Mesin Dc Ch1

8/17/2019 Mesin Dc Ch1

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MESIN ARUS SEARAH DANMESIN SINKRON (ECI 640)

PRIANDA

[email protected]

Referensi :

Electric machinery fundamentals, Stephen J. hapman, !c "ra# $ill, %th edition.


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MESIN ARUS SEARAH


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Chapter 1 Introducton to Machner!"rncp#e$

1%1 E#ectrca# Machne$& 'ran$orer$& and Da#! *e

1%+ Rotatona# Moton& Ne,ton-$ *a,& and "o,er Re#aton$hp$An.u#ar "o$ton / An.u#ar e#oct! / An.u#ar

Acce#eraton / 'orue ' / Ne,ton -$ *a, o Rotaton / 2or3 2 "o,er "

I%4 'he Ma.netc e#d"roducton o a Ma.netc e#d / Ma.netc Crcut$ /

1%5 arada!-$ *a,Induced o#ta.e ro a 'eChan.n.Ma.netc e#d

1%6 "roducton o Induced orce on a 2re

1%7 Induced o#ta.e on a Conductor Mo8n. n a Ma.netc e#d

I%9 'he *near DC Machne A Sp#e E:ap#eStartn. the *near DC Machne / 'he #near DCMachne a$ a Motor / 'he *near DC Machne a$ a

;enerator / Startn. "ro


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CHA"'ER 1IN'RODUC'ION 'O

MACHINER= "RINCI"*ES


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5% &'RANSORMERS& AND

DAI*= *IE An electrical machine is a de&ice that can con&ert eithermechanical ener'y to electrical ener'y or electrical ener'y to

mechanical ener'y. (hen such a de&ice is used to con&ert

mechanical ener'y to electrical ener'y, it is called a 'enerator.

(hen it con&erts electrical ener'y to mechanical ener'y, it iscalled a motor. Since any 'i&en electrical machine can con&ert

po#er in either direction, any machine can )e used as either a

'enerator or a motor. Almost all practical motors and

'enerators con&ert ener'y from one form to another throu'h

the action of a ma'netic field, and only machines usin'ma'netic fields to perform such con&ersions are considered in

this )oo*.


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+he transformer is an electrical de&ice that is closely related to electrical

machines. It con&erts ac electrical ener'y at one &olta'e le&el to ac

electrical ener'y at another &olta'e le&el. Since transformers operate onthe same principles as 'enerators and motors, dependin' on the action of

a ma'netic field to accomplish the chan'e in &olta'e le&el, they are usually

studied to'ether #ith 'enerators and motors.

+hese three types of electric de&ices are u)iuitous in modern daily life.

Electric motors in the home run refri'erators, free-ers, &acuum cleaners,)lenders, air conditioners, fans, and many similar appliances. In the

#or*place, motors pro&ide the moti&e po#er for almost all tools. f course,

'enerators are necessary to supply the po#er used )y all these motors.

(hy are electric motors and 'enerators so common/ +he ans#er is &ery

simple: Electric po#er is a clean and efficient ener'y source that is easy totransmit o&er lon' distances, and easy to control. An electric motor does

not reuire constant &entilation and fuel the #ay that an internal0

com)ustion en'ine does, so the motor is &ery #ell suited for use in

en&ironments #here the pollutants associated #ith com)ustion are not

desira)le.

E*EC'RICA* MACHINES& 'RANSORMERS& AND DAI*=*IE


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Instead, heat or mechanical ener'y can )e con&erted to electrical form at

a distant location, the ener'y can )e transmitted o&er lon' distances to the

place #here it is to )e used, and it can )e used cleanly in any home,office, or factory. +ransformers aid this process )y reducin' the ener'y

loss )et#een the point of electric po#er 'eneration and the point of its use.

E*EC'RICA* MACHINES& 'RANSORMERS& AND DAI*=*IE


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8 NE2'ON-S*A2& AND "O2ER

RE*A'IONSHI"S Almost all electric machines rotate a)out an a1is, called the

shaft of the machine. 2ecause of the rotational nature of

machinery, it is important to ha&e a )asic understandin' of

rotational motion. +his section contains a )rief re&ie# of the

concepts of distance, &elocity, acceleration, Ne#ton3s la#, and

po#er as they apply to rotatin' machinery.In 'eneral, a three0dimensional &ector is reuired to

completely descri)e the rotation of an o)4ect in space.

