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บทที่ -...
48
1 (Magnetic Circuit) 4 1. 2. 3. 4. 1.1 () (Magnetic Flux ; ) 1.1 . . . 1.1
Transcript of บทที่ -...
1
(Magnetic
Circuit)
4
1.
2.
3.
4.
1.1
( ) (Magnetic Flux ; )
1.1
. . .
1.1
1 2
. . .
. (CROSS ; )
. (DOT ;
)
(Ampare’s
Law) 1.1
1.2
= (1.1)
(Magnetic Field Intensity)
1.1
=
2
1.2 ( ) ( )
(2 ) = = 2
(Core) 1.3
( )
1 3
1.3
1.3
(N)
(S)
( )
( ) (Magnetomotive
Force ; , )
= (Amp turn) (1.2)
1.2
1.4 (Coil)
(Core) (Ferromagnetic Material)
2
(Leakage Flux ; ) Magnetizing Flux ( )
Magnetizing Flux ( )
Magnetic Flux ( )
1 4
1.1
( )
MagneticFlux
Mean core
Cross-sectional
Permeability Winding
i+
- N
. 3 . 2
length l
N turns
area A
i
l A
1.4
( ) ( )
( ) ( )
1.2
( )
=
( )
=
= = (Amp turn m , Amp m) (1.3)
(Magnetic Flux Density ; )
( ) 1.4
= (Weber m , Tesla) (1.4)
1 5
=
(Permeability)
= 4 × 10 (Weber/Amp-turn-m, Henry/m)
(Relative Permeability)
( )
(Ferromagnetic Material)
2,000 80,000
(Magnetic Flux ; )
(Magnetic Flux Density ; )
=
1.4
( )
1.5
= (Weber ; Wb) (1.5)
( )
( ) Flux Linkage ( )
= (Wb turn, Wb) (1.6)
Flux Linkage ( ) ( )
(Inductance ; )
= (Wb turn Amp , Henry) (1.7)
Flux Linkage ( ) ( )
(Induced Voltage ; )
1 6
= (Volt) (1.8)
1.8 =
(Faraday’s Law)
1.5
1.5
1.3
1.1
1.6
1 7
1.1
Voltage Volt Magnetomotive Force A t Current Amp Magnetic Flux Wb
Resistance Ohm ( ) Reluctance A t Wb
Conductance Siemen (S) Permeance Wb A t
Permittivity C N m Permeability H/m
= = 1
= = 1
= 8.854 × 10 C N m = 4 × 10 H m
1.6 . 1.6 .
(Reluctance ; ) =
- (+)
- (-)
Reluctance ( )
(+)
( ) ( )
= = = = = = (1.9)
1.
2.
l
A
1 8
1.1 ( ) 50
( ) 2 3,000 2
1
2
. ( )
. 1 ( ) 2 ( )
. 1 ( ) 2 ( )
. Flux Linkage ( )
. ( )
= 8 + 8.5 + 8.5 = 25 cm = 0.25 m
= 3 × 2.5 = 7.5 cm = 7.5 × 10 m
= 8 cm = 0.08 m
= 2 × 2.5 = 5 cm = 5 × 10 m
= = = .( × )( , )( . × )
= 88,419.41
= = = .( × )( , )( × )
= 42,441.32
mmf +-
+
-
1R
R 2
mmf 1
mmf 2
+ -
(= Ni)
1 9
. ( )
= = = ( )( ), . , .
= 7.64 × 10 Wb
= .
. 1 ( ) 2 ( )
=
= = = . × ( , . ).
= 270.21
= .
= = = . × ( , . ).
= 405.31
= .
. 1 ( ) 2 ( )
=
= = . ×. ×
= 1.018 T
= . = = . ×
×= 1.528 T
= .
. Flux Linkage ( )
= = (50)(0.764 × 10 ) = 0.0382 Wb t
= .
. ( )
1 = = 0.0382 2 = 0.0191 H
2
= = =, . , .
= 0.0191 H
= .
