Download - 1. Fundamentals of Magnetics
![Page 1: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/1.jpg)
Fundamentals ofMAGNETICS
Revised October 6, 2008
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 1
![Page 2: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/2.jpg)
Magnetic fields provide the fundamental mechanism by whichMagnetic fields provide the fundamental mechanism by which energy is converted from one from to another by means of:
•motors (electrical energy mechanical energy )
•generators (mechanical energy electrical energy)g ( gy gy)
•transformers (electrical energy electrical energy )
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 2
![Page 3: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/3.jpg)
Four basic principles:
1. A current flowing in a wire produces a magnetic field in the vicinity of the wire.y
2. A wire in the presence of a time-varying magnetic field will induce a voltage across the wire (transformer action).
3 A current-carrying wire in the presence of a magnetic field3. A current-carrying wire in the presence of a magnetic field experiences a force exerted on the wire (motor action)
4. A a wire moving in the presence of a magnetic field has a l i d d i ( i )voltage induced upon it (generator action).
Each of these will be considered in detail.
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 3
![Page 4: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/4.jpg)
The fundamental postulate governing the electrical behavior of i fi ld i i b A ’ lmagnetic fields is given by Ampere’s law:
enclosedC
H d I
: Vector Magnetic Field (amperes/meter)H
g ( p )
: Vector Closed Path of Integration (meters)N t C t l d bI C
: Net Current ecnlosed by enclosedI C
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 4
![Page 5: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/5.jpg)
I
CC
H d I
I H
CC
Side view (along axis) H( g )
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 5
![Page 6: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/6.jpg)
C
I H r
I
rH
Right Hand Rule Convention
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 6
![Page 7: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/7.jpg)
C
I H r
rr
I 22C
IH d rH r I H rr
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 7
![Page 8: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/8.jpg)
We assumed that field produced by the wire exists in air. The h f h i fi ld d d l d d hstrength of the magnetic field produced also depends on the
material in which the field exists. Certain materials, called ferromagnetic materials, have the property of confining the f g , p p y gfield inside of it. Ferromagnetic materials are characterized by their permeability, .
The for air (or free space):
7 Henrys74 10oHenrysmeter
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 8
![Page 9: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/9.jpg)
The ratio of the permeability for a given material to that of free i k h l b lspace is known as the relative permeability,
r
o
For modern ferromagnetic materials,
1000 8000r
For an idealized case, r
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 9
![Page 10: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/10.jpg)
Note the similarity:
From circuit theory, the magnetic energy stored by an inductor is
212mW L I2
I L
From electromagnetic theory, the magnetic energy stored by a magnetic field is
212mW H
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 10
![Page 11: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/11.jpg)
The relationship that described the effect of the permeability ( h i l) h h i fi ld i i b h(the material) has on the magnetic field H is given by the relationship
B H
where B is the magnetic flux density.
Units:
2
Henry Ampere Ampere-Henrymeter meter meter
B H H
2
Weber Teslameter
H H
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 11
![Page 12: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/12.jpg)
B is the magnetic flux density. If we integrate the density over b i h l fl an area we obtain the total magnetic flux
ˆ ˆB d B d B d d
A A AB da B nda B ndxdy
n̂
dB
A dx
dy
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 12
![Page 13: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/13.jpg)
EXAMPLE
Mean path length c
Current i i
Cross sectional
N turns
Cross-sectionalarea AMagnetic
Core
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 13
![Page 14: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/14.jpg)
EXAMPLE
enclosed ccC
NiH d I H Ni H
cC
NiB B H H
c
A
NiAB da
i
c
Ac
iiiiiiii
xx
xxxxxx ix
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 14
![Page 15: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/15.jpg)
Circuit analogy for Magnetic circuits:
The simplifications and idealizations discussed imply that i l i i d l b d dsimple circuit models can be used to study magnetic circuits.
This idea is used extensively in the design of motors and y ggenerators to simplify what would otherwise be an extremely complicated electromagnetic problem.