$o#e&er, machines normally turn on a fi1ed shaft, so their

rotation is restricted to one an'ular dimension. Relati&e to a'i&en end of the machine3s shaft, the direction of rotation can

)e descri)ed as either cloc*#ise 5(6 or countercloc*#ise

5(6. 7or the purpose of this &olume, a countercloc*#ise

an'le of rotation is assumed to )e positi&e, and a cloc*#ise

one is assumed to )e ne'ati&e.


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7or rotation a)out a fi1ed shaft, all the concepts in this section reduce to

scalars. Each ma4or concept of rotational motion is defined )elo# and is

related to the correspondin' idea from linear motion.Angular Position >

+he an'ular position > of an o)4ect is the an'le at #hich it is oriented,measured from some ar)itrary reference point. An'ular position is usually

measured in radians or de'rees. It corresponds to the linear concept of

distance alon' a line.Angular Velocity ?

An'ular &elocity 5or speed6 is the rate of chan'e in an'ular position #ith

respect to time. It is assumed positi&e if the rotation is in a

countercloc*#ise direction. An'ular &elocity is the rotational analo' of the

concept of &elocity on a line. ne dimensional linear &elocity alon' a line isdefined as the rate of chan'e of the displacement alon' the line 5r6 #ith

respect to time

RO'A'IONA* MO'ION& NE2'ON-S *A2& AND "O2ERRE*A'IONSHI"S


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Similarly, an'ular &elocity ? is defined as the rate of chan'e of the an'ulardisplacement > #ith respect to time.


If the units of an'ular position are radians, then an'ular &elocity is

measured in radians per second.

In dealin' #ith ordinary electric machines, en'ineers often use units other

than radians per second to descri)e shaft speed. 7reuently, the speed is'i&en in re&olutions per second or re&olutions per minute. 2ecause speed

is such an important uantity in the study of machines, it is customary to

use different sym)ols for speed #hen it is e1pressed in different units. 2y

usin' these different sym)ols, any possi)le confusion as to the units

intended is minimi-ed. +he follo#in' sym)ols are used in this )oo* to

descri)e an'ular &elocity:


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+he su)script m on these sym)ols indicates a mechanical uantity, as

opposed to an electrical uantity. If there is no possi)ility of confusion

)et#een mechanical and electrical uantities, the su)script is often left out.+hese measures of shaft speed are related to each other )y the follo#in'

euations:


Angular Acceleration α

An'ular acceleration is the rate of chan'e in an'ular &elocity #ith respect

to time. It is assumed positi&e if the an'ular &elocity is increasin' in an

al'e)raic sense. An'ular acceleration is the rotational analo' of the

concept of acceleration on a line. Just as one0dimensional linearacceleration is defined )y the euation

an'ular acceleration is defined )y


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If the units of an'ular &elocity are radians per second, then an'ular

acceleration is measured in radians per second suared.

Torque

In linear motion, a force applied to an o)4ect causes its &elocity to chan'e.

In the a)sence of a net force on the o)4ect, its &elocity is constant. +he

'reater the force applied to the o)4ect, the more rapidly its &elocity

chan'es.

+here e1ists a similar concept for rotation. (hen an o)4ect is rotatin', itsan'ular &elocity is constant unless a torue is present on it. +he 'reater

the torue on the o)4ect, the more rapidly the an'ular &elocity of the o)4ect

chan'es.

(hat is torue/ It can loosely )e called the 8t#istin' force8 on an o)4ect.

Intuiti&ely, torue is fairly easy to understand. Ima'ine a cylinder that isfree to rotate a)out its a1is. If a force is applied to the cylinder in such a

#ay that its line of action passes throu'h the a1is 57i'ure 909a6, then the

cylinder #ill not rotate.

