Chapter 1 - Magnetic Circuits

8/19/2019 Chapter 1 - Magnetic Circuits

1/35

1

PLT 205PLT 205

Chapter 1Chapter 1

ElectromagnetismElectromagnetism

ByBy

Shuhaimi ZakariaShuhaimi Zakaria


2/35

2

Magnetic Materials an CircuitsMagnetic Materials an Circuits

!ntrouction!ntrouction

Magnet contains a north pole an south pole"Magnet contains a north pole an south pole"

Magnet #lu$ lea%es the magnet as the northMagnet #lu$ lea%es the magnet as the north

pole an the place &here the #lu$ returns topole an the place &here the #lu$ returns to

the magnet as the south pole"the magnet as the south pole"


3/35

3

T&o types o# magnet'T&o types o# magnet'

• Permanent magnetPermanent magnet

• ElectromagnetElectromagnet


4/35

4

(ight )an (ule an *mpere+s La&(ight )an (ule an *mpere+s La&

,hen a conuctor carries current a magnetic,hen a conuctor carries current a magnetic#iel is prouce aroun it"#iel is prouce aroun it"


5/35

5

-ingers. inicate current irection-ingers. inicate current irection

Thum/ . inicate the irection o# magnetic #lu$Thum/ . inicate the irection o# magnetic #lu$is &rapping aroun the &ireis &rapping aroun the &ire


6/35

6

The relationship /et&een current anThe relationship /et&een current an

magnetic #iel intensity can /e o/taine /ymagnetic #iel intensity can /e o/taine /y

using *mpere+s La&"using *mpere+s La&"

*mpere+s La& states that the line integral o#*mpere+s La& states that the line integral o#

the magnetic #iel intensity ) aroun athe magnetic #iel intensity ) aroun a

close path is eual to the total currentclose path is eual to the total current

linke /y the contour"linke /y the contour"



7/35

7

∫ ∑= idl . H


8/35

8

)')' the magnetic #iel intensity at a point on thethe magnetic #iel intensity at a point on the

contour"contour"

dl:dl: the incremental length at that point"the incremental length at that point"

!#!# θ θ == the angle /et&een %ectorsthe angle /et&een %ectors HH anan dldl

thenthen

∫ ∑=θ icos Hdl


9/35

9


Consier a rectangular core &ith &iningConsier a rectangular core &ith &ining

Ni Hl c =∴

Nii =∑cl dl =

There#oreThere#ore

cl

Ni H =


10/35

10

(elationship /et&een B3)(elationship /et&een B3)

The magnetic #iel intensity ) prouces aThe magnetic #iel intensity ) prouces amagnetic #lu$ ensity B e%ery&here itmagnetic #lu$ ensity B e%ery&here it

e$ists"e$ists"

Tesla2 or )m / weber ( H B µ=

T

2

0 or )m / wb( H B r µµ=


11/35

11

µ 3 Permea/ility o# the meium3 Permea/ility o# the meium

0µ 3 Permea/ility o# #ree space3 Permea/ility o# #ree space m.t . Awb-710x4π

0 µ

µ µ =r 3 (elati%e permea/ility o# the meium3 (elati%e permea/ility o# the meium

-or #ree space or electrical conuctor 4*l or Cu-or #ree space or electrical conuctor 4*l or Cu

or insulators is unityor insulators is unityr µ


12/35

12

M*6ET!C E78!9*LET C!(C8!TM*6ET!C E78!9*LET C!(C8!T

* simple magnetic circuit* simple magnetic circuitha%ing a ring shapeha%ing a ring shape

magnetic core 4toroimagnetic core 4toroi

an a coil that e$tensan a coil that e$tens

aroun the entirearoun the entirecircum#erencecircum#erence

,hen current i #lo&s,hen current i #lo&s

through the coil o# through the coil o# turns a magnetic #lu$ isturns a magnetic #lu$ is

prouce"prouce"


13/35

13

*ssumption*ssumption

o*ll #lu$es are con#ine to the core*ll #lu$es are con#ine to the core

oThe #lu$es are uni#ormly istri/ute in the coreThe #lu$es are uni#ormly istri/ute in the core

