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TRANSFORMERS AND
MAGNETIC CIRCUITS
PhD, IIT Roorkee
ss s an ro essorDepartment of Electrical Engineering
School of Engineering
Shiv Nadar Universit , Noida
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Magnetic Fields and Circuits
A current ithrough a coil produces a
magnetic flux,f, in webers, Wb.
BA =A
d =
B Ai
B = magnetic flux density in Wb/m2.
H = magnetic field intensity in A/m.
=
m = magnetic permeability
l Ni=Ampere's Law: d i= H li reluctance
2Ni=FMagnetomotive force =
F R
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4Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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Right-Hand Rule
5Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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Magnetic Flux
Magnetic flux,f, in webers, Wb. Current entering"dots" produce
1v 2v
21
2N1N
.
11 flux in coil 1 produced by current in coil 1 =
flux in coil 1 roduced b current in coil 2=
21 flux in coil 2 produced by current in coil 1 =
22 ux n co pro uce y current n co=
1 11 12total flux in coil 1 = = +
62 21 22total flux in coil 2
= = +
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Faraday's Law
2i
1i
1v
2v
2N
1Ni
1 1 1
Farada 's Law: induced volta e in coil 1 is
1 1d dv t N
= =
Sign of induced voltage v1 is such that the current ithrough
dt dt
an external resistor would be opposite to the current i1that produces the fluxf1.
7Example of Lenz's law Symbol L of inductance from Lenz
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Mutual Inductance
2i1i
1v
2v
2N1N
1 11 121 1 1 1( )
d d dv t N N N dt dt dt
= = +
ara ay s aw
di di
In linear range, flux is proportional to current
1 11 12vdt dt
=
8self-inductance
mutual inductance
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Mutual Inductance
2i
1i
1v
2v
2N
1N
1 2di div t L L= +
t t1 2
2 21 22( ) di di
v t L L= +1 2
1 1( ) di di
v t L M dt dt
= +
Linear media 1 22 2( ) di di
v t M L= +
12 21L M= =
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2 22 1 11
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Ideal Transformer - Voltage
1 1( ) dv t Ndt=2i1i
The input AC voltage, v1,
produces a flux
1v 2v2N1N
f 11
1( )v t d t
N =
2 2( ) dv t N
dt=
coil 2 induces a voltage, v2across coil 2
d1 1 1
2 2 2
v dtdv N N
= =2
2 1
1
v vN
=
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Ideal Transformer - Impedance
2i1i Load impedance
2
L =ZI
1v 221
11 2
N=V V
Input impedance
2N1
i = V
Z
2
1i L
N = Z Z
1 2
1N1 2
2
Li
n=Z 2
1
Nn
N=Turns ratio
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Ideal Transformer - Power
2i1i
vi=1v 221
Power delivered to primary Power delivered to load
1 1 1P v i= 2 2 2P v i=
12 1
Ni i
N=
22 1
1
Nv v
N
= 2 2 2 1 1 1P v i v i P= = =
Power delivered to an ideal transformer by the source
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s trans erre to t e oa .
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Application: L.V.D.T.
near ar a e eren a rans ormer
Position transducer
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LVDT's are often used on clutch actuationand for monitoring brake disc wear
'
sensors in an automotive
active sus ension s stem
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Voltages Induced in Field-Cutting Conductors
Blue =16Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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18Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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Magnetic Field Around a Long Straight Wire
IHB ==
r2
19Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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20Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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Flux Density in a Toroidal Core
B =
21Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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22Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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23Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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24Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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25Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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Magnetization Circuits with Air-gap
wli
lN
d
cc
cc
A
l
= g
g
l=
Ni
+
=
ggcc lHlHNi += )( fringingNeglectingwdAA gc ==
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Fringing
wlci
N
With large air-gaps, flux tends to leak outside the air gap. This is
approximate this increase is:
dwlddlww =+=+=
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Fringing
r ng ng s approx ma e y coun e y a ng e eng o e gap o
the depth and width in computing effective gap area.
28Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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29Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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A Magnetic Circuit with Reluctances in Series and Parallel
30Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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31Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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A Magnetic Circuit using Dot-conversion
32Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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34Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
E C id ti
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Energy Considerations
dBHW
W
B
==
Al 0
35Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
Core Loss
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Core Loss
Power loss due to hysteresis is proportional to frequency, assuming constantpeak flux.
36Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
Core Loss
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Core Loss
37Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
M i i C
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Magnetization Curves
saturation
B
Ma netization curve Ma netization curve(linear) (Ideal) (non-linear) (Actual)
(see also Fig. 1.6 in the text)
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40Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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41Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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42Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
REAL TRANSFORMERS
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REAL TRANSFORMERS
43Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
REAL TRANSFORMERS
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REAL TRANSFORMERS
44Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
REAL TRANSFORMERS
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REAL TRANSFORMERS
Regulation and Efficiency
%100regulationpercent loadload-no
=V
%1001%100efficiencypowerin
oss
in
oa
==PP
45Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
REAL TRANSFORMERS
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REAL TRANSFORMERS
46Electrical Engineering: Principles and Applications, 3rd edition by Allan R. Hambley, Pearson publication, 2005
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