$o#e&er, if the same force is placed so that its line of action passes to the

ri'ht of the a1is 57i'ure 909)6, then the cylinder #ill tend to rotate in acountercloc*#ise direction.



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+he torue or t#istin' action on the cylinder depends on 596 the ma'nitude

of the applied force and 56 the distance )et#een the a1is of rotation and

the line of action of the force.


7I";RE 909

5a6 A force applied to a cylinder so that it passes throu'h the a1is of

rotation. < . 5)6 A force applied to a cylinder so that its line of action Ʈ

misses the a1is of rotation. $ere is countercloc*#ise. Ʈ

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+he torue on an o)4ect is defined as the product of the force applied to

the o)4ect and the smallest distance )et#een the line of action of the force

and the o)4ect3s a1is of rotation. If r is a &ector pointin' from the a1is of

rotation to the point of application of the force, and if F is the applied force,

then the torue can )e descri)ed as


#here > is the an'le )et#een the &ector r and the &ector F. +he directionof the torue is cloc*#ise if it #ould tend to cause a cloc*#ise rotation and

countercloc*#ise if it #ouId tend to cause a countercloc*#ise rotation

57i'ure 906.

+he units of torue are ne#ton0meters in SI units and pound0feet in the

En'lish system.

Newton's Law of Rotation

Ne#ton3s la# for o)4ects mo&in' alon' a strai'ht line descri)es the

relationship )et#een the force applied to an o)4ect and its resultin'

acceleration. +his relationship is 'i&en )y the euation

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15RO'A'IONA* MO'ION& NE2'ON-S *A2& AND "O2ER

RE*A'IONSHI"S

7I";RE 90

Deri&ation of the euation for the torue on an

o)4ect.

In SI units, force is measured in ne#tons,mass in *ilo'rams, and acceleration in

meters per second suared. In the

En'lish system. force is measured in

pounds, mass in slu's, and acceleration

in feet per second suared. A similar euation descri)es the

relationship )et#een the torue applied

to an o)4ect and its resultin' an'ular

acceleration. +his relationship, called

Newton’s law of rotation, is 'i&en )y theeuation

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RE*A'IONSHI"S

#here is the net applied torue in ne#ton0meters or pound0feet and α is

the resultin' an'ular acceleration in radians per second suared. +he term

J ser&es the same purpose as an o)4ect3s mass in linear motion. It is

called the moment of inertia of the o)4ect and is measured in *ilo'ram0

meters suared or slu'0 feet suared. alculation of the moment of inertia

of an o)4ect is )eyond the scope of this )oo*.

Work W

7or linear motion, #or* is defined as the application of a force throu'h a

distance. In euation form,

#here it is assumed that the force is collinear #ith the direction of motion.7or the special case of a constant force applied collinearly #ith the

direction of motion, this euation )ecomes 4ust

+he units of #or* are 4oules in SI and foot0pounds in the En'lish system.

7or rotational motion, #or* is the application of a torque throu'h an angle.$ere the e uation for #or* is

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RE*A'IONSHI"S

and if the torue is constant,

Power P

Po#er is the rate of doin' #or*, or the increase in #or* per unit time. +he

euation for po#er is

It is usually measured in 4oules per second 5#atts6, )ut also can )e

measured in foot0pounds per second or in horsepo#er.

2y this definition, and assumin' that force is constant and collinear #ith

the direction of motion, po#er is 'i&en )y

Similarly, assumin' constant torue, po#er in rotational motion is 'i&en )y

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RE*A'IONSHI"S

Euation 5909=6 is &ery important in the study of electric machinery,)ecause it can descri)e the mechanical po#er on the shaft of a motor or

'enerator.

Euation 590I=6 is the correct relationship amon' po#er, torue, and speed

if po#er is measured in #atts, torue in ne#ton0meters, and speed in

radians per second. If other units are used to measure any of the a)o&euantities, then a constant must )e introduced into the euation for unit

con&ersion factors. It is still common in ;.S. en'ineerin' practice to

measure torue in pound0feet, speed in re&olutions per minute, and po#er

in either #atts or horsepo#er. If the appropriate con&ersion factors are

included in each term, then Euation 5909=6 )ecomes

#here torue is measured in pound0feet and speed is measured inre&olutions per minute.

