Chap 1 Colour

Electrical Machines ELX 311 Chapter 1 2013/02/05

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ELX 311 Chapter 1

Magnetic Circuits and Materials

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1.1 INTRODUCTION TO MAGNETIC CIRCUITS




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1.1 INTRODUCTION TO MAGNETIC CIRCUITS

Primary way by which energy is converted from one form to another in Generators, Motors and TransformersFour basic principles of how magnetic fields are used

1. Current in a wire causes a magnetic field.2. Transformer action (Time changing magnetic field through

a coil induces a voltage in the coil)3. Motor action (Current carrying wire in a magnetic field has

a force induced on it)4. Generator action (Moving wire in a magnetic field has a

voltage induced in it)

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1.1 Intro to Magnetic Circuits (2)




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How a coil creates a magnetic field

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DEFINITIONSH: Magnetic Field Intensity. A measure of the effort that a current is putting into the establishment of a magnetic field. [At/m] = Ampereturns / meter: Magnetic Permeability of material. The relative ease of establishing a magnetic field in a given material. [H/m] = Henry / meterB: Magnetic Flux Density. Measure of the number of flux lines per unit area. [Wb/m2] = Weber / square meter or [T] = Tesla




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Amperes Law

0

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The magnetomotive force (mmf )in ampere-turns around any closed path is equal to the net current in amperes enclosed by the path.

Value of H is the component of

H tangent to ds.

Hx

Hx

Hx


Amperes Law




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Magnetic circuit

Valid Assumptions:All the current remains within the conductorAll Magnetic Flux () remain within the high permeability

magnetic core, i.e. negligible leakage fluxThe cross sectional area Ac is perpendicular to the flux lines

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Magnetic circuit

because path length of any flux line is close to the mean core length lc

Since = Ni Hclc = Ni = F Or




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Permeability and Flux

= 0r where:0: Permeability of free space ( air)

0 = 4pi 10-7 H/mr: Relative permeability of the specific material

r range from 2000 80000: Flux in Weber [Wb]

= BAA = perpendicular cross sectional area through which flux lines are cutting

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Magnetic Circuit with air gap

/




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Magnetic Circuit with air gap

RRRR: Reluctance in [At/Wb]

: Flux in [Wb]

FFFF: Magnetomotive Force (MMF) in [At]

= Ni

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Electrical Circuit vs Magnetic Circuit

= Ni




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1.1 Intro to Magnetic Circuits (11b)

Polarity of the MMF

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Magnetic Circuits

Magnetic Circuit Calculations are always an approximation because of:1. Leakage flux2. Assumption of a mean path length3. Nonlinearity of since = f() (See Sec. 1.3)4. Fringing (Ignored in this book)




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Example 1-1

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Example 1-1




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Do:Example 1.2 our way

Practice Problems 1.1 and 1.2

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1.2 FLUX LINKAGE, INDUCTANCE AND

ENERGY




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1.2 Flux linkage, inductance and energy (1)

Faradays Law

!" # $$& #$$&

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Problem with Faradays equation It assumes that the same amount of flux is present in each

turn of the coil.

NOT the case due to flux leakage.! $$& !" *!

+, *$$&

+, $$&*

+,

!" $$& where *

+,[Wbt]


Flux LinkageFlux that links the coils




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Inductance

If {(r = constant) or (RRRRgggg >> RRRRcccc )} thenRelation between and i = linear

Inductance:

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Inductance

Inductance is measure in Henrys (H)

or weber-turns per ampere

Look at proportionality of Inductance for the case (RRRRgggg >> RRRRcccc )} :




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Example 1.3

Given: N-turns, current = i, r =, g1 , A1, g2 , A2.Find: a) L b) B1

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Example 1.4

Given: Based on ex. 1.1, r = 72300 2900Find: Inductance for two r values.




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Inductance (Self and Mutual)

Resultant Core Flux:

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Inductance (Self and Mutual)

Flux linkage in coil 1 due to i1 and i2 respectively:

Self Inductance

Mutual Inductance




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Energy

For static magnetic circuit

Power [W= J/s]:

W [J]:


Energy




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1.3 PROPERTIES OF MAGNETIC MATERIALS

1.3 Properties of Magnetic Materials (1)

Importance of Magnetic Materials

Magnetic Material are important in the context of

electromechanical energy conversion devices:

1. Large B with low F - high B means high energy density. Like choosing between different types of conductors,

certain conductors conduct better than other.

2. MM constrain and direct magnetic fields in well-defined

paths

Transformers: maximise coupling and lower excitation

current.

Rotating Machines: shape the fields to obtain desired

torque-production and electrical terminal

characteristics.





