Chapter1 part1

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06/10/22 Dr Awang Jusoh/Dr Makbul 1 Chapter 1 Introduction to Electromechanical Energy Conversion

description

Magnetic circuit

Transcript of Chapter1 part1

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Chapter 1

Introduction to Electromechanical Energy

Conversion

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1.1 Magnetic Circuits

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Magnetic Field Concept

Magnetic Fields:• Magnetic fields are the fundamental

mechanism by which energy is converted (transferred) from one form to another in electrical machines.

Magnetic Material• Definition : A material that has potential to attract other

materials toward it, materials such as iron, cobalt, nickel• Function: Act as a medium to shape and direct the

magnetic field in the energy conversion process

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Magnetic Field Concept

• Magnetic field around a bar magnet

• Two “poles” dictated by direction of the field

• Opposite poles attract (aligned magnetic field)

• Same poles repel (opposing magnetic field)

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Magnetic Field Concept

1. Outside - Leaves the north pole (N) and enters the south pole (S) of a magnet. Inside - Leaves the south pole (S) and enters the north pole (N) of a magnet.

2. Like (NN, SS) magnetic poles repel each other.3. Unlike (NS) magnetic poles attracts each other.4. Magnetic lines of force (flux) are always

continuous (closed) loops, and try to make as shortest distance loop.

5. Flux line never cross each others

Magnetic Flux/ Flux Line Characteristic

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Magnetic Field Concept

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Machines Basic Requirements

• Presence of a “magnetic fields” can be produced by:– Use of permanent magnets– Use of electromagnets

• Then one of the following method is needed:– Motion to produce electric current

(generator)– Electric current to produce motion

(motor)

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Ampere’s Law

• Any current carrying wire will produce magnetic field around itself.

Magnetic field around a wire:• Thumb indicates direction of current flow• Finger curl indicates the direction of field

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Ampere’s Law

Ampere’s law: the line integral of magnetic field intensity around a closed path is equal to the sum of the currents flowing through the surface bounded by the path

H • dl = i∑∫

Recall that the vector dot product is given by

H • dl = Hdlcos(θ)

dl

H

I1 I2

in which is the angle between H and dl.

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Ampere’s Law

If the magnetic intensity has constant magnitude and points in the same direction as the incremental length dl everywhere along the path, Ampere’s law reduces to

in which l is the length of the path.€

Hl = i∑

Examples of such cases: (i) Magnetic field around a long straight wire, (ii) Solenoid

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Example 1:( a long straightWire)

Example 2:

(Solenoid)

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Flux Density

• Number of lines of magnetic force (flux) passing through unit area

or Wb/m2

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Field Intensity

• The effort made by the current in the wire to setup a magnetic field.

• Magnetomotive force (mmf) per unit length is known as the “magnetizing force” H

• Magnetizing force and flux density related by:

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Permeability

• Permeability is a measure of the ease by which a magnetic flux can pass through a material (Wb/Am). The higher the better flux can flow in the magnetic materials.

• Permeability of free space o = 4 x 10-7 (Wb/Am)

• Relative permeability, r :

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Reluctance

• Reluctance, which is similar to resistance, is the opposition to the establishment of a magnetic field, i.e." resistance” to flow of magnetic flux. Depends on length of magnetic path

, cross-section area A and permeability of material .

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Magnetomotive Force

• The product of the number of turns and the current in the wire wrapped around the core’s arm. (The ability of a coil to produce flux)

N

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Magnetomotive Force

• The MMF is generated by the coil• Strength related to number of turns and

current, Symbol F, measured in Ampere turns (At)

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Magnetization Curve

B(T)

H(A/m)

Magnetization curve (B-H characteristic)

Saturation

HB r0

Behavior of flux density compared with magnetic field strength, if magnetic intensity H increases by increase of current I, the flux density B in the core changes as shown.

flux ()

current (I)

linear

Near saturation

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Magnetic Equivalent Circuit

lci

N+F-

E R

i

Analogy between magnetic circuit and electric circuit

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Magnetic Circuit with Air Gaplc

i

N lg

+F-

c

g

g

gg

c

cc

ggccgC

g0

gg

cc

cc

AB

AB

densityFlux

lHlHNiNi

A

l

A

l

;

;

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Parallel Magnetic Circuit

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l2l1

l3

IN

S1S2S3

+- NI

13 2

I

IILoop I

NI = S33 + S11

= H3l3 + H1l1

Loop II

NI = S33 + S22

= H3l3 + H2l2

Loop III

0 = S11 + S22

= H1l1 + H2l2

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Electric vs Magnetic Circuit

Term Symbol Term Symbol

Magnetic flux Electric current I

Flux density B Current density J

Magnetomotive force

F Electromotive force

E

Permeability Permitivity

Reluctance Resistance R

Magnetic circuit Electric circuit

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Leakage Flux

• Part of the flux generated by a current-carrying coil wrapped around a leg of a magnetic core stays outside the core. This flux is called leakage flux.

Useful flux

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Fringing Effect• The effective area provided for the flow of

lines of magnetic force (flux) in an air gap is larger than the cross-sectional area of the core. This is due to a phenomenon known as fringing effect.