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4% 'HE MA;NE'IC IE*D

!a'netic fields are the fundamental mechanism )y #hichener'y is con&erted from one form to another in motors,

'enerators, and transformers. 7our )asic principles descri)e

ho# ma'netic fields are used in these de&ices:

9.A current0carryin' #ire produces a ma'netic field in the area

around it.

.A time0chan'in' ma'netic field induces a &olta'e in a coil of

#ire if it passes throu'h that coil. 5+his is the )asis of

transformer action.6

>.A current0carryin' #ire in the presence of a ma'netic field

has a force induced on it. 5+his is the )asis of motor action.6

%.A mo&in' #ire in the presence of a ma'netic field has a

&olta'e induced in it. 5+his is the )asis of 'enerator action.6

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+he )asic la# 'o&ernin' the production of a ma'netic field )y a current is

Ampere3s la#:

'HE MA;NE'IC IE*D

"roducton o a Ma.netc e#d

#here H is the ma'netic field intensity produced )y the current I net and dl

is a differential element of len'th alon' the path of inte'ration. In SI units, I

is measured in amperes and H is measured in ampere0turns per meter. +o

)etter understand the meanin' of this euation, it is helpful to apply it to

the simple e1ample in 7i'ure 90>. 7i'ure 90> sho#s a rectan'ular core #ith

a #indin' of N turns of #ire #rapped a)out one le' of the core. If the core

is composed of iron or certain other similar metals 5collecti&ely called

ferromagnetic materials6, essentially all the ma'netic field produced )ythe current #ill remain inside the core, so the path of inte'ration in

Ampere3s la# is the mean path len'th of the core l c . +he current passin'

#ithin the path of inte'ration I net is then Ni , since the coil of #ire cuts the

path of inte'ration N times #hile carryin' current i . Ampere3s la# thus

)ecomes

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'HE MA;NE'IC IE*D


7I";RE 90>

A simple ma'netic core.

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'HE MA;NE'IC IE*D


$ere H is the ma'nitude of the ma'netic field intensity &ector . +herefore,

the ma'nitude or the ma'netic field intensity in the core due to the applied

current is

+he ma'netic field intensity H is in a sense a measure of the 8effort8 that a

current is puttin' into the esta)lishment of a ma'netic field. +he stren'th of

the ma'netic field flu1 produced in the core also depends on the material

of the core. +he relationship )et#een the ma'netic field intensity H and the

resultin' ma'netic flu1 density B produced #ithin a material is 'i&en )y

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'HE MA;NE'IC IE*D


+he actual ma'netic flu1 density produced in a piece of material is thus

'i&en )y a product of t#o terms:H , representin' the effort e1erted )y the current to esta)lish a

ma'netic field

μ, representin' the relati&e ease of esta)lishin' a ma'netic field in a'i&en

+he permea)ility of any other material compared to the permea)ility of free

space is called its relative permeability :

+he units of ma'netic field intensity are ampere0turns per meter, the unitsof permea)ility are henrys per meter, and the units of the resultin' flu1

density are #e)ers per suare meter, *no#n as teslas 5+6.

+he permea)ility of free space is called μ o and its &alue is

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'HE MA;NE'IC IE*D


Relati&e permea)ility is a con&enient #ay to compare the ma'neti-a)ility

of materials. 7or e1ample, the steels used in modern machines ha&erelati&e permea)ilities of ??? to ??? or e&en more. +his means that, for

a 'i&en amount of current, ??? to ??? times more flu1 is esta)lished in a

piece of steel than in a correspondin' area of air. 5+he permea)ility of air is

essentially the same as the permea)ility of free space.6 )&iously, the

metals in a transformer or motor core play an e1tremely important part in

increasin' and concentratin' the ma'netic flu1 in the de&ice.

Also, )ecause the permea)ility of iron is so much hi'her than that of air,

the 'reat ma4ority of the flu1 in an iron core li*e that in 7i'ure 90> remains

inside the core instead of tra&elin' throu'h the surroundin' air, #hich has

much lo#er permea)ility. +he small lea*a'e flu1 that does lea&e the iron

core is &ery important in determinin' the flu1 lin*a'es )et#een coils and

the self0inductances of coils in transformers and motors.