The #lu$ outsie the toroi 4calle leakage #lu$The #lu$ outsie the toroi 4calle leakage #lu$

is so small 4can /e neglecteis so small 4can /e neglecte

8se *mpere+s La&8se *mpere+s La&

∫ = Nidl . H

Nir . H =π2

Ni Hl =

F Ni Hl ==

- : Magnetomoti%e #orce 4mm#- : Magnetomoti%e #orce 4mm#


14/35

14

)m / At (

l

Ni H =

H B µ=

)T (

l

Ni B

µ=

,here,here

N N – no of turns of coil – no of turns of coil

ii – – current in the coilcurrent in the coil

H H – magnetic field intensity – magnetic field intensity

ll – mean length of the core – mean length of the core


15/35

15

Magnetic -lu$ ;ensity 4B anMagnetic -lu$ ;ensity 4B an

Magneti


16/35

16

Case 2Case 2

-erromagnetic material core 4iron steel #errite-erromagnetic material core 4iron steel #errite

H H B r µµ=µ= 0

)T (

l

Ni B r

µµ= 0

The magnetic #lu$ ensity BThe magnetic #lu$ ensity B

increases almost linear inincreases almost linear in

the region o# lo& %alues o#the region o# lo& %alues o#

magnetic intensity )"magnetic intensity )"

*t higher %alue o# ) the*t higher %alue o# ) the

change o# B is nonlinear"change o# B is nonlinear"


17/35

17

M*6ET!C E78!9*LET C!(C8!TM*6ET!C E78!9*LET C!(C8!T

The flux in the coil,The flux in the coil, )weber ( BA=φ

R

F

R

Ni

Al

Ni A

l

Ni==

µ

=µ

=φWhereWhere

ФФ – flux in the coil (wb) – flux in the coil (wb)

F – magnetomotive force (mmf)F – magnetomotive force (mmf)

R – 1R – 1!!" # 1$ ,Reluctance" # 1$ ,Reluctance

$ # %ermeance$ # %ermeance

" – cross sectional area" – cross sectional area


18/35

18

**L=6> BET,EE M*6ET!C**L=6> BET,EE M*6ET!C

C!(C8!T *; ELECT(!C C!(C8!TC!(C8!T *; ELECT(!C C!(C8!T

a Magnetic eui%alent circuita Magnetic eui%alent circuit / Electric eui%alent circuit/ Electric eui%alent circuit

To sol%e magnetic eui%alent circuit . ?irchho##To sol%e magnetic eui%alent circuit . ?irchho##

9oltage an Current La&s 4?9L @ ?CL9oltage an Current La&s 4?9L @ ?CL

&lectric circuit&lectric circuit 'agnetic circuit'agnetic circuit

riving forceriving force &'F (&)&'F (&) ''F (F)''F (F)

$roduces$roduces urrent (i)urrent (i) Flux (*)Flux (*)

+imited by+imited by Resistance (R)Resistance (R) Reluctance (R)Reluctance (R)


19/35

19

M*6ET!C C!(C8!T ,!T) *!( 6*PM*6ET!C C!(C8!T ,!T) *!( 6*P

!n electric machines the!n electric machines the

rotor is physically isolaterotor is physically isolate#rom the stator /y the air#rom the stator /y the air

gap"gap"

Practically the same #lu$ isPractically the same #lu$ ispresent in the poles 4maepresent in the poles 4mae

/y magnetic core an the air/y magnetic core an the air

gap"gap"

To maintain the same #lu$To maintain the same #lu$

ensity the air gap &illensity the air gap &ill

reuire much more mm# thanreuire much more mm# than

the core"the core"


20/35

20

cc

cc

A

l R

µ=

g g

g

g A

l

R µ=

g c R R

Ni

+=φ

g g cc I H l H Ni +=

,here,here l l cc . mean length o# the core . mean length o# the core

l l g g . the length o# the air gap . the length o# the air gap

c

cc A

B φ=

g

g

g A B

φ=


21/35

21

-(!6!6 E--ECT-(!6!6 E--ECT

-ringing E##ect : Bulging o# the #lu$ lines in-ringing E##ect : Bulging o# the #lu$ lines in

the air gap"the air gap"

The e##ecti%e cross section area o# air gapThe e##ecti%e cross section area o# air gap

increase so the reluctance o# the air gapincrease so the reluctance o# the air gapecrease" The #lu$ ensity Becrease" The #lu$ ensity Bgg A BA Bcc " B" Bcc is theis the

#lu$ ensity in the core"#lu$ ensity in the core"