1 10
1.2 . ( ) 500 ( )
0.0012 . ( ) (Square Wave)
50 . (Steady State)
. ( ) ( )
. ( )
( ) 1.2
v+
-t (s)
v
1/501/100
E = 90 V
-E = -90 V
. . v
N 0
i
. ( ) ( )
= = = , =
(Steady State)
= 90 V, = 500 turn
= =
90 V, = 500 turn
=
1 11
max
max
1
= =
, 0 1 100
=
| = | , 2 =
=( )( )
=( )( )( )
= 0.9 × 10 Wb
2
= = = (2)(100)
= = =
(2)(100) =
=( )( )
=( )( )( )
= 0.9 × 10 Wb
= . , .
. ( ) ( )
1.2 =
2 = , 2 =
= 2(100) = 2(100)(1.2)(0.0012)(500) = 144 V
144
t (s)1/1000
max
- maxt
1 12
1.4
(Air gap) -
(Reluctance) 2 Fringing Field
Fringing Field
1.7
1.7 . Fringing Field ( )
( ) = , =
( = )
1.7 . Fringing Field ( )
( ) > , >
( > )
1.3 ( ) 600
8,000 ( ) 30 . ( ) 0.05 .
( ) 10 . Fringing Field
( ) 5%
( ) 1
( )
( )
N
i
lg
lc
1 13
= = ×( × )( , )( × )
= 29,841.6
= = . ×( × )( . × × )
= 378,940.3
= = , =
= = ×. × ×
× 1 = 0.952 T
= .
= =
= = ( ) × ( , . , . ) = 0.68 A
= .
1.3
( ) -
( )
1.4 2 = 700 = 200
2 = 0.05 . = 0.1 . ( ) 0.5
Fringing Field
+-
cR
R gNi
1 14
. Air gap 1 ( , )
. Air gap 2 ( , )
. Flux Linkage ( , )
= = . ×( × )( . × . × )
= 636,619.8
= = . ×( × )( × . × )
= 1,061,033
. Air gap 1 ( , )
= , = = ( )( . ), .
= 5.5 × 10 Wb
= = . ×. × . ×
= 0.88 T
= . , = .
. Air gap 2 ( , )
= , = = ( )( . ) ( )( . ), ,
= 2.356 × 10 Wb
= , = = ( )( . ) ( )( . ). ×
= 250 × 10
= . , =
. Flux Linkage ( , )
= ( + ) = (700)(5.5 × 10 + 2.356 × 10 ) = 0.55 Wb t = = (200)(2.356 × 10 ) = 0.047 Wb t
= . , = .
+-
+
-
+-
1 2
R 21R 2R 2
N i2N i1
+ 21
1 15
1.5
=
(Slope) 1.8 .
(Linear)
(Ferromagnetic Material) 1.8 .
( ) Region II (
) Region I Region III (Non-linear)
1.8
B-H Curve DC Magnetization Curve
. Region III ) )
) )
(Saturate) )
)
(Air gap)
( )
1 16
B-H Curve
1.9 B-H Curve 3
(Silicon sheet steel) (Cast steel) (Cast iron)
B, T
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0 200 400 600 800 1,000 H, A-t/m
1.9 B-H Curve DC Magnetization Curve
1 17
3
1.9
Silicon sheet steel Cast steel
Cast iron
1.5 2
1 Cast steel = 40 cm 1 2
2 Cast iron = 10 cm
= 2 ( ) 200
1 ( )
0.69 ( )
Relative Permeability ( ) 2
= =
=
= 2
= (2 )
= 2 = 0.69 2 = 0.345 T
Cast steel = 0.69 T B-H Curve 1.9 450 t/m
Cast iron = 0.345 T B-H Curve 1.9 800 t/m = +
= ( + ) = [(450)(0.4) + (800)(0.1)] 200 = 1.3 A
= .
= = , =
Cast steel = . , = . ( × )( ) = ,
Cast iron = . , = . ( × )( ) = .
+
-
+ -Ni+
-
1H l1
2 2H l
N
i
l 2
l1
1 18
1.6 C I 2
Silicon sheet steel 2
( ) 57.3 .
2 ( ) 0.8 .
