lecture 4 - magnetics.pdf

7/30/2019 Lecture 4 - Magnetics.pdf

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Magnetics

Lecture 4

20 August 2003

MMME2104

Design & Selection of Mining Equipment

Electrical Component


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Lecture Outline

Magnetic Circuits Magnetic field intensity, flux density, relative

permeability and reluctance

B-H curves

Magnetic Materials

Electromagnetism Magnetic fields produced by a conductor

Faradays and Lenzs laws

Electromagnetic Effects Hysteresis Eddy currents

Inductance


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Magnetic Circuit Nomenclature

n/aNumber of turns/loopsN

Weber (Wb)Magnetic flux

Henries (H)Mutual inductanceM

Henries (H)Self-inductanceL

n/aRelative permeabilityr

Henries per metre (H/m)Permeability of free spaceo

Tesla (T)Magnetic flux densityB

Amp-turns per metre (A/m)Magnetic field strengthH

Amp-turns (A)Magneto-motive forceMMF

UnitsRepresentsSymbol


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Magnetic Flux and Flux Density

Whenever a magnetic flux () exists in a body or component, it is dueto the presence of a magnetic field intensity (H), given by:

H = MMF / l

where l is the length of the component and MMF is the magneto-

motive-force.

The resulting magnetic flux density (B) is given by:

B = / A

where A is the cross sectional area of the component.


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Reluctance

The magnetic flux () and magneto-motive-force (MMF) are relatedthrough the reluctance () of the magnetic circuit, as follows:

MMF = or

H l = B A

This relationship is analogous to the voltage/current relationship for a

resistor in an electrical circuit:

V = i R


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Reluctance

So how do we determine reluctance?

The reluctance () is defined in terms of magneticpermeability ():

A

l

=


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Magnetic Circuits: SummaryMagnetic quantities

Electrical analogies

Henries per metre (H/m)Permeability

1/ Henries (H-1)Reluctance

Telsa (T) or Webers per square

meter (Wb/m2)

Magnetic flux densityB

Webers (Wb)Magnetic flux

Amp-turns per metre (A/m)Magnetic field intensityH

Amp-turns (A)Magneto-motive forceMMF


1 / Ohm-metres (-1m-1)Conductivity

Ohms ()ResistanceR

Amps per square meter (A/m2)Current densityJ

Amps (A)CurrentI

Volts per metre (V/m)Electric field intensityE

Volts (V)Voltage or Electro-motive-forceV or EMF



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Magnetic Circuits: B-H Curves

All these magnetic quantities are very confusing!

Fortunately, magnetic materials and circuits can be

described far more easily in terms of B H curves:

B = H

(This relationship can be derived from previous

expressions. The electrical analogy would be a J E

curve, but this serves no useful purpose for analysingelectrical circuits.)


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B-H Curve of Vacuum

The magnetic flux density in vacuum

is directly proportional to the

magnetic field intensity:B = o H

where o = permeability constant

= 4 x 10-7 H/m

Non-magnetic materials such as

copper, paper, rubber and air

have B-H curves almost identicalto that of vacuum. These

materials never saturate!

Source: T. Wildi, Electrical Machines, Drives and Power Systems, 5th Edition, Prentice-Hall, 2002


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B-H Curve of Magnetic Materials

The magnetic flux density in magnetic materials also depends upon the

magnetic field intensity:

B = ro H

where r= relative permeability of magnetic material

However, relative permeability is not constant and varies with the flux

density in the material. Consequently, the B-H relationship for

magnetic materials is non-linear and this also explains why B-H

curves are so useful in magnetics. This non-linear magneticphenomenon is known as saturation


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B-H Curve of Magnetic Materials


r= 1120 r= 560

r= 160


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Magnetic Materials

Commonly used magnetic materials:

Permanent magnets Rare-earth magnets (highest flux/field and high cost)

Ferrite magnets (low-medium flux/field and low cost)

Laminated materials

Iron-silicon alloys High relative permeability but low conductivity

Used to minimise eddy currents at power frequencies (Hz)

Ferrites Sintered manganese or nickel alloys Low saturation flux density and very low conductivity

Suited to RF frequencies (MHz)


