

S#all *usto#ers

1phase Motors

+lectric Power

utput

-istriution

'ine


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Magnetic CircuitsMagnetic circuits are an essential part in any transformer or electromechanical #evice' "t

is the common lin .etween any two or more win#ings' "t services as an electricalisolation me#ium .etween two or more electronic circuits for noise re#uction or

elimination'

The simple magnetic circuit shown in figure 1-1 is use# to e/plain the concepts of themagnetic circuits'

Mean path len)th lc

*rosssectional area /c

Ma)netic core,

per#eaility

i

+

v

0in(in) with

turns

'ines o" "lu

Figure 1-1 Magnetic Circuit

Assumptions0

1' The permea.ility of the core is much greater than the permea.ility of the surroun#ingair'

' The core is ma#e of uniform cross-sectional area'

Amperes 2aw

Amperes law is the .asic relation .etween the flu/ intensity in the core 345 an# the

electric current in the coil 3i5'

=NildH' Ampere-turn

Consi#ering uniform magnetic #ensity across the core cross-sectional area6

cclHldH' Ampere-turn

7here cl 8 mean path length of the core'

cH 8 Magnetic flu/ intensity in the core'

9nits of 4c

!


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9sing Amperes law e:uation6 the units of the flu/ intensity 3 cH 5 are0

c

cl

NiH =

m

Aor

m

turnAmp

Direction of 4c

9se the right han# rule to #etermine the #irection of the magnetic fiel#' "f the fingerspoint to the #irection of the current6 the thum. will .e the #irection of the magnetic

fiel# intensity'

Magneto-Motive Force 3MMF5

"t is the force that pro#uces the magnetic flu/ in the core

cclHNiF == A-t

Flu/ Density ;

a' "n Air The relation .etween 4 3flu/ intensity5 an# ; 3flu/ #ensity5 in air is linear'

o

/ir *ore

53b53a

H

i

B

H

Figure 1- 3a5 Coil with air core6 3.5


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H

Bo =

Am

weber

mA

m

weber

=

$ince6 Li=

7here is the flu/ linage in we.er-turn or simply we.eri is the current in A

L is the in#uctance in 4enry

iL

= w.=A8 4enry

Therefore6 the units of the permea.ility are in 4=m'The permea.ility of air is constant an# e:uals to0

+1>( = o 4=m

.' "n a Magnetic Material with Constant ,ermea.ility

Figure 1-! >> to 1>>> for

materials use# in transformers an# rotating machines'Due to the high permea.ility of the magnetic core6 the magnetic flu/ is confine#

almost entirely to the core'

)


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Magnetic Flu/ 3 5

=s

daB' w.

?eglecting leaage flu/ an# using assumption 6 the flu/ will .e

ccc AB= we.er

Magnetic


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DC Electric Circuit Magnetic Circuit

@aria.le $ym.ol 9nits @aria.le $ym.ol 9nits

Current " Amps Flu/ 7.

EMF E or @ @olts MMF F A-t

Con#uctivity 1=3-m5 ,ermea.ility 4=m


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E/ample 1-1

Determine the flu/ #ensity in each leg of the magnetic circuit shown in the figure' The

#epth of the core is ) cm an# the relative permea.ility is 1>>'

Figure 1-) Magnetic circuit for E/ample 1-1

Magnetic Circuits with $mall Air-gap

Devices with moving elements must have air-gap

Mean path len)th lc

*rosssectional area /c

Ma)netic core,

per#eaility

i

+

v

0in(in) with

turns

'ines o" "lu

/ir)apper#eaility o

gl

5

Figure 1-* Magnetic circuit with a small air-gapAssumptions0

1' The permea.ility of the core is much greater than the permea.ility of the surroun#ing

air'

%


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' The core is ma#e of uniform cross-sectional area'

!' The gap length is much smaller than any of the core si#es6 ie Dan#E


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( )gc

g

g

c

c

gg

c

go

gc

c

ggcc

RR

A

l

A

l

lA

l

l

Bl

BlHlH

Hll.dHNiF

+=

+

=

+=

+=+=

===

A

>

>c+

Ni

c

cR

gR

Figure 1-% DC e:uivalent of figure 1-*

E/ample 1-

Determine the current in the win#ing that will pro#uce a flu/ of ! mw. in the center leg

of the magnetic circuit shown in the figure' The #epth of the core is ) cm an# Hr8(>>>'

Figure 1-G Magnetic circuit for E/ample 1-

Ferromagnetic Materials

To un#erstan# the phenomenon of the ferromagnetic materials6 it is necessary to analyIe

the atom of this type of materials' Any atom is ma#e of stationary nucleus an# electronsin constant motion in or.its aroun# it' Also6 each electron is spinning aroun# its own a/is'