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 15
![Page 16: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/16.jpg)
Circuit analogy for Magnetic circuits:
10 turns
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 16
![Page 17: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/17.jpg)
Circuit analogy for Magnetic circuits:
I Flux
+_V R +
_NiY c
Ae
Magnetomotive Reluctance
Flux
Force Reluctance
VIR
c c
NiA ANi
c c
Ye
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 17
e
![Page 18: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/18.jpg)
Circuit analogy for Magnetic circuits:
Electrical Circuit Magnetic Circuit
Electromotive Force (emf)
VMagnetomotive Force (mmf)
NiYCurrent
IFlux
Y
eResistance ReluctanceVR
I I
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 18
![Page 19: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/19.jpg)
Circuit analogy for Magnetic circuits:
Note the polarity – determined via a modified right-hand-rule
I
+_V+_V
I
+_
YII
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 19
![Page 20: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/20.jpg)
Jargon – Just as the conductance is the reciprocal of the resistance,
1GR
the permeance, is the reciprocal of the reluctance.
R
cp p
1c ce
This circuit analogy is always an approximation to the real system.
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 20
![Page 21: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/21.jpg)
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 21
![Page 22: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/22.jpg)
mean path5 + 20 + 2.5 = 27.5
7.5 + 15 + 7.5 = 300
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 22
![Page 23: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/23.jpg)
ANi
Ye
1c Y
e
left top right bottom
eY
e e e e eleft top right bottom
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 23
![Page 24: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/24.jpg)
top bottome e
e
rightelefte
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 24
![Page 25: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/25.jpg)
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 25
![Page 26: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/26.jpg)
Magnetic cores often require “gaps”.
N
Ae
“ideal” gapgg
A
e
Sgap
o A e
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 26
![Page 27: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/27.jpg)
Fringing fields in a gap. Note that the cross-sectional area of the gap is greater than that of the metal.g p g
N “real” gapgapg
Ae
eff
eff
o A
A A
g
S, 1.05ffTypically A A, 1.05ffeTypically A A
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 27
![Page 28: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/28.jpg)
i
Model:core
c
Ae
+_NiY
gape
core A
gapg p
o effAe
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 28
![Page 29: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/29.jpg)
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 29
![Page 30: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/30.jpg)
Mean path0.37 m 0.37 m
0.37 m0.37 m
0.37 m 0.37 m
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 30
![Page 31: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/31.jpg)
+_
NiY
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 31
![Page 32: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/32.jpg)
1 / 3e 3 / 3e
e
5e1 / 6e 3 / 6e
left right
+_
2e4e
Ni/ 3e/ 6e / 3e / 6e
fgap
ggap
1 / 3e1 / 6e3 / 3e 3 / 6e
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 32
![Page 33: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/33.jpg)
1e 4e5e
+_
2e e2 3e
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 33
![Page 34: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/34.jpg)
1e 4e 5e
+_
2e eNi
2 3e
TOT 5 1 2 3 4 e e e e e e
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 34
![Page 35: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/35.jpg)
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 35
![Page 36: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/36.jpg)
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 36
![Page 37: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/37.jpg)
Example – A Simple Synchronous Machine – Assume that h d h i fi i bili i d h ithe rotor and stator have infinite permeability. Find the air-gap
flux and the flux density Bg. Use I = 10 A, N = 1000 turns, gap length g = 1 cm, and Ag = 2000 cm2.g p g g , g
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 37
![Page 38: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/38.jpg)
Example – A Simple Synchronous Machine
gap0, ,2 2
gap o gapcgap
o gap gap gap
NI AA A
Y
e ee
gape
o gap gap gap
gap
erotore
e+_
statore statore
Y
gape
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 38
![Page 39: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/39.jpg)
Example – A Simple Synchronous Machine
2 2o gap
gap
NI A
Ye
7
2 2
1000 10 4 10 0.2 0 13
gapgap gap
Webers
e
0.132 0.01
Webers
0.13 0.650 2
gapgapB Tesla
A
0.2gapA
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 39
![Page 40: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/40.jpg)
Real Magnetic Materials
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 40
![Page 41: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/41.jpg)
Recall from Slide 14:
NiH Y
c c
H
and
andNiBA A AHA Y
cc
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 41
![Page 42: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/42.jpg)
Saturation or Magnetization Curve
Saturated Region
A Y
Linear
Y
NiY
LinearRegion
NiY
The permeability is constant only for air (o).
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 42
![Page 43: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/43.jpg)
Recall from Slide 14:
NiH Y
c c
H
and
an
BA H
dNi AA A Y
c c
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 43
![Page 44: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/44.jpg)
Saturation or Magnetization Curve
B Saturated Region
Slope = o
Linear
The slope of this curve definesthe permeability of a given material
Slope =
H
LinearRegion
HSince generators/motors rely on the flux to produce voltage/torque, we wish to produce as much flux as possible. In practice we operate near the knee of the curve
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 44
but not in the saturated region.