Fringing Field ( )
534 ( ) 3.1
( )
= =
=
= = = + 2
= + 2 = + 2
2
+
× . ×( . × )
+ × ( )( . )( . × )
4.5 × 10 + 1.3 ………………………………… (A)
(A) B-H Curve Silicon sheet steel 2
1 Load Line
(A) B-H Curve 1.9
4.5 × 10 + 1.3 = +
(Slope) 4.5 × 10 ( ) = 1.3 T
= 1,000 )
= 1,000 4.5 × 10 (1,000) + 1.3 = 0.85 T
= 0, = 1.3 = 1,000, = 0.85
B-H Curve 1.9 Load line
N
i lg
lc
lg
+
-
+ -Ni+
-
c cH l
g2H lg
1 19
Load line Silicon sheet steel
.
2 (Trial and error method)
BC
Load line
1 20
1.
2. B-H Curve =
3. = + 2 )
4. =
5. 4
2 - 4
( 3.1 A)
(A) = 0 = 1.3 T 1.3 T
= 1.2 T
=
1 1.2 400 9.5493 x 105 1757.088 3.29
2 1.1 270 8.7535 x 105 1555.27 2.91
3 1.15 330 9.1514 x 105 1653.314 3.096
3.1 )
1 = 1.2 T 3.29 3.1
2 1.1 T 2.91 3.1
3 1.1 1.2 T = 1.15 T
2.096 3.1
.
1.7 . B-H Curve
2 . = 200 turns, = 100 turns
. = 1 A
( ) 0.6
. = 0.5 A, = 1.96 A ( )
1 21
= 7.5 × 2 = 15 cm, = 1 × 1.5 cm
= 7 × 2 = 14 cm, = 1.5 × 1.5 cm
= , =
= = (1.5 × 1.5) (1.5) = 1.5 = 1.5
B-H Curve 2 .)
=2 × 10 , 0 < 0.8 T
(10 + 2), > 0.8
= 500 , 0 < 0.8 T(3 2) × 10 , > 0.8 ………………………………… (A)
. = 1 A, = 0.6 T = 1.5 = (1.5)(0.6) = 0.9 T
= 0.6 T (A) = 500 = 300 t/m
= 0.9 T (A) = (3 2) × 10 = 700 t/m
+
-
+ -
+- N i1 1
N i2 2 +-
hor horH l
ver verH l
1 22
+ = +
= ( + )
= [(700)(0.15) + (300)(0.14) (200)(1)] 100 0.53
.
. = 0.5 A, = 1.96 A ( )
+ = +
(200)(0.5) + (100)(1.96) = 0.15 + 0.14
0.15 + 0.14 = 296 ………………………………… (B)
(B) (B) 296
1. = 1.5
2. , B-H Curve (A)
3. , (B)
4. (B)
1-3 (B)
0.15 + 0.14
1 0.6 0.9 300 700 147
2 0.7 1.05 350 1150 221.5
3 0.8 1.2 400 1600 296
= 0.8 T, = 1.2 T
= = (0.8)(1.5 × 1.5 × 10 ) = 1.8 × 10 Wb
= = (1.2)(1 × 1.5 × 10 ) = 1.8 × 10 Wb
= .
1 23
1.6
(Relative Permeability ; )
1. = 1 1
1.1 Nonmagnetic (Free Space, Vacuum) = 1
1.2 Diamagnetic < 1 1)
1.3 Paramagnetic > 1 1)
2. (Magnetic Material) 1
2.1 Ferromagnetic
- Hard Ferromagnetic Alnicos, Chromium steel, Copper-nickel
Alloys
- Soft Ferromagnetic (Iron)
Nickel, Cobalt, Tungsten, Aluminium Silicon steel
Cast steel
2.2 Ferrimagnetic Ferrite (Iron oxide)
- Hard Ferrite
- Soft Ferrite Nickel-zinc, Manganese-zinc
3. Super-paramagnetic Permalloy
(Molybdenum-nickel-iron powder)
(Inductor Core)
1.2 (
Electric Machines, M.S.Sarma, M.K.Pathak)
1 24
1.2
1.7 (Hysteresis)
1.10 .