In a core such as the one sho#n in 7i'ure 90>, the ma'nitude of the flu1

density is 'i&en )y

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'HE MA;NE'IC IE*D


#here d A is the differential unit of area. If the flu1 density &ector isperpendicular to a plane of area A, and if the flu1 density is constant

throu'hout the area, then this euation reduces to

+hus, the total flu1 in the core in 7i'ure 90> due to the current i in the

#indin' is

#here A is the cross0sectional area of the core.

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'HE MA;NE'IC IE*D

Ma.netc Crcut$

In Euation 5906 #e see that the current in a coil of #ire #rapped around

a core produces a ma'netic flu1 in the core. +his is in some senseanalo'ous to a &olta'e in an electric circuit producin' a current no#. It is

possi)le to define a 8ma'netic circuit8 #hose )eha&ior is 'o&erned )y

euations analo'ous to those for an electric circuit. +he ma'netic circuit

model of ma'netic )eha&ior is often used in the desi'n of electric

machines and transformers to simplify the other#ise uite comple1 desi'nprocess.

In a simple electric circuit such as the one sho#n in 7i'ure 90%a, the

&olta'e source V dri&es a current I around the circuit throu'h a resistance

R . +he relationship )et#een these uantities is 'i&en )y hm3s la#:

In the electric circuit, it is the &olta'e or electromoti&e force that dri&es the

current no#. 2y analo'y, the correspondin' uantity in the ma'netic circuit

is called the magneto motive force 5mmf6. +he ma'neto moti&e force of

the ma'netic circuit is eual to the effecti&e current no# applied to the

core, or

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'HE MA;NE'IC IE*D

7I";RE 90%

5a6 A simple electric circuit. 5)6 +he ma'netic circuit

analo' to a transformer core.

#here the sym)ol for ma'neto moti&e force, measured in ampere0turns.

i*e the &olta'e source in the electric circuit, the ma'neto moti&e force in

the ma'netic circuit has a polarity associated #ith it. +he positive end of

the mmf source is the end from #hich the flu1 e1its, and the negative end

of the mmf source is the end at #hich the flu1 reenters.

Ma.netc Crcut$

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'HE MA;NE'IC IE*D

+he polarity of the mmf from a coil of #ire can )e determined from a

modification of the ri'ht0hand rule: If the fin'ers of the ri'ht hand curl in thedirection of the current no# in a coil of #ire, then the thum) #ill point in the

direction of the positi&e mmf 5see 7i'ure 90 =6.

In an electric circuit, the applied &olta'e causes a current I to flo#.

Similarly, in a ma'netic circuit, the applied ma'neto moti&e force causes

flu1 ϕ to )e produced. +he relationship )et#een &olta'e and current in anelectric circuit is hm3s la# 5B < IR6C similarly, the relationship )et#een

ma'netomoti&e force and flu1 is

Ma.netc Crcut$

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'HE MA;NE'IC IE*D

Ma.netc Crcut$

7I";RE 90=Determinin' the polarity of a

ma'netomoti&e force source

in a ma'netic circuit.

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'HE MA;NE'IC IE*D

+he reluctance of a ma'netic circuit is the counterpart of electrical

resistance, and its units are ampere0turns per #e)er.+here is also a ma'netic analo' of conductance. Just as the conductance

of an electric circuit is the reciprocal of its resistance, the permeance of a

ma'netic circuit is the reciprocal of its reluctance:

Ma.netc Crcut$

+he relationship )et#een ma'neto moti&e force and flu1 can thus )e

e1pressed as

;nder some circumstances, it is easier to #or* #ith the permeance of a

ma'netic circuit than #ith its reluctance.