![Page 45: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/45.jpg)
Saturation or Magnetization Curve
B Saturated Region
Linear
The slope of this curve definesthe permeability of a given material
H
LinearRegion
HSince generators/motors rely on the flux to produce voltage/torque, we wish to produce as much flux as possible. In practice we operate near the knee of the curve
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 45
but not in the saturated region. Why?
![Page 46: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/46.jpg)
.B vs HH
.B vs Ho H
SaturatedRegion
LinearRegion
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 46
![Page 47: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/47.jpg)
The advantage of using a ferromagnetic material for cores in electric machines and transformers is that one gets many times more flux formachines and transformers is that one gets many times more flux for a given magnetomotive force (mmf) with iron than with air. However, if the resulting flux has to be (approximately) proportional
h li d f h h b d i h dto the applied mmf, then the core must be operated in the unsaturated region of the magnetization curve.
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 47
![Page 48: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/48.jpg)
Hysteresis Effects in Magnetic Materials
i2
t1
2
, B2
, HY
2
1 , HY
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 48
![Page 49: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/49.jpg)
Hysteresis Effects in Magnetic Materials
i2
t1 3
, B2
, HY
2
3
, HY1
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 49
![Page 50: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/50.jpg)
Hysteresis Effects in Magnetic Materials ii
2
t1 3
4
, B2
, HY3
11
4
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 50
4
![Page 51: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/51.jpg)
Hysteresis Effects in Magnetic Materials
i2
6
t1 3
45
4, B
26
, HY
3
1
5
1
4
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 51
![Page 52: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/52.jpg)
Hysteresis Effects in Magnetic Materials
Residual Flux
Coercive mmf
Y
Coercive mmf
A Typical Hysteresis Loop – Present flux in the core is determined not only by the applied mmf but also on the previous history of the fluxin the core.
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 52
![Page 53: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/53.jpg)
What causes hysteresis effects?MODELMODEL
The permanent magnetic moment of an atom comes about from:
Electron orbitElectron spin (primary contributor in solids)Nuclear dipole momentNuclear dipole moment
Ferromagnetic materials (Fe, Ni, Co, Cd., & alloys)These tend to be high conductivity metals (like iron and steel) and also
it lquite lossy.
On a macroscopic scale, a piece of ferromagnetic material is made up of magnetic domains where each domain is in effect a smallof magnetic domains, where each domain is in effect a small permanent magnet. These domains are randomly oriented.
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 53
![Page 54: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/54.jpg)
What causes hysteresis effects? With no externally applied mmf or magneticapplied mmf or magnetic field, individual magnetic domains are randomly li d d h flaligned and the net flux
is zero.
Wi h llWith an externally applied mmf or magnetic field, individual magnetic , gdomains align and a net flux occurs.
When no further alignment is possible,
iEEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 54
saturation occurs.
![Page 55: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/55.jpg)
First Major Effectj
Faraday’s Law – Induced voltage due to a time-changing magnetic fieldtime-changing magnetic field.
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 55
![Page 56: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/56.jpg)
Faraday’s Law – Induced voltage due to a time-changing magnetic field. g
i t
t
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 56
![Page 57: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/57.jpg)
Faraday’s Law – Induced voltage due to a time-changing magnetic field. g
i t
t
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 57
![Page 58: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/58.jpg)
Faraday’s Law – Induced voltage due to a time-changing magnetic field. g
A look ahead: Transformer Action
Pi t Si t
+ Pv t Sv t
PN SN P S
PN SN
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 58
![Page 59: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/59.jpg)
Faraday’s Law: If flux passes through a turn of a coilFaraday s Law: If flux passes through a turn of a coil of wire, a voltage will be induced in the turn of wire that is directly proportional to the rate of change of the flux.
appliedinduced
de
d
applied t
AppliedFlux:
einduced is the induced voltage
induced dt+
-inducede
is the flux passing through the loop
li dd t 0appliedd t
dt
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 59
![Page 60: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/60.jpg)
Lenz’s Law (explains the minus sign in Faraday’s Law)
First, a slight variant of the right-hand-rule:
II I
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 60
![Page 61: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/61.jpg)
Lenz’s Law: For a short-circuited loop, the applied change of flux produces a current that would cause a flux opposing the
flux produces a current that would cause a flux opposing the original flux change. This a statement of the conservation of energy. Applied
applied t inducedB t
Flux:
i
0appliedd t
dt
induced
i dtInduced
Flux: induced tThe induced current is such as to OPPOSE the CHANGE inapplied field.
