Lecture 21Lecture 21Magnetic Circuits, Materialsg ,
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Magnetic Circuits and TransformersMagnetic Circuits and Transformers
1. Understand magnetic fields and their interactionswith moving charges.
2 Use the right hand rule to determine the direction2. Use the right-hand rule to determine the directionof the magnetic field around a current-carryingwire or coilwire or coil.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
3. Calculate forces on moving charges and current carrying wires due to magnetic fieldscarrying wires due to magnetic fields.
4 Calculate the voltage induced in a coil by a4. Calculate the voltage induced in a coil by a changing magnetic flux or in a conductor cutting through a magnetic fieldcutting through a magnetic field.
5 Use Lenz’s law to determine the polarities of5. Use Lenz s law to determine the polarities of induced voltages.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
6. Apply magnetic-circuit concepts to determine the magnetic fields in practical devices.
7. Determine the inductance and mutual inductance of coils given their physical parameters.
8. Understand hysteresis, saturation, core loss, d dd i dand eddy currents in cores composed of
magnetic materials such as iron.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
9. Understand ideal transformers and solve circuits that include transformers.
10. Use the equivalent circuits of real transformers to determine their regulations gand power efficiencies.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Magnetic Field Linesg
Magnetic fields The fluxMagnetic fields can be visualized as lines of flux h f l d
The flux density vector B is tangent to h li f flthat form closed
pathsthe lines of flux
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.density flux MagneticB =
Magnetic Fieldsg
• Magnetic flux lines form closed paths that g pare close together where the field is strong and farther apart where the field is weak.p
• Flux lines leave the north-seeking end of a gmagnet and enter the south-seeking end.
• When placed in a magnetic field, a compass indicates north in the direction of the flux
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
lines.
Right-Hand Ruleg
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Forces on Charges Moving in g gMagnetic Fields
Buf ×= quq
( )θsinquBf =
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Forces on Current-Carrying Wires
Blf ×=ddqd
y g
ddqdt
q
Bl ×= ddtdq
Bl ×= idForce on straight wire of length l in a constant
( )θiilBf
Force on straight wire of length l in a constant magnetic field
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
( )θsinilBf =
Force on a Current Carrying Wirey g
l 1Ai
ml101
==
TBAi5.0
10=
90=θ o
NTmAilBf 5)5.0)(1)(10()sin( === θ
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Flux Linkages and Faraday’s LLaw
Magnetic flux passing through a surface area A:
φ d⋅= ∫ ABA∫
For a constant magnetic flux density perpendicular
BA=to the surface:
The flux linking a coil with N turns:
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
φλ N=
Faraday’s Lawy
Faraday’s law of magnetic induction:
dtde λ
=dt
The voltage induced in a coil whenever its gflux linkages are changing. Changes occur from:
• Magnetic field changing in time
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
• Coil moving relative to magnetic field
Lenz’s Law
Lenz’s law states that the polarity of the induced voltage is such that the voltagewould produce a current (through an external resistance) that opposes the original change in flux linkages.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Lenz’s Law
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
US5,975,714 Renewable Energy Flashlight
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
US5,975,714 Renewable Energy Flashlight
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
US5,975,714 Renewable Energy Flashlight
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Voltages Induced inField-Cutting Conductors
BludxBldeBlxBA ===→==λλ
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Bludt
Bldt
eBlxBA ===→==λ
Magnetic Field Intensity and g yAmpère’s Law
intensityfieldMagneticH == HB μ7
0 104πμ ×= − AmWb
ytpermeabiliRelativer0μ
μμ =
Ampère’s Law: ∑∫ =⋅ idlH
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Ampère’s Law
The line integral of the magnetic field g gintensity around a closed path is equal to the sum of the currents flowing through the
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
g gsurface bounded by the path.
Magnetic Field Intensity and g yAmpère’s Law
productdotHdld =• lH )cos(θ productdotHdldlH )cos(θ
directionsametheinpointsandmagnitudeconstanta hasHfieldmagnetic the If
iHldlength lincrementa the asdirection sametheinpointsand magnitudeΣ= l
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Magnetic Field Around a Long g gStraight Wire
IIrHHl π2 ==
rIHπ2
=
IHB
rμ
π2
rIHB
πμμ
2==
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Flux Density in a Toroidal Corey
NIRHHl π2 ==
RNIHπ2
=
NIB
Rμπ2
RB
πμ2
=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Flux Density in a Toroidal Corey
rR
NIBA2
2ππ
μφ ==
NIrR2
2μπ
=
INR2
22
=
RIrNN
2
22μφλ ==
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Example 15.7
Σ=•∫ idlH
1path for A10=∫
3pathforA102path for 0
==
3path for A10−=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Example 15.8Find the force between these two wires if theythese two wires if they are 1 m long and separated by 0.1 m:p y
rI
mBπ
μ2
)1.0(
7
101 =
Tm
Axπ
π
20)1.0(2
)10)(104( 7=
−
TmAlBif
T
μθ
μ
)20)(1)(10()sin(
20
1221 ==
→
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
repulsiveNTmA
μμ
200)20)(1)(10(
==
Magnetic CircuitsMagnetic Circuits
I i i li ti d tIn many engineering applications, we need to compute the magnetic fields for structures that l k ffi i t t f t i ht f dlack sufficient symmetry for straight-forward application of Ampère’s law. Then, we use an
i t th d k ti i itapproximate method known as magnetic-circuitanalysis.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
magnetomotive force (mmf) of an N-turn current carrying coilcurrent-carrying coil
IN=F Analog: Voltage (emf)
reluctance of a path for magnetic flux
g g ( )
l=R
Aμ=R
Analog: Resistance
φRF =
g
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
φRFAnalog: Ohm’s Law
Reluctance ↔Resistance
⎟⎠⎞
⎜⎝⎛=↔⎟
⎠⎞
⎜⎝⎛=
AlR
Al ρ1
R
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
⎠⎝⎠⎝ AAρ
μ
Magnetic Circuit for Toroidal Coilg
rARl 2 2ππ
⎞⎛⎞⎛⎞⎛
==
rR
rR
Al 21211
22 μππ
μμ ⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛=⎟
⎠⎞
⎜⎝⎛=R
IN
NI2
=⎠⎝⎠⎝
F
F
RINr
2
2μφ ==RF
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Advantage of theAdvantage of theMagnetic-Circuit Approachg pp
Th d f h i i i hThe advantage of the magnetic-circuit approach is that it can be applied to unsymmetrical
ti ith lti l ilmagnetic cores with multiple coils.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Magnetic Circuit with an Air Gapg
Find what current is required to generate a flux density
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
of Bgap=0.25 T in the air gap.