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Electromagnetism

Magnetic field produced by a conductor:

i

B

r

iB

2=

r


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Electromagnetism

Magnetic field produced by a solenoid:ni

l

NiB ==


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Electromagnetic Induction

Voltage induced in a wire loop by a changing magnetic field (Faradays

Law):

= magnetic flux enclosedwithin loop (flux linkage)

Vdt

dNV

=

N turns


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Two special cases of Faradays law need to be considered:

1. When a coil is stationary and the flux linking it changes with time(produced by an AC current). This produces an equation for the induced

voltage (V) called the flux linking equation (useful in transformers):

V = 4.44 BpA N f

where:

Bp is the peak flux density linking the coil

A is the cross-sectional area of the coil

N is the number of turns of the coil

f is the frequency (Hz)


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2. When the flux does not change with time but a conductor moves

through the magnetic field. This produces an equation for the

induced voltage (V) called the flux cutting equation (useful in

electric motors and generators):

V = B l v

where:

l is the (active) length of the conductor in the magnetic fieldv is the velocity of the moving conductor


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Lenzs Law is used in conjunction with Faradays Law to

define the direction of the induced voltage:

the direction of the induced voltage is such that if a currentflows as a consequence, the flux produced must

oppose the flux change inducing it


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Electromagnetic Force

Lorentz force on a conductor: BilF =



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Electromagnetic Force



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Electromagnetic Effects

In rotating electric machines, the electromagnetic force is

used to create torque.

In transformers and electric motors/generators, there are

three kinds of losses that may be attributed to

electromagnetic effects: Hysteresis

Eddy currents

Magnetostriction

Electromagnetism also gives rise to the property of

inductance.


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Hysteresis

Hysteresis losses occur when the

flux changes continuously

both in value and direction.

The magnetic material absorbs

energy each cycle and

dissipates it as heat.

To reduce hysteresis losses,

magnetic materials areselected that have a narrow

hysteresis loop.


C


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Eddy Currents

Eddy currents occur when AC voltages are induced in a conductorby a changing magnetic field

Eddy currents dissipate power as resistive losses in the conductor

To reduce eddy current losses, magnetic materials are laminated

(for a given core size, eddy current losses decrease in proportionto the square of the number of laminations).


M t t i ti


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Magnetostriction

When a magnetic field is established in a ferromagnetic material the

dimensions of the crystal structure change the atomic spacing in

the direction of the field increasing and that perpendicular to the

field decreasing.

In addition to being another source of power loss, this minute change in

the size of the material occurs at twice the supply frequency and

causes a characteristic hum at 100 Hz for devices supplied at

normal mains frequency.

This noise cannot be overcome and can be the source of complaints byresidents living close to distribution transformers.

I d t


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Inductance

To better understand inductance, we combine Faradays Law with thevoltage/current relationship for an inductor:

Therefore, the inductance is the rate of change of flux linkages in a coilproduced by the current in that coil. This characteristic of a coil is knownas self-inductance. We may further write:

For a linear magnetic circuit (constant permeability) the inductance is constantbut for a ferromagnetic material (non-linear due to changing relativepermeability) the inductance falls as the material experiences saturation.

dt

diL

dt

dNV =

=

di

dNL

=

l

ANN

di

dNL

or

22

=

=

=

M t l I d t


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Mutual Inductance

Two coils that are wound on the same magnetic core (and are therefore

linked by the same magnetic flux) will induce voltages/currents in

each other and are said to exhibit mutual inductance (M). For

example, if all of the flux produced by the current in coil 1 linkswith coil 2, then the mutual inductance with coil 2 (M21) is given

as:

Similarly:

Therefore:

=

=12

1

221

NN

di

dNM

=

= 212

112

NN

di

dNM

211212LLMMM ===

M t l I d t


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Mutual Inductance

However, in practice there is some flux produced by coil 1 which leaks

from the core (and is therefore called leakage flux) and does not

link with coil 2 so M < (L1L2).

This leads us to define an inductance called the leakage inductance

which for coil 1 is L1M12 and for coil 2 is L2M21.

Self- and mutual-inductances are most conveniently presented as a

matrix equation:

=

2

1

221

121

2

1

i

i

dt

d

LM

ML

V

V

lecture 4 - magnetics.pdf

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