( = 0, = 0) 0
B-H Curve . ( ) 0
( )
(Residual Flux Density ; )
Coercivity Coercive Force ( ) ( = )
B-H Curve 2
B-H Curve 2-3
1 25
B-H Curve . Hysteresis Loop
1 1
.
. . Hysteresis Loop
1.10 Hysteresis Loop
( ) ( )
(Domain)
1.11 .
1.11 .
B (or )
H (or i)
rB
c-H
1 26
. .
1.11
( ) Coercive
Force ( )
Hysteresis Loop 1.10 .
Hysteresis Loop 1.12
1.12 Hysteresis Loop
1 27
Hysteresis Loop
Hysteresis Loop 1 ( 1.12) Magnetization Curve
1.5 2 Demagnetization
Curve
Hysteresis Loop 1.13
Hysteresis Loop (Deltamax Core) 50%
Nickel 50% (50% Fe and 50% Ni)
1.13 Hysteresis Loop
Hysteresis Loop
Magnetic Switch
1.7.1
(Hysteresis Loss)
1.10 .
=
= = = =
= , =
1 28
= ( ) = =
= ( ) (1.10)
Hysteresis Loop 1.14
1.14 Hysteresis Loop
Hysteresis Loop 1
1 ( | )
| = ( ) = ( )( ) =
(Energy Density)
( ) 1.11
= (1.11)
Hysteresis Loop ( )
1.11
1 29
General Electric Hysteresis Loop
1.12
(1.12)
1.11 1.12
= (Watt) (1.13)
1.5 2.5 ( 1.6)
1.7.2
( )
1.15 . (Eddy Current Loss)
2
1.
2. (Laminated Core) (
Varnish) 1.15 .
. .
1.15
1 30
(Actual Iron
Area) (Gross Area) Stacking Factor
= (1.14)
1.15
= (Watt) (1.15)
Laminated Core
Laminated Core 0.5 5 .
0.01 0.5 .
1.7.3
(Core Loss ; ) 2
(Hysteresis Loss ; ) (Eddy Current Loss ; )
= + (Watt) (1.16)
1.16 Static Loop Dynamic Loop
( )
-
B-H Curve ( 1.16) B-H Curve
1 31
Hysteresis Loop Static Loop
( ) B-H Curve
( 1.16) B-H Curve Dynamic Loop
Dynamic Loop
= ( )( )( )
= (1.17)
1.16 1.17
1.8 50 ( )
1,500 60 ( ) 2,040
( ) ( )
( ) 50 60
= + = + = +
= 50 Hz ; (50) + (50) = 1,500, + 50 = 30 ……………….. (1)
= 60 Hz ; (60) + (60) = 2,040, + 60 = 34 ……………….. (2)
(2) - (1) ; 10 = 4, = 0.4
= 0.4 (1) ; + 50(0.4) = 30, = 10 = + = + = 10 + 0.4
= 10 , = 0.4
= ; = = ( ) =
= . = . ( ) =
= ; = = ( ) =
= . = . ( ) = .
1 32
1.8
1.17 . ( )
1.18
= sin (1.18)
Wb
= 2 rad s
Hz
1.17
= = ( sin )
= cos = cos (1.19)
1.18 1.19 . .
90°
= 2 = 2 = 2 2
= = = . (1.20)
1.20
1.17 .
Exciting Current ( ) - Curve ( B-H Curve)
1 33
- Curve (
)
1.8.1
- Curve
1.18 .
. . - Curve
1.18
- Curve
- Curve
- Curve
( 90°)
i3
i3
i9
i9
9
3
3
9
t9 t3
1 34
1.18 . 1 ( )
, 1.19 .
1.19
.
1.19
1.8.2
- Curve
1.20 .
. . - Curve
1.20
i3
i3
i9
i9
9
3
3
9
t9 t3
1 35
1.8.1
- Curve
- Curve
2
( 1.18 .)
1.20 . 1.21 .
1.21
1.21
1.21 .
(Induction Machine)
(Core Loss)
(Magnetizing Flux)
1 36
1.9 ( ) 380
Laminated Core Silicon sheet steel B-H Curve 1.9
( ) 100 . (Gross Area) 22 .