1>


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1' Electron r.ital Motion

The or.ital motion of an electron aroun# the nucleus in a certain #irection

constitutes a current in the opposite #irection' The flow of a current in any or.it

creates a magnetic moment at the center of the or.it'

Figure 1-1> Electron or.ital motion

,mois the magnetic moment #ue to the or.ital motion of the electron'

' Electron $pinning Motion

Figure 1-11 Electron spinning motion

,msis the magnetic moment #ue to the spinning motion of the electron'

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"n atoms of a non-magnetic material6 the electrons are arrange# symmetrically so that the

magnetic moments #ue to the or.ital an# spinning motions cancel at the atom level

leaving the atom magnetically neutral'"n a ferromagnetic material6 the atoms are arrange# with their moments in parallel

forming magnetic #omains'

Figure 1-1 Magnetic #omains of a ferromagnetic material

7ith no magnetic fiel# applie# to the ferromagnetic material6 the net magnetic moment is

Iero an# the material is magnetically neutral'

The application of a magnetic fiel# 345 to the magnetic material forces some of themagnetic #omains to orient themselves in the #irection of the applie# fiel# creating a

magnetic fiel# in the core of the ferromagnetic material' As 4 is increase#6 more #omains

move in the #irection of the applie# fiel# creating more flu/ in the core an# hence more

flu/ #ensity' After all the #omains have .een move# in the #irection of the applie# fiel#6further increase of 4 will result in a very small increase in the flu/ #ensity an# that is #ue

to the fact that the material is saturate#' The plot of the flu/ #ensity 3;5 versus the flu/

intensity 345 is nown as the magnetiIation curve of the magnetic material'

M543; >c +=M is the magnetic polariIation of the material'

4mxM =

mx is the magnetic suscepti.ility of the material

1

4

5314

4543;

r

>

>

>c

m

r

m

m

x

x

x

+=

= +=

+=


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Jr )> *>> >>> )>>> 1>>6>>> %>>6>>>

Magnetization curve

Figure 1-1! MagnetiIation curve

'

E/ample 1-!

The #epth of the magnetic core shown in the figure is ) cm' Determine the current in the

win#ing that will pro#uce ) mwe. flu/ in the center leg of the core' 9se the

magnetiIation curve shown in figure 1-1>c of the te/t.oo'

1!


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Figure 1-1( Magnetic circuit of E/ample 1-!

E/ample 1-(Determine the current 3i5 that will pro#uce >') mw. flu/ in the core of the magnetic

circuit shown in the figure' The #epth of the core is ( cm' 9se the magnetiIation curve of

figure 1'1>c of the te/t.oo' Also #etermine the in#uctance of the coil'

Figure 1-1) Magnetic circuit of E/ample 1-(

Hysteresis

"f a coil is woun# on an iron core an# e/cite# with an alternating current6 the flu/intensity will alternate .etween K4man# 4m' ,lotting flu/ #ensity 3;5 as a function of

flu/ intensity 345 yiel#s the 4ysteresis loop of the magnetic material'

1(


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Figure 1-1* 4ysteresis loop of a magnetic material

Flux linkage and Inductance

The flu/ linage 3in we.-turn5 is the pro#uct of the flu/ 3 5 an# the num.er of turns

of the win#ing that the flu/ lins'

1)


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N= we.er-turn

coil'.y theseencoretheofreluctancetheis


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i

1 ??

+

1e

+

1v ,pen

+

e

Figure 1-1% Fara#ays law

"f a flu/6 6 passes through a turn of a coil6 a voltage will .e in#uce# in that turn that is

#irectly proportional to the rate of change of the flu/'

ne turndt

dte

153 =

Two turnsdt

dte

53 =

?1turnsdt

dNte

11 53 =

dt

d

dt

dNte 111 53

==

?turnsdt

d

dt

dNte 53

==

For two-port networs with .oth 1 an# ii are into the #otte# terminals

11

11111

iLiL

iLiL

+=

+=

7here0

2118 self in#uctance of coil 1 8cR

N1

218218 mutual in#uctance .etween coils 1 an# 8cR

NN 1

$ince port is left open6 i8> an# 111111 6 iLiL ==

dt

diL

dt

dte 111

11 53 ==

an#

dt

diL

dt

dte

11

53 ==

E/ample 1-)

1+


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For a #epth of ) cm6 relative permea.ility of (>>>6 an# a current A1>>cos)53 tti = 6

#etermine the following0

3a5 Flu/6 3.5 Flu/ linage6 3c5 "n#uctance6 an# 3#5 "n#uce# @oltage

Figure 1-1G Magnetic circuit for E/ample 1-)

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1. Magnetic Circuits