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 61
![Page 62: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/62.jpg)
Lenz’s Law: The polarity of the induced voltage at the terminals of the coil is such that the current produced would cause a flux in a pdirection opposing the original flux.
Applied
i d dB t
ppFlux: applied t
+
_inducede
i
inducedB t
iInduced
Flux:
0appliedd t
dt
induced tAn induced electromotive force generates a current that induces a counter magnetic field that opposes the magnetic field generating the current
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 62
the current.
![Page 63: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/63.jpg)
For N turns all with the same flux,
i t Applied Flux
0d t
dt
+
inducedde Ndt
dt
_
Induced (opposing) flux
N t h thi i l i l GENERATOREEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 63
Note how this is also a simple GENERATOR.
![Page 64: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/64.jpg)
Flux Linkage:
d de N inducede N
dt dt
Flux Linkage: N
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 64
![Page 65: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/65.jpg)
In practice the polarity of the induced voltage isIn practice, the polarity of the induced voltage is determined from physical considerations. As a result, the minus sign in Faraday’s law is often omitted. The
(Ch i f i d ) i h i itext (Chapman is from industry) omits the minus sign for the remainder of the book. This could be confusing.
inducedde Ndt
Text omits the minus sig .n
dt
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 65
![Page 66: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/66.jpg)
Example – Consider a coil of wire (10 turns) wrapped around an ironcore. If a flux has a peak value of 0.1 Weber and varies sinusoidally at ap yfrequency of 60 Hz. What is the value and polarity of the inducedvoltage (flux is increasing in the direction shown).
10 turns
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 66
![Page 67: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/67.jpg)
Solution –
0.1sin 120t t
10 0.1 120 cos 120inducedde N tdt
120 cos 120dt
t
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 67
![Page 68: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/68.jpg)
Solution – (Note how this one-half of a transformer – the secondary)
ii +
inducede10 turns_
120 cos 120inducede t
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 68
![Page 69: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/69.jpg)
Example – A Primitive AC Generator
B Fi ld Cross-Sectional Area A = 0.4 m2B – Field(1 Tesla)
10-Turn Coil
Axis of RotationS d 3600
Find the induced voltageat the terminals.
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 69
Speed = 3600 rpm
![Page 70: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/70.jpg)
Example – A Primitive AC Generator
A = 0.4 m2
B = 1 T
Prc
ojectedos
AB da B A
ProjectedArea
t
3600 rpmAngular Speed:
t
2 1360060 sec
radians revolutions radians minutesecond minute revolution onds
120 377 60radians Hertzsecond
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 70
![Page 71: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/71.jpg)
Example – A Primitive AC Generator
B Fi ldB – Field(1 Tesla)
A
A
cosB A B A
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 71
![Page 72: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/72.jpg)
Example – A Primitive AC Generator
B = 1 T
A = 0.4 m2 cosAB da B A
t
cost BA t
cos
sininduced
t BA tde t N NBA tdt
1508sin377dt
t volts
What is the polarity of the induced emf?EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 72
What is the polarity of the induced emf?
![Page 73: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/73.jpg)
Polarity of induced emf:
d B Ade t N N
inducede t N Ndt dt
:
?
Two questions
is B A
?disdt
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 73
![Page 74: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/74.jpg)
Polarity of induced emf: A mathematical approach
A
Convention for positive loop orientation
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 74
![Page 75: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/75.jpg)
Polarity of induced emf: induced
d B Ae t N
dt
90 0B A
d B Ad
90 0B A
d B Ad
0 increasing
d B Addt dt
0 decreasingd B Ad
dt dt
A
A
B
B
+ _
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 75
_ +
![Page 76: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/76.jpg)
Example – A Primitive AC Generator
In this example, is constant but is changing with time.B
B A
A
A
B
B
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 76
B cos cos
![Page 77: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/77.jpg)
Example – A Primitive AC Generator
cosB A B A
d0B A
d
cos sind t tdt
sin
sin
d B A tdt
B A t
sin
induced
B A t
d B Ae t N
sin
induced dtN B A
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 77
![Page 78: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/78.jpg)
Example – A Primitive AC Generator
iiLoopOrientation
_ +
Orientation
+ sininducede t N B A
inducede t
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 78
![Page 79: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/79.jpg)
Example – A Primitive AC Generator
Approach used in text:Current direction opposesoriginal change in flux.
cos
sin
B A
dN N B A t
i
g gsin
decreasing
N N B A tdt
i
_ +
B
sininducede t N B A +
inducede t
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 79
![Page 80: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/80.jpg)
Second Major Effectj
Lorentz Force Law: A magnetic field induces a force on a current-carrying wireinduces a force on a current-carrying wire in the presence of the field.