Magnetic Domains

Iron (Fe = Ferrous) and alloys of iron with other metals

(cobalt, tungsten, nickel, aluminium)

http://mujiholic-technoholic.blogspot.com/2008/01/do-

you-know-magnet-works.html

Magnetic Domains:

http://hyperphysics.phy-astr.gsu.edu/hbase/solids/ferro.html


Hysteresis and the B-H Curve

http://hyperphysics.phy-astr.gsu.edu/hbase/solids/hyst.html




35B-H loops for M-5 grain-oriented electrical steel 0.012 in thick. Only the top halves of the loops are shown here. (Armco Inc.) Figure 1.9


Hysteresis and the B-H Curve

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DC or Normal Magnetization Curve

http://info.ee.surrey.ac.uk/Workshop/advice/coils/BHCkt/index.html




37B-H loops for M-5 grain-oriented electrical steel 0.012 in thick. Only the top halves of the loops are shown here. (Armco Inc.) Figure 1.9



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Dc magnetization curve for M-5 grain-oriented electrical steel 0.012 in thick. (Armco Inc.)Figure 1.10




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Practice Problem 1.6

1.6 T

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1.4

AC EXCITATION




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1.4 AC Excitation (1)

Induced Voltage

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Induced Flux and Voltage How??

http://skelectricals1997.hpage.co.in/ac_alternator_-_basic_1802439.html




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Excitation Current i

B = /A and H = Ni/l, Hence B-H curve can be replace by a -I curve

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Magnetisation Current

From: Electric Machinery Fundamentals, SJ Chapman, 4ed




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Excitation rms VAsAC excitation characteristic of core materials

VAs required to excite a core:

Core

VolumeExcitation rms Vas pu mass:


Excitation rms VAsAC excitation characteristic of core materials

Normalised rms excitation VAs a material property, ie

independent of shape and size

Depends only on Bmax because Hrms = function of Bmaxbased on BH curve at a specific frequency.





Excitation rms required for a magnetic material per unit weight

Exciting rms voltamperes per kilogram at 60 Hz for M-5 grain-oriented electrical steel 0.012 in thick. (Armco Inc.)Figure 1.12


Excitation Current ComponentsThe excitation current supplies:

a. the mmf required to produce the core flux

(magnetization current) and

b. the power input associated with the energy in

the magnetic field in the core.

POWER

Active Reactive

Core (Heat)

Losses

Eddy

CurrentsHysteresis

Magne-

tization

iex = icore_loss + imagnetization





Eddy Current Losses

Peddy = k B2max f

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Hysteresis Losses





Pcore = Peddy + Physteresis

Core loss at 60 Hz in watts per kilogram for M-5 grain-oriented electrical steel 0.012 in thick. (Armco Inc.) Figure 1.14

Core loss depend on:

Metallurgy of the material, ie type of material

Flux density

Frequency

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Grain-Oriented (Silicon) Steel

Nonoriented Steel Oriented Steel




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Grain-Oriented Silicon Steel

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Grain-Oriented Silicon Steel

www.learnabout-electronics.org




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Example 1.8

1 in = 2.54 cm

Mean flux

path length lc

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1.5 PERMANENT MAGNETS




1.5 Permanent Magnets (1)

Residual flux density Br and coercivity Hc


Residual flux density Br and coercivity Hc

(a) Second quadrant of hysteresis loop for Alnico 5;(b) second quadrant of hysteresis loop for M-5 electrical steel; (c) hysteresis loop for M-5 electrical steel expanded for small B. (Armco Inc.)Figure 1.16





Example 1.9

0.25

0.3


Coercivity Hc

Coercivity Hc is a measure of: the magnitude of the mmf required to demagnetize the

material

The capability of the material to produce flus in a

magnetic circuit which includes an air gap (ex 1.9)





Maximum Energy Product

Useful measure of the

capability of a PM

material.

Corresponds to the

largest B-H product,

which in turn corres-

ponds to a point on the

second quadrant of the

hysteresis loop.

This point results in the

smallest volume of that

material required to

produce a given flux

density in an air gap.

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Maximum Energy Product

[Joule]

[m3] [J/m3]

Choosing a material with the largest available

maximum energy product can result in the smallest

required magnet volume.




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Maximum Energy ProductObjective: Find an equation that will give me the volume of

the magnetic material required to produce a specific

magnetic flux density inside a given sized airgap.

From Example 1.9:

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Maximum Energy ProductObjective: Find an equation that will give me the volume of

the magnetic material required to produce a specific

magnetic flux density inside a given sized airgap.

1. Minimize Vm by operating

the magnet at point of

maximum energy product, ie

the load line must go

through (HmBm)max

2. The larger (HmBm), the

smaller the required magnet

to produce the required Bg.





Example 1.10

Find: The Vg,min for Bg = 0.8 T

Step 1: Graphically estimate the point of maximum energy product on

the BH curve.

Pt1: H = -30, B 1.1 HB = -33 kJ/m3

Pt2: H -40, B 1.0 HB = -40 kJ/m3

Pt3: H -45, B 0.8 HB = -36 kJ/m3

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Example 1.10

Find: The Vg,min for Bg = 0.8 T

Step 2: From Eq 1.57 calculate Am:

= AmBm = AgBg so

Am = AgBg/ Bm= 2cm20.8/1

= 1.6cm2Step 3: From Eq 1.58 calculate lm:

F = 0 = Hmlm + Hglg so

lm = -Hglg/ Hm = -Bglg/0Hm= -0.80.2cm/(4pi10-7 -40 103)= 3.18 cm

Step 4: Hence the minimum magnet volume required is:

Vm = lm Am= 3.18 x 1.6

= 5.09 cm3

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