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Unformatted text preview: 51 Chapter 5
Magnetic Circuits Exit 2001 by N. Mohan Print TOC " ! 52 Magnetic Field
❑ Magnetic field, H, produced by current carrying conductor ❏ Ampere’s Law dl
H "
∫ closed path Exit 2001 by N. Mohan H d! = ∑i i3 i1
i2 TOC " ! 53 H in a Toroid
i
rm ID OD ID
OD 1 ID + OD Mean radius, rm = 2
2 lm = 2π rm
Ni
Ni
=
Ampere's Law ⇒ H m =
2π rm lm
Exit 2001 by N. Mohan TOC " ! 54 Flux Density B
❏ Units:Weber / meter 2 [Wb / m 2 ] or Tesla [T ] henries µo = 4π × 10 −7 ❏ In air B = µo H , m ❏ Ferromagnetic materials
Bm Bm µo Bsat µm
µo
Hm Hm x Linear approximation Bm = µm H m
x Bsat ~ 1.6  1.8 Tesla
x In saturation µm approaches µo
Exit 2001 by N. Mohan TOC " ! 55 Flux, Flux Linkage, and MMF
❏ Flux fm [Wb]
[assuming uniform flux density] Am φm = Bm Am
φm Ni
Bm = µ m H m and H m =
!m Ni Ni
F
∴ φm = Am µ m
=
=
! m ! m ℜm µ A m m
!m
µ m Am ❏ Reluctance
❏ Flux Linkage λm = Nφm ❏ MMF
Exit ℜm = F = Ni 2001 by N. Mohan TOC " ! 56 Magnetic Structures with Air Gaps
Hm !m + H g ! g = N i
Bm = µ m H m , Bg = µo H g φm = φ g = φ
i Bg Bm
!m +
!g = N i
µm
µo !g N φ = Am Bm = Ag Bg
φ
Am Bm = Bg = φ
Ag !g
!m
)= N i
φm (
+
Am µ m Ag µo
#$% # %
$
ℜm φm =
Exit 2001 by N. Mohan To account for fringing Ag = ( w + ! g )(d + ! g )
ℜ = ℜm + ℜ g ℜg F
ℜ
TOC " ! 57 Inductance
i φm Am N λm = Lm i i N
× lm Hm µm × ( µm ) Bm × ( Am ) N2
× Lm =
lm µm Am φm ×( N ) λm N2
N2
λm N Lm =
= µ m Am N =
=
i lm lm ℜ µ m Am • For linear magnetic conditions inductance depends only on magnetic circuit ❏ Energy stored in magnetic circuits
❏ Energy density
Exit 2001 by N. Mohan W
1
2
w=
Bm
=
volume 2 µ m 1
1
2
W = Li 2 =
Bm Amlm
&
2
2 µm
volume TOC " ! 58 Faraday’s Law  Induced Voltage
❏ Induced voltage dλ
dφ
e=
=N
dt
dt ❏ Current direction is into positive polarity
voltage → flux direction φ (t )
i (t )
+
e(t ) N
− ❏ Lenz’s law: Polarity of induced voltage
x When current and flux directions are consistent (a current
as indicated would create a flux as indicated), then the
voltage should be labeled positive where the current enters
the coil. Exit 2001 by N. Mohan TOC " ! 59 Coil in Sinusoidal SteadyState
❑ Induced voltage under sinusoidal steadystate
Given φ (t ), i (t ) e(t ) ˆ
φ (t ) = φ sin ω t
e (t ) = N i (t ) t dφ
ˆ
= N φ ω cos ω t
dt φ (t ) +
e(t ) N
− ❑ Relating e(t ), φ (t ), and i (t )
λ Nφ
L= =
i
i
N
⇒ i (t ) = φ (t ) L dφ(t) & e(t) = N dt
Exit 2001 by N. Mohan di (t )
⇒ e(t ) = L
dt TOC " ! 510 Leakage and Magnetizing Inductances
φm i
+ ⇒ e
− i
+
e
− i (t ) e (t ) −
+ l
+
+
Ll φl e(t )
− N φ!
λ = Nφ = N φm + &
&
λ!
λm φ = φ m + φ! λ λm λ!
=
+
i
i
i ⇒ Lself = Lm + L! λ = Lself i = Lm i + L! i em (t )
− Lm φm
R
+
v (t )
− Ll i (t ) + el (t )
+
em (t )
e(t )
−
− di
di
di
e = Lm + L!
= em + L!
dt
dt
dt
& &
em
Exit 2001 by N. Mohan e!