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 80
![Page 81: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/81.jpg)
Lorentz Force Law – Force exerted on charges
EF qE
of little interest here
MF q v B
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 81
![Page 82: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/82.jpg)
BB
B
C C
B
CA
A
Right-Hand Rule for determining the direction of A B C
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 82
![Page 83: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/83.jpg)
Right-Hand Rule for determining the direction of F q v B
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 83
![Page 84: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/84.jpg)
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 84
![Page 85: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/85.jpg)
Lorentz Force Law – a more useful form for our application
is the velocity of the charge MF F q v B v q
s t e ve oc ty o t e c a geM q v v q
qq B B
q B Bt t
i B
i B
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 85
![Page 86: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/86.jpg)
Lorentz Force Law
F i B
:F force
::
F forcei current
:
:
oriented conductor path length
B magnetic flux density
:B magnetic flux density
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 86
![Page 87: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/87.jpg)
Lorentz Force Law BF i
x x x x x x x x x x x x x xx wire
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
xB into page
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x F
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x x x x x x x x x x x x x xx
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 87
![Page 88: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/88.jpg)
Lorentz Force Law
F i B
F
B
sinF F i B i
Where is the angle between vectors and B
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 88
![Page 89: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/89.jpg)
Lorentz Force Law
Recall Faraday’s Law: induceddedt
andB A
For constant B (with some hand-waving)
d dA d x dxB B B B vdt dt dt dt
What is the polarity?
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 89
![Page 90: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/90.jpg)
Lorentz Force Law
d B B vdt
What is the polarity?
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 90
![Page 91: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/91.jpg)
Lorentz Force Law
Recall: q v BFF qE Eq q
q q
F B
E v B
For constant B,
inducede E d v B induced
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 91
![Page 92: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/92.jpg)
Lorentz Force Law
B inducede v B
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 92
![Page 93: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/93.jpg)
Lorentz Force Law – An intuitive approach
wire x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x wire
+++++
F q v B
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
B into page v
B
+
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
xB into page inducede v B
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x_
____ F q v B
x x x x x x x x x x x x x xx
The direction for is such that is positive. (upwards in B v
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 93
this example)
![Page 94: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/94.jpg)
Example – A Primitive Generator Revisit our previous example.
i Coil End LengthendF
B
iL
gCoil Side Length
i i
ix
endF
sin 90F Ni B
Bx
FEEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 94
sin 90endF Ni B endF
![Page 95: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/95.jpg)
Example – A Primitive Generator
i Coil End Length iL
gCoil Side Length
F Bi i
sideFsideF
i
sideF NiLB
B
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 95
![Page 96: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/96.jpg)
Example – A Primitive Generator
Axial view:
BdF NiLB
d
F
sideF NiLB
sideF xt
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 96
![Page 97: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/97.jpg)
Torque
r
F
F
sin
T r F
T r F Newton Meters
sinT r F Newton Meters
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 97
![Page 98: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/98.jpg)
Example – A Primitive Generator
B F NiLB d
F
sideF NiLB
sideF x
t
2 sin2
sin
T F
NBiL
sinsin ,
NBiLNBiA A L loop area
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 98
![Page 99: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/99.jpg)
Observations:
•As the coil turns, the forces on the ends will vary with position, but will be equal and opposite, thus creating no net force on the coil as a whole, nor any torque about the axis of rotation.
•The forces on the sides of the coil will also be equal and oppositeThe forces on the sides of the coil will also be equal and opposite in direction, and will not vary with position, but will exert a torque about the axis of rotation.
•Note that the torque resists motion in the given direction, which generates a current. In other words, mechanical energy is being converted to electrical energy as the coil is turned.
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 99
![Page 100: 1. Fundamentals of Magnetics](https://reader030.vdocuments.us/reader030/viewer/2022012616/619e6d8ab387b3611f6f6be4/html5/thumbnails/100.jpg)
Example – A Primitive Generator
Mechanical Power: sinmP T i NBA t
Electrical Power: sininducede t i i N B A
Mechanical Power = Electrical Power
M ch more on this laterMuch more on this later...
EEL 3211© 2008, Henry Zmuda
1. Fundamentals of Magnetics 100