(hat is the reluctance of the core in 7i'ure 90>/ +he resultin' flu1 in this

core is 'i&en )y Euation 5906:

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'HE MA;NE'IC IE*D

Ma.netc Crcut$

2y comparin' Euation 590>96 #ith Euation 5906, #e see that the

reluctance of the core is

Reluctances in a ma'netic circuit o)ey the same rules as resistances in an

electric circuit. +he eui&alent reluctance of a num)er of reluctances in

series is 4ust the sum of the indi&idual reluctances:

Similarly, reluctances in parallel com)ine accordin' to the euation

Permeances in series and parallel o)ey the same rules as electricalconductances.

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'HE MA;NE'IC IE*D

Ma.netc Crcut$alculations of the flu1 in a core performed )y usin' the ma'netic circuit

concepts are always appro1imations0at )est, they are accurate to #ithin

a)out = percent of the real ans#er. +here are a num)er of reasons for this

inherent inaccuracy:

9.+he ma'netic circuit concept assumes that all flu1 is confined #ithin a

ma'netic core. ;nfortunately, this is not uite true. +he permea)ility of a

ferroma'netic core is ??? to ??? times that of air, )ut a small fraction of

the flu1 escapes from the core into the surroundin' lo#0permea)ility air.

+his flu1 outside the core is called leakage flux , and it plays a &ery

important role in electric machine desi'n.

.+he calculation of reluctance assumes a certain mean path len'th and

cross sectional area for the core. +hese assumptions are not really &ery

'ood, especially at corners.

>.In ferroma'netic materials, the permea)ility &aries #ith the amount of

flu1 already in the material. +his nonlinear effect is descri)ed in detail. It

adds yet another source of error to ma'netic circuit analysis, since the

reluctances used in ma'netic circuit calculations depend on the

permea)ility of the material.

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'HE MA;NE'IC IE*DMa.netc Crcut$

%. If there are air 'aps in the flu1 path in a core, the effecti&e cross0

sectional area of the air 'ap #ill )e lar'er than the cross0sectional

area of the iron core on either side. +he e1tra effecti&e area is caused

)y the 8fringing effect8 of the ma'netic field at the air 'ap 57i'ure 90

6.

It is possi)le to partially offset these inherent sources of error )y usin' a

8corrected or 8effecti&e8 mean path len'th and the cross0sectional

area instead of the actual physical len'th and area in the calculations.

+here are many inherent limitations to the concept of a ma'netic circuit,

)ut it is still the easiest desi'n tool a&aila)le for calculatin' flu1es in

7I";RE 90

+he frin'in' effect of a ma'netic field at an air 'ap. Note

the increased cross0sectional area of the air 'ap

compared #ith the cross0sectional area of the metal.

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'HE MA;NE'IC IE*DMa.netc Crcut$E1act calculations usin' !a1#ell3s euations are 4ust too difficult, and they

are not needed any#ay, since satisfactory results may )e achie&ed #ith

this appro1imate method.

+he follo#in' e1amples illustrate )asic ma'netic circuit calculations. Note

that in these e1amples the ans#ers are 'i&en to three si'nificant di'its.

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7I";RE 90F

5a6 +he ferroma'netic core of E1ample 909. 5)6 +he ma'netic circuit

correspondin' to 5a6.

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405% ARADA=-S *A2INDUCED O*'A;E


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5% ARADA= S *A2 INDUCED O*'A;EROM A 'IMECHAN;IN; MA;NE'IC

IE*D+he first ma4or effect to )e considered is called 7aradayGs la#.

It is the )asis of transformer operation. 7araday3s la# states

that if a flu1 passes throu'h a turn of a coil of #ire, a &olta'e

#ill )e induced in the turn of #ire that is directly proportional to

the rate of change in the flu1 #ith respect to time. In euation

form,

#here ein! is the &olta'e induced in the turn of the coil and @

is the flu1 passin' throu'h the turn. If a coil has N turns and if

the same flu1 passes throu'h all of them, then the &olta'einduced across the #hole coil is 'i&en )y

41ARADA=-S *A2INDUCED O*'A;E ROM A 'IME


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ARADA= S *A2INDUCED O*'A;E ROM A 'IMECHAN;IN; MA;NE'IC IE*D

+he minus si'n in the euations is an e1pression of en-Gs la#. en-3s la#

states that the direction of the &olta'e )uildup in the coil is such that if the

coil ends #ere short circuited, it #ould produce current that #ould cause a

flu1 opposin' the ori'inal flu1 chan'e. Since the induced &olta'e opposesthe chan'e that causes it, a minus si'n is included in Euation 590>6. +o

understand this concept clearly, e1amine 7i'ure 909%. If the flu1 sho#n in

the fi'ure is increasin' in stren'th, then the &olta'e )uilt up in the coil #ill

tend to esta)lish a flu1 that #ill oppose the increase.