!# the air gaps is small the #ringing e##ect can!# the air gaps is small the #ringing e##ect can

/e neglecte"/e neglecte"


22/35

22

c g A A =

c

c g A

B B φ==

!n practical large air gap &ill /e!n practical large air gap &ill /ei%ie into se%eral small airi%ie into se%eral small air

gaps to reuce the #ringinggaps to reuce the #ringing

e##ect"e##ect"


23/35

23

!;8CT*CE!;8CT*CE

* coil &oun on a magnetic* coil &oun on a magnetic

core as sho&n in #igure a/o%ecore as sho&n in #igure a/o%eis #reuently use in electricis #reuently use in electric

circuits"circuits" This coil may /eThis coil may /e

represente /y an ieal circuitrepresente /y an ieal circuit

element calle inuctanceelement calle inuctance

&hich is e#ine as the #lu$&hich is e#ine as the #lu$

linkage o# the coil per amperelinkage o# the coil per ampere

o# its current"o# its current"

φ=λ N

i L λ=

-lu$ linkage-lu$ linkage

!nuctance!nuctance

R

N

A

l

N

N

Hl

HA N

i

HA N

i

NBA

i

N L

22

=

µ

=µ

=µ

==φ

=


24/35

24

15cm

30cm

15cm

15cm 10cm

l 2

30cm

30cm15cm 10cm

N =200turns

φ

i

E$ample 1E$ample 1

l1

* #erromagnetic core is* #erromagnetic core is

sho&n in -igure" Threesho&n in -igure" Threesies o# this core are o#sies o# this core are o#

uni#orm &ith &hile theuni#orm &ith &hile the

#ourth sie is some&hat#ourth sie is some&hat

thinner" The epth o# thethinner" The epth o# the

core 4into the page is 10cmcore 4into the page is 10cm

an the other imensionsan the other imensions

are sho&n in #igure" There isare sho&n in #igure" There is

200 turn coil &rappe200 turn coil &rappe

aroun the le#t sie o# thearoun the le#t sie o# thecore" *ssuming relati%ecore" *ssuming relati%e

permea/ilitypermea/ility r r o# 2500 ho&o# 2500 ho&

much #lu$ &ill /e proucemuch #lu$ &ill /e prouce

/y a 1* input current/y a 1* input current


25/35

Wbturns A

m x

m

A

l

A

l R

or

/.300,14

)01.0)(104)(2500(

45.027

1

1

1

11

=

=

==

−π

µ µ µ

Wbturns A

m x

m

Al

Al R

or

/.600,27

)015.0)(104)(2500(

3.127

2

2

2

22

=

=

==

−π

µ µ µ

25

Solution 1Solution 1

The mean path length o# region 1 is 5cm an theThe mean path length o# region 1 is 5cm an the

cross3sectional area is 10 $ 10 cm : 100cmcross3sectional area is 10 $ 10 cm : 100cm2"2" There#oreThere#ore

the reluctance in the #irst region is'the reluctance in the #irst region is'

The mean path length o#The mean path length o#

region 2 is 1D0cm an theregion 2 is 1D0cm an the

cross3sectional area is 15 $cross3sectional area is 15 $10 cm : 150cm10 cm : 150cm2"2" There#oreThere#ore

the reluctance in the seconthe reluctance in the secon

region is'region is'


26/35

Wbturns A

Wbturns AWbturns A

R R Req

/.900,41

/.600,27/.300,14

21

=

+=

+=

turns A Aturns NI F .200)0.1)(200( ===

Wb

Wbturns A

turns A

R

F

0048.0

/.900,41

.200

=

==φ

26

There#ore the total reluctance in the core is'There#ore the total reluctance in the core is'

The total magnetomoti%e #orce 4MM- is'The total magnetomoti%e #orce 4MM- is'

The total #lu$ in the core is gi%en /y'The total #lu$ in the core is gi%en /y'


27/35

27

The dimensions for a core is shown in figure elow! There are 2The dimensions for a core is shown in figure elow! There are 2

t"#es of material used to form this core! $ts cross sectional area is 7 %t"#es of material used to form this core! $ts cross sectional area is 7 %

1010&4&4 mm22 and there are 50 turns on the center leg! 'ssume thatand there are 50 turns on the center leg! 'ssume that H H forfor

(heet (teel and )ast $ron is 70 't*m and 290 't*m!(heet (teel and )ast $ron is 70 't*m and 290 't*m!