Magnetic Circuit with an Air Gapg
)5.064(11 cmxlR −==
20
105.231
)3)(2(
mx
cmcmAR
rcore ==
−
μμμ
4
47
101955
106)104)(6000(
x
mxx
=
=−−π
10195.5 x=
6000=rμ
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
r
Fringingg gWe approximately account for fringing by adding the length of the gap to the depth and g g g p pwidth in computing effective gap area.
2410758
)5.03()5.02( cmcmxcmcmAgap ++=−
70
24
104
1075.8
x
mx
gap =≈
=−
−
πμμ
24
2
7 1075.8105.0
1041
mxmx
xRgap =
−
−
−π
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
610547.4 x=
Magnetic Circuit with an Air Gapg
RRR +
xxx
RRR gapcoretotal
10600.410547.410195.524
664 =+=
+=
Wbx
mxTAB gapgap
10188.2
)1075.8)(25.0(4
24
=
==−
−φ
turnsAxxRF
1006)10600.4)(10188.2( 64
=== −φ
AturnsAFi
Ni
012.21006===
=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
AturnsN
i 012.2500
Exercise 15.9
Determine the current required to establish a flux
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Determine the current required to establish a flux density of 0.5T in the air gap
Exercise 15.9
)12()12( cmcmxcmcmAgap ++=24109
)12()12(
mx
cmcmxcmcmAgap
=
++−
2
70
1011
104
mx
xgap =≈−
−πμμ
6
247
1084281075.8
1011041
mxmx
xRgap =
−−π610842.8 x=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Exercise 15.9)16282(11 cmxxlRcore
−+==
20
10271
)2)(2(
mx
cmcmAR
rcore
−
μμμ
3
47
104107
104)104)(5000( mxx=
−−π3104.107 x=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Exercise 15.9RRR coregaptotal +=
mxTAB
Rxx
gapgap
gap
)109)(5.0(
10107.010842.824
66
==
≈+=−φ
Ri
mWb
total
45.0
=
=φ
xxxN
i
1000)1045.0)(10107.010842.8( 366 +
=−
A027.41000
=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
A Magnetic Circuit with ReluctancesA Magnetic Circuit with Reluctances in Series and Parallel
Find the flux density in each gap
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Find the flux density in each gap
A Magnetic Circuit with Reluctances
t t l RR +=1
in Series and Parallel
ba
ctotal
NiRR
RR+
+11
b
totalc
dividercurrentR
RNi
φφ
φ
=
=
)(
ca
b
cba
a
RRR
dividercurrentRR
φφ
φφ
=
+= )(
a
aa
cba
b
AB
RRφ
=
+
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.a
aa
a
AB
Aφ
=
Magnetic Materialsg
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Magnetic MaterialsMagnetic Materials
The relationship between B and H is not linear for the types of iron used in motors and transformers.yp
B-H curves exhibits “h t i ”
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
“hysterysis”
Magnetic MaterialsT t l li t
gTotal alignmentResidual
alignment at H=0
Alignment
• Magnetic field of atoms within small domains are aligned
Linear 1-22-3
Magnetic field of atoms within small domains are aligned
• Magnetic fields of the small domains are initially randomly oriented
• As the magnetic field intensity increases, the domains tend to align, leaving a residual alignment even when the applied field
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
is reduced to zero
Energy Considerationsgy
φ
∫∫∫ ===tt
dNidtidtdNdtviW
000
φφφ
∫∫ ==
==BB
dBHVdBAlHW
AdBdandHlNi φ
∫
∫∫ ==
B
core
dBHWW
dBHVdBAlHW00
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
∫== V dBHWVW
0
Energy Considerationsgy
W B
dBHAlWW
B
v ∫==0Al 0
The area between the B-H curve and the B axis represents
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
pthe volumetric energy supplied to the core
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Core LossCore Loss Power loss due to hysteresis is proportional to y p pfrequency, assuming constant peak flux.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Eddy-Current Lossy
As the magnetic field changes in a material, it g gcauses “eddy currents” to flow. Power loss due to eddy currents is proportional to the square of y p p qfrequency, assuming constant peak flux.
vP2
RP =
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Energy Stored in theEnergy Stored in theMagnetic Fieldg
2B
μμ 2
2BdBBWB
v == ∫ μμ 20∫
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.