transformers and magnetic circuits

8/13/2019 Transformers and Magnetic Circuits

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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

1


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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



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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= =

9

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

=

10


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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

12


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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

13

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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Magnetic Field Around a Long Straight Wire

IHB ==

r2



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Flux Density in a Toroidal Core

B =



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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.



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A Magnetic Circuit with Reluctances in Series and Parallel



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A Magnetic Circuit using Dot-conversion



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E C id ti


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Energy Considerations

dBHW

W

B

==

Al 0


Core Loss


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Core Loss

Power loss due to hysteresis is proportional to frequency, assuming constantpeak flux.


Core Loss


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Core Loss


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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REAL TRANSFORMERS


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REAL TRANSFORMERS


REAL TRANSFORMERS


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REAL TRANSFORMERS


REAL TRANSFORMERS


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REAL TRANSFORMERS

Regulation and Efficiency

%100regulationpercent loadload-no

=V

%1001%100efficiencypowerin

oss

in

oa

==PP


REAL TRANSFORMERS


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REAL TRANSFORMERS



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