TOC " ! 511 Transformers
❏ Tightly coupled coils (low leakage inductance)
❏ Essential for power transmission and distribution
❏ Helpful in understanding induction machines Exit 2001 by N. Mohan TOC " ! 512 Transformers  Development
❏ Single coil
Assuming zero resistance and zero
leakage inductance
e1 = N1 φm +
e1
− N1 dφm
dt φm determined completely by
1
applied voltage: φm = N ∫ e1 dτ
1
im depends on Lm
❏ Two coils dφm
dt
e (t )
N
⇒ 1
= 1
e2 (t )
N2 e2 (t ) = N 2 Exit 2001 by N. Mohan & e1 (t ) = N1 dφ m
dt +
e1 im
Lm − φm +
e1
− N1
N2
+
e2 −
TOC " ! 513 Transformer Model
+ im + e1 Lm e2
− −
N1 N 2
#' %
$'
Ideal
Transformer ❏ Dot polarity
❏ Magnetizing inductance Exit 2001 by N. Mohan TOC " ! 514 Transformer with Secondary Loaded
❏ φm determined by e1 alone
hence i2 in secondary induces φm i1 (t ) +
e1
− N1
N2 i2 (t )
+
e2 i2 ' in the primary such that − ′
N1 i2 = N 2 i2
i1 (t ) i′ N
⇒ 2 = 2
i2 N1 + i1 (t ) = i2 '(t ) + im (t )
#$% # $'
' %
relflected
load
current Exit 2001 by N. Mohan i2 (t ) i2 '(t ) magnetizing
current im + e1 Lm e2
− −
N1 N 2
#' %
$'
Ideal
Transformer TOC " ! 515 Real Transformers
i1 (t ) ❏ Add leakages
+
❏ Core loss
v1
 hysteresis
−
 eddy currents
❏ Winding resistances R1 i2 '(t ) Ll1
+
e1 Ll2 R2 i2 (t ) + Lm − v2 − Rhe + e2 im − TOC " N1 N 2
#' %
$'
Ideal
Transformer Real
Transformer ❏ Laminations to reduce eddy current loss
i φm
circulating
currents circulating
currents φm Exit 2001 by N. Mohan ! 516 Determining Transformer Model Parameters
i1 (t ) R1 i2 '(t ) Ll1 + + v1 e1 − − Real
Transformer i2 (t ) + v2 − Lm + e2 im
Rhe R2 Ll2 − N1 N 2
#' %
$'
Ideal
Transformer ❏ Open circuit test
x Core loss, Rhe
x Magnetizing inductance, Lm
❏ Short circuit test
x Winding resistance, R1 , R2
x Leakage inductance, Ll1 , Ll2
Exit 2001 by N. Mohan TOC " ! 517 Open Circuit Test
❏ Secondary unloaded (open circuit)
❏ Rated voltage applied to primary
❏ Measure
x To find Rhe
Rhe I oc 2
Voc
= Poc x To find Lm +
Voc jX m Rhe − V
= oc
Rhe jX m
I oc Exit 2001 by N. Mohan TOC " ! 518 Short Circuit Test
❏ One winding shortened
small voltage applied to other winding
❏ Measure VSC , and I SC , and PSC
x To find R1 and R2
I1 1P
R2 = SC
2
2 I SC N R1 = R2 1 N2 R1 + 2 2R2 + j2 X l 2 = 2001 by N. Mohan +
VSC − − N2
I SC
2 N jX l1 2 + jX l 2 N1 VSC
I SC
2 I SC E2 −
N1 N = Xl2 1 X l1 N2 R2 + E1 x To find Ll1 and Ll2 Exit jX l 2 jX l1 2 N R1 2 + R2 N1 +
VSC
− N1 N2 TOC " ! 519 Permanent Magnets
❏ Typically used in smaller motors
❏ Applicable power range increasing due to new materials
❏ In simplest analysis, treated simply as a source of magnetic
flux
Bm (T)
−1.4
−1.2
−1.0
e
F
Nd B
Sm Co −0.8
−0.6 Alnico 0.4 Ferrite 
1000 
800 
500
− H m ( kA / m ) 
200 Figure 520 Characteristics of various permanent magnet materials. Exit 2001 by N. Mohan TOC " ! 520 Summary
❏ What is the role of magnetic circuits? Why are magnetic
materials with very high permeabilities desirable? What
is the permeability of air? What is the typical range of
the relative permeabilities of ferromagnetic materials like
iron?
❏ Why can "leakage" be ignored in electric circuits but not
in magnetic circuits?
❏ What is Ampere's Law and what quantity is usually
calculated by using it?
❏ What is the definition of the mmf F?
❏ What is meant by "magnetic saturation"?
❏ What is the relationship between φ and B?
❏ How can magnetic reluctance ℜ be calculated? What
field quantity is calculated by dividing the mmf F by
the reluctance ℜ ?
Exit
TOC
"
2001 by N. Mohan ! 521 Summary
❏ In magnetic circuits with an air gap, what usually dominates
the total reluctance in the flux path: the air gap or the rest
of the magnetic structure?
❏ What is the meaning of the flux linkage λ of a coil?
❏ Which law allows us to calculate the induced emf?
What is the relationship between the induced voltage and the
flux linkage?
❏ How is the polarity of the induced emf established?
❏ Assuming sinusoidal variations with time at a frequency f,
how are the rms value of the induced emf, the peak of the
flux linking a coil, and the frequency of variation f related?
❏ How does the inductance L of a coil relate Faraday's Law
to Ampere's Law?
❏ In a linear magnetic structure, define the inductance of a
coil in terms of its geometry.
Exit
TOC
" !
2001 by N. Mohan 522 Summary
❏ What is leakage inductance? How can the voltage drop
across it be represented separate from the emf induced by
the main flux in the magnetic core?
❏ In linear magnetic structures, how is energy storage defined?
In magnetic structures with air gaps, where is energy mainly
stored?
❏ What is the meaning of "mutual inductance"?
❏ What is the role of transformers? How is an ideal
transformer defined? What parasitic elements must be
included in the model of an ideal transformer for it to
represent a real transformer?
❏ What are the advantages of using permanent magnets?
Exit 2001 by N. Mohan TOC " ! ...
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 Fall '06
 Scalzo

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