Stacking Factor 0.95
220 50
( ) ( , )
= ( )( ) = (22 × 10 )(0.95) = 20.9 × 10 m
1.20 ; = = 2
= =( )( )
= 2.606 × 10 Wb
= = . ×. ×
= 1.247 T
B-H Curve 1.9 = 1.247 T Silicon sheet steel 500
=
, = , = ( )( ) = 1.32 A
= . , , = .
1.9 (Permanent Magnet ; PM)
(Iron) (Nickel)
(Cobalt) Hard Iron Soft Iron
2 B-H Curve
Demagnetization Curve 1.22 2 Alnico 5 (
Iron, Aluminum, Nickel, Cobalt) Samarium-cobalt
N+
-
v +-e
i
+
-Ni+
- c cH l
1 37
. Alnico 5
. Samarium-cobalt
1.22 Demagnetization Curve
(BH)max
B (tesla)
Br
-Hc
0.96
- 42
H
H
1 38
1.22 Alnico 5 Demagnetization Curve
Samarium-cobalt Demagnetization Curve
(Residual
Flux Density) (Alnico 5 1.24 T, Samarium-
cobalt 0.87 T)
(Coercivity Coercive Force) (Alnico 5 49 ,
Samarium-cobalt 640 )
Demagnetization Curve
BH Product Energy Product . (J m )
1.22 . BH Product 3 20 k J m , 30 k J m 40 k J m
Alnico 5 BH Product (Maximum Energy Product ; (BH)max) 40 k J m
0.96 T, 42 k A m
Energy Product
(BH)max
(BH)max
1.10 Demagnetization Curve
(BH)max 2 2
1 39
= +
= + ………….. (1)
; = +
( ) = 0 ; ( ) = + = 0
2 + = 0, 2 + 1 = 0, 2 1,
(1) ; = + =
(BH)max
1.11 (PM)
( ) = 4 cm , = 0.5 cm, = 2.5 cm
( )
. Alnico 5 = 3.3 cm
. Samarium-cobalt = 1.5 cm
= 0
+ = 0, …………… (1)
lg
PM
1 40
= = , = ………………… (2)
=
= = = , = ………………… (3)
(1) (3) ………………… (4)
. Alnico 5 = 3.3 cm
(4) ; . ..
4.125
5.184 × 10 = +
50 kA m ; 5.184 × 10 ( 50 × 10 ) 0.26 T
Load Line (-50 kA/m, 0.26 T) 0 Demagnetization
Curve Alnico 5
0.25 T
(2) ; = =.
(0.25) = 0.4 T
( ) .
0.26 0.25
Load Line
1 41
. Samarium-cobalt = 1.5 cm
(4) ; . ..
2.356 × 10
. × ………………… (5)
Samarium-cobalt Demagnetization Curve ( 1.22 .)
Load Line
= + = .×
+ 0.87 ………………… (6)
(5) (6) ; = .× . ×
+ 0.87, = 0.5517 T
(2) ; = =.
(0.5517) = 0.883 T
( ) .
1.12 Samarium-cobalt Soft Iron
Soft Iron ( )
= 2 cm , = 0.6 cm (BH)max
(BH)max Samarium-cobalt
( , ) = 0.8 T
Samarium-cobalt 1.22 . = 2 = 0.87 2 = 0.435 T, = 2 640 2 320 kA m
( ) = = (0.435 )(320) = 139.2 kJ m
( ) = . = . ,
= 0.435 T, 320 kA m
+ = 0, + = 0, = =
(0.8)(0.6) (4 × 10 )( 320 × 10 ) = 1.2 cm
= = (0.8)(2) 0.435 = 3.68 cm
= . , = .
lglm PMAg
Am Soft Iron
Soft Iron
1 42
1
1. ( ) 70 ( )
0.8 6,000
. ( )
. ( )
. ( )
. Flux Linkage ( )
. ( )
0.754 mWb, 200 , 1.508 T, 0.0528 Wb t, 66 mH
2. Toroid 1,000
( ) 150
= 8 cm
= 10 cm = 2 cm
( )
20 mH
3. . ( ) 500 ( ) 11 .
( ) (Square
Wave) ( ) .
