1_Introduction to Magnetic Circuits

8/11/2019 1_Introduction to Magnetic Circuits

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PowerPoint Slidesto accompany

Electr ic MachinerySixth Edition

A.E. F itzgerald

Char les Kingsley, Jr.Stephen D. Umans

Chapter 1

Magnetic Circuits andMagnetic Materials


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Introduction

The transformer, although not an

electromechanical energy conversion device, is

an important component of the overall energy-

conversion process.


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Faradays Law of Induction

The induced emf in any closed circuit is equal

to the rate of change of the magnetic flux

through the circuit.

Unit of magnetic flux: weber, Wb.

1 Wb= 1 Tm2


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This drawing shows the variables in the flux

equation:



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Magnetic flux:



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The magnetic flux is analogous to the electric

fluxit is proportional to the total number of

lines passing through the loop.



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The minus sign gives the direction of the

induced emf:

The induced electromotive force in any closed

circuit is equal to the negative of the time rate ofchange of the magnetic flux through the circuit.



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The minus sign gives the direction of the

induced emf:

When an emf is generated by a change inmagnetic flux according to Faraday's Law, the

polarity of the induced emf is such that it

produces a current whose magnetic field

opposes the change which produces it so as torestore the changed field.

This obeys Newtons 3rdlaw & the law of

conservation of energy.

Lenzs Law


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Maxwells Equations

(Magneto-quasistatic approximation)

The line integral of the tangentialcomponents of the magnetic field

intensity H around a closed contour C

is equal to the total current passing

through any surface S linking that

contour. The source of H is the

current density J.


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Maxwells Equations

(Magneto-quasistatic approximation)

The magnetic flux density B is

conserved, that no net flux enters or

leaves a closed surface.


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A second simplifying assumption

involves the concept of the magneticcircuit. The general solution for the

magnetic field intensity H and the

magnetic flux density B in a structureof complex geometry is extremely

difficult. However, a three-dimensional

field problem can often be reduced to

what is essentially a one-dimensional

circuit equivalent, yielding solutions

of acceptable engineering accuracy.


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Simple magnetic circuit.

>> 0


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The magnetic flux crossing a surface S is the

surface integral of the normal component of B.

Due to the assumption that the magnetic fluxdensity is uniform across the cross section of a

magnetic circuit, the equation can be reduced to

this simple scalar equation:


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The relationship between the mmf acting on a magnetic

circuit and the magnetic field intensity is given as:

The relationship between the magnetic field intensity H and

the magnetic field intensity B is a property of the material in

which the field exists. It is common to assume a linearrelationship, thus

,

where is known as the magnetic permeability.


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A Few Definitions Related to

Electromagnetic Field

(Unit is Weber (Wb)) = Magnetic Flux Crossing a Surface of

Area A in m2.

B (Unit is Tesla (T)) = Magnetic Flux Density = /AH (Unit is Amp/m) = Magnetic Field Intensity =

B

= permeability of the material

o = 4*10-7H/m (H Henry) = Permeability of free space (air)

r = Relative Permeability

r>> 1 for Magnetic Material

0

r


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For the present, we assume that r is a

known constant, although it variesappreciably with the magnitude of the

magnetic flux density.

Transformers are wound on closed cores.However, energy conversion devices which

incorporate a moving element must have

air gaps in their magnetic circuits. When

the air-gap length g is much smaller thanthe dimensions of the adjacent core faces,

the same analysis techniques could still be

applied.


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,

(from )

(from B-H relationship)


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Analogy between electric and magnetic circuits.

(a) Electric circuit, (b) magnetic circuit.


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Hopkinson's law: the magnetic analogy to Ohm's law


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Fringing Field Effect

In practical systems, the magnetic field lines fringeoutward somewhat as they cross the air gap.

Provided this fringing effect is not excessive, the

magnetic circuit concept remains applicable. The

effect of these fringing fields is to increase theeffective cross-sectional area of the air gap.


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Air-gap fringing fields.


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The magnetic circuit shown has dimensions Ac = Ag = 9 cm2,g = 0.050 cm, lc = 30 cm, and N = 500 turns. Assume the r=

70,000 for core material. (a) Find the reluctances Rc and Rg.

For the condition that the magnetic circuit is operating with Bc

= 1.0 T, find (b) the flux and (c) the current i.


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Find the flux and current from the previous

problem if (a) the number of turns is doubled

to N = 1000 turns while the circuit dimensionsremain the same and (b) if the number of

turns is equal to N = 500 and the gap is

reduced to 0.040 cm.


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The magnetic structure of a synchronous machine isshown schematically in the figure. Assuming that the

rotor and stator iron have infinite permeability, find the

air-gap flux and flux density Bg. For this example I = 10

A N = 1000 turns g = 1 cm and Ag = 2000 cm2

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1_Introduction to Magnetic Circuits