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Magnetic Circuits and Transformers
Discussion D10.1
Chapter 6
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Hans Christian Oersted (1777 – 1851)
Ref: http://chem.ch.huji.ac.il/~eugeniik/history/oersted.htm
1822
In 1820 he showed that a current produces a magnetic field.
X
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André-Marie Ampère (1775 – 1836)French mathematics professor who only a week after learning of Oersted’s discoveries in Sept. 1820 demonstrated that parallel wires carrying currents attract and repel each other.
attract
repel
A moving charge of 1 coulomb per second is a current of 1 ampere (amp).
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Michael Faraday (1791 – 1867)Self-taught English chemist and physicist discovered electromagnetic induction in 1831 by which a changing magnetic field induces an electric field.
Faraday’s electromagneticinduction ring
A capacitance of 1 coulomb per voltis called a farad (F)
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Joseph Henry (1797 – 1878)American scientist, Princeton University professor, and first Secretary of the Smithsonian Institution.
Discovered self-induction
Built the largest electromagnets of his day
Unit of inductance, L, is the “Henry”
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Magnetic Fields and Circuits
A current i through a coil produces amagnetic flux, , in webers, Wb.
BA A
d B A
H = magnetic field intensity in A/m.
v
i
+
-
NB = magnetic flux density in Wb/m2.
B H
= magnetic permeability
Ampere's Law: d iH l
Hl Ni
NiFMagnetomotive force F R
reluctance
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Magnetic Flux
Magnetic flux, , in webers, Wb.
1v 2v
2i1i
+ +
- -2N1N
11 flux in coil 1 produced by current in coil 1
12 flux in coil 1 produced by current in coil 2
21 flux in coil 2 produced by current in coil 1
22 flux in coil 2 produced by current in coil 2
1 11 12 total flux in coil 1
2 21 22 total flux in coil 2
Current entering "dots" produce fluxes that add.
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Faraday's Law
1v 2v
2i1i
+ +
- -2N1N
1 1 1N
Faraday's Law: induced voltage in coil 1 is
Sign of induced voltage v1 is such that the current i through an external resistor would be opposite to the current i1 that produces the flux 1.
Total flux linking coil 1:
1 11 1( )
d dv t N
dt dt
i
Example of Lenz's law Symbol L of inductance from Lenz
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Mutual Inductance
1v 2v
2i1i
+ +
- -2N1N
1 11 121 1 1 1( )
d d dv t N N N
dt dt dt
Faraday's Law
1 21 11 12( )
di div t L L
dt dt
In linear range, flux is proportional to current
self-inductance mutual inductance
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Mutual Inductance
1v 2v
2i1i
+ +
- -2N1N
1 21 11 12( )
di div t L L
dt dt
1 22 21 22( )
di div t L L
dt dt
12 21L L M Linear media
1 21 1( )
di div t L M
dt dt
1 22 2( )
di div t M L
dt dt
2 22L L 1 11L LLet
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Ideal Transformer - Voltage
1 1( )d
v t Ndt
2 2( )d
v t Ndt
1 1 1
2 2 2
dv N Ndt
dv N Ndt
22 1
1
Nv v
N
11
1( )v t dt
N
The input AC voltage, v1, produces a flux
This changing flux through coil 2 induces a voltage, v2 across coil 2
1v 2v
2i1i
+ +
- -2N1NAC Load
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Ideal Transformer - Current
12 1
2
Ni i
N
The total mmf applied to core is
NiF
Magnetomotive force, mmf
1 1 2 2N i N i F R
For ideal transformer, the reluctance R is zero.
1 1 2 2N i N i
1v 2v
2i1i
+ +
- -2N1NAC Load
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Ideal Transformer - Impedance
11 2
2
N
NV V
Input impedance
2
2L
VZ
I
21 2
1
N
NI I
1v 2v
2i1i
+ +
- -2N1NAC Load
Load impedance
1
1i
VZ
I
2
1
2i L
N
N
Z Z
2L
i n
ZZ 2
1
Nn
NTurns ratio
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Ideal Transformer - Power
12 1
2
Ni i
N
Power delivered to primary
P vi
22 1
1
Nv v
N
1 1 1P v i
1v 2v
2i1i
+ +
- -2N1NAC Load
Power delivered to load
2 2 2P v i
2 2 2 1 1 1P v i v i P
Power delivered to an ideal transformer by the source is transferred to the load.
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L.V.D.T. Linear Variable Differential Transformer
http://www.rdpelectronics.com/displacement/lvdt/lvdt-principles.htm
Position transducer
http://www.efunda.com/DesignStandards/sensors/lvdt/lvdt_theory.cfm
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LVDT's are often used on clutch actuationand for monitoring brake disc wear
LVDT's are also used for sensors in an automotive active suspension system