(Steady State)
ra
N
r1
r2
1 43
. ( ) 0.8 mWb ( )
. ( ) 1 ( )
50 ( )
. . ( ) 60
( )
max
max
. 60 Hz . 110 V . 132 V
4. Toroid 5,000
( ) 480 Fringing Field -
( ) 5%
= 4 cm
= 6 cm = 1 mm
( )
0.28 mWb ( )
1.5 A
5. = 1,000 turns, = 12 A
= 1 cm, = 1,990 cm -
( ) ( )
( )
0.15 Wb, 0.754 T
Ni Rotor
Stator
l g
l gAg
Ag
N
ilg
lc
r
r1r2
1 44
6. 2 ( , ) = 300 turns
= 300 turns, = 0.5 cm 1,200 2
( ) 10 Fringing Field
( ) ( ) ( )
= 0.31 mWb, = 0.62 T, = 4.934 × 10
7. 3 ( , , ) 2 (gap 1
gap 2) = 120 turns, = 620 turns, = 300 turns, = 1 mm
= 1.1 mm, = 9 cm , = 16 cm 3 ( )
2 Fringing Field
,
= 0.95 mWb, 1.17 mWb, = 0.22 mWb
1 45
8. Cast steel Toroid
( ) 200 = 6 cm, = 10 cm
. ( ) ( ) 1.28 mWb
.
( ) 2 .
( )
1.28 mWb
Fringing Field
. 1.76 A . 9.87 A
9. Cast iron Toroid
( ) 301 = 6 cm, = 10 cm, = 2 mm
( ) 2.1
( )
( )
Fringing Field
0.24 T, 0.288 mWb
N
il
r
r1r2
A A
BC
Load line
N
ilg
lc
r
r1r2
3 cm
1 46
10. 2 (2-pole)
- (Pole) Cast steel ( ) 10 .
( ) 400 .
- (Air gap) ( ) 1 .
( ) 400 .
- (Armature) Silicon
sheet steel ( ) 20 .
( ) 400 .
- (Yoke) Cast steel
( ) 160 .
( ) 200 .
( ) ( ) 1.12 T
( ) ( )
1,170 t/pole, 44.8 mWb, 22.4 mWb
11. . 2 1 ( ) 18 .
2 ( ) 9 . 2 B-H Curve
2 . = 200 turns, = 82 turns
. = 2.5 A ( ) 0.28 mWb
. = 1.5 A, = 1.67 A ( )
1 cm 2 cm
1 cm
1 cm
2 cm
N2N1 i2
i1
B, T
H, A-t/m . . B-H Curve
1.2
1.0
400 12000
l2
l1
1 2
. 1.4 A . 0.22 mWb
Ni Ni
Armature
Yoke
PolePole
Air gap
1 47
12. 50
( ) 1,350 60 ( ) 1,800
( )
( ) 960
40 Hz
13. ( )
( )
1 220 60
2 200 50
. 1 2 [ ( ) ( )]
. 1 2 [ ( ) ( )]
. 1.008 . 1.21
14. = 396 turns, = 103.5 cm
= 20 cm , = 2,000 220 50
1.3 sin 100 A
15. ( ) 329
Laminated Core Silicon sheet steel B-H Curve 1.9
( ) 103 . (Gross Area) 24 . Stacking
Factor 0.95 50
( , )
1.25
( )
( )
1.2 T, 200 V
N+
-
v +-e
i
1 48
16. (PM)
( ) = = 4 cm
. Alnico 5 = 2.7 cm
= 0.55 cm -
( )
. Samarium-cobalt
= 1.3 cm -
( ) 0.5 T
. 0.3 T
. 0.9 cm
17. (PM) Soft
Iron ( ) = 3.5 cm , = 0.5 cm -
( ) 0.7 T
. Alnico 5
( , )
. Samarium-cobalt
(BH)max (BH)max
( , )
. = 6.63 cm, = 2.55 cm
. = 0.435 T, 320 kA m, ( ) = 139.2 kJ m
= 0.87 cm, = 5.63 cm
lg PM
0.31
Load Line
B (tesla)
lglm PMAg
Am Soft Iron
Soft Iron
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