1

Magnetic Circuits and Transformers

Discussion D10.1

Chapter 6

2

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

3

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).

4

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)

5

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