Air gap

– to avoid flux saturation when too much current flows

- To increase reluctance

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Example 1

Refer to Figure below, calculate:-1) Flux 2) Flux density 3) Magnetic intensity Given r = 1,000; no of turn, N = 500; current, i = 0.1 A.

cross sectional area, A = 0.0001m2 , and means length core lC = 0.36 m.

iN

lc

1. 1.75x10-5 Wb2. 0.175 Wb/m2

3. 139 AT/Wb

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Example 2

• The Figure represents the magnetic circuit of a relay. The coil has 500 turns and the mean core path is lc = 400 mm. When the air-gap lengths are 2 mm each, a flux density of 1.0 Tesla is required to actuate the relay. The core is cast steel.

a. Find the current in the coil. (6.93 A) b. Compute the values of permeability and relative

permeability of the core. (1.14 x 103

, 1.27) c. If the air-gap is zero, find the current in the coil for the

same flux density (1 T) in the core. ( 0.6 A)

i

N

Movablepart

lg

Pg 8 : SEN

Data- 1T – 700 at/m

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Electromagnetic Induction

• An emf can be induced in a coil if the magnetic flux through the coil is changed. This phenomenon is known as electromagnetic induction.

• The induced emf is given by

• Faraday’s law: The induced emf is proportional to the rate of change of the magnetic flux.

• This law is a basic law of electromagnetism relating to the operating principles of transformers, inductors, and many types of electrical motors and generators.

dt

dNe

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Electromagnetic Induction

• Faraday's law is a single equation describing two different phenomena: The motional emf generated by a magnetic force on a moving wire, and the transformer emf generated by an electric force due to a changing magnetic field.

• The negative sign in Faraday's law comes from the fact that the emf induced in the coil acts to oppose any change in the magnetic flux.

• Lenz's law: The induced emf generates a current that sets up a magnetic field which acts to oppose the change in magnetic flux.

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Lenz’s LawAn induced current has a direction such that the magnetic field due to the induced current opposes the change in the magnetic flux that induces the current.

As the magnet is moved toward the loop, the B through the loop increases, therefore a counter-clockwise current is induced in the loop. The current produces its own magnetic field to oppose the motion of the magnet

If we pull the magnet away from the loop, the B through the loop decreases, inducing a current in the loop. In this case, the loop will have a south pole facing the retreating north pole of the magnet as to oppose the retreat. Therefore, the induced current will be clockwise.

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Self-Inductance

• From Faraday’s law

• Where is the flux linkage of the winding is defined as

• For a magnetic circuit composed of constant magnetic permeability, the relationship between and i will be linear and we can define the inductance L as

• It can be shown later that

e = Ndφ

dt=

dt

≡Nφ

L ≡λ

i

L =N 2

ℜ eq

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Self-Inductance

• For a magnetic circuit having constant magnetic permeability

• So,

φ=F

ℜ=

Nilμoμ r A

L ≡Nφ

i=

N

i.

Nilμoμ r A

=μoμ r AN 2

l=

N 2

ℜHenry

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Mutual Inductance

+ +

--

ii

N Nturns turns

g

Magnetic corePermeability ,

Mean core length lc,

Cross-sectional area Ac

Notice the current i1 and i2 have been chosen to produce the flux in the same direction. It is also assumed that the flux is confined solely to the core and its air gap.

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Mutual Inductance

The total mmf is therefore

with the reluctance of the core neglected and assuming that Ac = Ag the core flux is

φ=(N1i1 + N2i2)μoAc

g

F = N1i1 + N2i2

If the equation is broken up into terms attributable to the individual current, the flux linkages of coil 1 can be expressed as

1 = N1φ = N12 μoAc

g

⎝ ⎜

⎠ ⎟i1 + N1N2

μoAc

g

⎝ ⎜

⎠ ⎟i2

⇒ λ1 ≡ L11i1 + L12i2

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Mutual Inductance

where

L11 = N12 μoAc

g

⎝ ⎜

⎠ ⎟ is the self-inductance of coil 1

and

L11i1 is the flux linkage of coil 1 due to its own current i1.

The mutual inductance between coils 1 and 2 is

L12 = N1N2

μoAc

g

⎝ ⎜

⎠ ⎟

and

L12i2 is the flux linkage of coil 1 due to current i2.

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Mutual Inductance

where

L21 = L12

is the self-inductance of coil 2.

Similarly, the flux linkage of coil 2 is

is the mutual inductance and

2 = N2φ = N1N2

μoAc

g

⎝ ⎜

⎠ ⎟i1 + N2

2 μoAc

g

⎝ ⎜

⎠ ⎟i2

⇒ λ 2 ≡ L21i1 + L22i2

L22 = N22 μoAc

g

⎝ ⎜

⎠ ⎟

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Mutual Inductance: Example

+ +

--

ii

N Nturns turns

g

Magnetic core

Permeability ,

Cross-sectional area Ac = Ag = 1 cm X 1.5916 cmAir gap length, g = 2 mm

N1 = 100 turns, N2 =200 turns Find L11, L22, and L12 = L21 = M

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Magnetic Stored Energy

We know that for a magnetic circuit with a single winding

=Nφ = Li

e =dλ

dt=

d

dt(Nφ) =

d

dt(Li)and

For a static magnetic circuit the inductance L is fixed

e = Ldi

dt

For a electromechanical energy device, L is time varying

e = Ldi

dt+ i

dL

dt

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Magnetic Stored Energy

The power p is

p = ie = idλ

dt

Thus the change in magnetic stored energy

ΔW = pdt = idλλ1

λ2

∫t1

t2

∫ =λ

Ldλ

λ1

λ2

∫ = 12L λ 2

2 − λ12

( )

The total stored energy at any is given by setting 1 = 0:

W =1

2Lλ2 =

L

2i2