+etermine,+etermine,

Given dimension:

lab

= lcd

= lef

= lfa

=

10 cm

lbc = lde = 2.5 cm

Ø = 3.5 x 10-4 Wb

E$ample 2E$ample 2


28/35

28

i! the magnetomoti-e force .mmf/i! the magnetomoti-e force .mmf/ Hl Hl on each legon each leg

At l H

At l H

m At H

m At H

bcdebcde

e!be!b

bcde

e!b

5.4315.0290

213.070

/290

/70

=×=×

=×=×

≈

≈

ii! the currentii! the current I I flows in the windingflows in the winding

At

At I

At l H l H NI bcdebcdee!be!b

29.150

5.64

5.645.4321

==

=+=+=

Solution 2Solution 2


29/35

29

iii" the total reluctances o# the core"iii" the total reluctances o# the core"

At Wb

At

Wb

F

R 6

4

10426.5

)29.1)(50(

105.3 −−

×=×

== φ


30/35

30

' core with three legs is shown in igure! $ts de#th is 8 cm and there are ' core with three legs is shown in igure! $ts de#th is 8 cm and there are

500 turns on the center leg! The dimensions are shown in the figure! The500 turns on the center leg! The dimensions are shown in the figure! The

core is com#osed of the steel ha-ing the & cur-e and the -alue of core is com#osed of the steel ha-ing the & cur-e and the -alue of

and are shown in Tale!and are shown in Tale! )alculate,)alculate,

E$ample DE$ample D


31/35

31

0.20.2 0.250.25 0.50.5 1.01.0

-- 4545 5050 7070 160160

B an ) %alueB an ) %alue

Wb x x BATot 004.008.010.05.0 ===φ

T mmx

Wb

A A B B 25.0

08.010.0

002.0

2

2

1

121 =====

φ φ

outer outer center center Tot l H l H F +=

t A x x F Tot .8.7402.15034.070 =+=

A

t

t A

N

F i Tot 15.0

500

.8.74===

Solution DSolution D

1! The current that is reuired to #roduce the flu% densit" of 0!5 T in1! The current that is reuired to #roduce the flu% densit" of 0!5 T in

the center leg of the core!the center leg of the core!


32/35

32

ii! The reluctances of the center and right legs of the core!ii! The reluctances of the center and right legs of the core!

Wb x x BATot 008.008.010.00.1 ===φ

Wbmm

Wb B B 50.0

)08.0)(10.0(

004.021 ===

outer outer center center Tot l H l H F +=

t A x x F Tot .8.12502.17034.0160 =+=

At

t A

N

F i Tot 25.0

500

.8.125===


33/35

33

iii! The current that is reuired to #roduce a flu% densit" ofiii! The current that is reuired to #roduce a flu% densit" of

1!0 T in the center leg of the core!1!0 T in the center leg of the core!

Wbt "AWb

mmt A F

Tot

Tot #ent /.95.5

004.0

)34.0)(/.70(===ℜ

φ

Wbt "AWb

mmt A F

Tot

Tot rig$t /.5.25002.0

)02.1)(/.50(

===ℜ φ


34/35

34

i-! the new reluctances of the center and right legs of the core!i-! the new reluctances of the center and right legs of the core!

Wbt "AWb

mmt A F

Tot

Tot #ent /.8.6

008.0

)34.0)(/.160(===ℜ

φ

Wbt "AWb

mmt A F

Tot

Tot

rig$t

/.85.17004.0

)02.1)(/.70(===ℜ

φ


35/35

35

*ssignment 1*ssignment 1

* #erromagnetic core is* #erromagnetic core issho&n in -igure a/o%e"sho&n in -igure a/o%e"

*ll sie o# this core are*ll sie o# this core are

uni#orm &ith" Theuni#orm &ith" The

epth o# the core isepth o# the core is10cm" *ssuming10cm" *ssuming

relati%e permea/ilityrelati%e permea/ility r r

o# 2500 ho& much #lu$o# 2500 ho& much #lu$

&ill /e prouce /y a 1*&ill /e prouce /y a 1*input currentinput current

15cm

30cm

15cm

15cm 15cm

l c

30cm

30cm15cm 15cm

N =200turns

φ

i

Documents

Chapter 1 - Magnetic Circuits