A current flo#in' as sho#n in 7i'ure 909%) #ould produce a flu1 opposin'the increase, so the &olta'e on the coil must )e )uilt up #ith the polarity

reuired to dri&e that current throu'h the e1ternal circuit. +herefore, the

&olta'e must )e )uildup #ith the polarity sho#n in the fi'ure. Since the

polarity of the resultin' &olta'e can )e determined from physical

considerations, the minus si'n in Euations 590>=6 and 590>6 is often left

out. It is left out of 7araday3s la# in the remainder of this )oo*.



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7I";RE 909%

+he meanin' of en-3s la#: 5a6 A coil enclosin' an increasin' ma'netic

flu1C 5)6 determinin' the resultin' &olta'e polarity.



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+here is one ma4or difficulty in&ol&ed in usin' Euation 590>6 in practical

pro)lems. +hat euation assumes that e1actly the same flu1 is present in

each turn of the coil. ;nfortunately, the flu1 lea*in' out of the core into thesurroundin' air pre&ents this from )ein' true. If the #indin's are ti'htly

coupled, so that the &ast ma4ority of the flu1 passin' throu'h one turn of

the coil does indeed pass throu'h all of them, then Euation 590>6 #ill

'i&e &alid ans#ers. 2ut if lea*a'e is uite hi'h or if e1treme accuracy is

reuired, a different e1pression that does not ma*e that assumption #ill )e

needed. +he ma'nitude of the &olta'e in the i th tum of the coil is al#ays

'i&e n )y

If there are N turns in the coil of #ire, the total &olta'e on the coil is



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ARADA= S *A2 INDUCED O*'A;E ROM A 'IMECHAN;IN; MA;NE'IC IE*D

+he term in parentheses in Euation 590%?6 is called the flu1 lin*a'e " of

the coil, and 7araday3s la# can )e re#ritten in terms of flu1 lin*a'e as

+he units of flu1 lin*a'e are #e)er0turns.

7araday3s la# is the fundamental property of ma'netic fields in&ol&ed in

transformer operation. +he effect of en-3s la# in transformers is to predict

the polarity of the &olta'es induced in transformer #indin's.

#$am%le &(. 7i'ure 909= sho#s a coil of #ire #rapped around an iron

core. If the flu1 in the core is 'i&en )y the euation



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If there are 9?? turns on the core. #hat &olta'e is produced at theterminals of the coil/ f #hat polarity is the &olta'e durin' the time #hen

flu1 is increasin' in the reference direction sho#n in the fi'ure/ Assume

that all the ma'netic flu1 stays #ithin the core 5i.e., assume that the flu1

lea*a'e is -ero6.

7I";RE 909=

+he core of E1ample 900.

Determination of the &olta'e

polarity at the terminals is

sho#n.



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)olution

2y the same reasonin' as in the discussion on pa'es H0 >?, the directionof the &olta'e #hile the flu1 is increasin' in the reference direction must )e

positi&e to ne'ati&e, as sho#n in 7i'ure 909=. +he ma'nitude of the

&olta'e is 'i&en )y

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%INDUCED ORCE ON A

2IRE A second ma4or effect of a ma'netic field on its surroundin'sis that it induces a force on a current0carryin' #ire #ithin the

field. +he )asic concept in&ol&ed is illustrated in 7i'ure 909.

+he fi'ure sho#s a conductor present in a uniform ma'netic

field of flu1 density B, pointin' into the pa'e. the conductor it

self is I meters lon' and contains a current of i amperes. +heforce induced on the conductor is 'i&en )y

48

"RODUC'ION O INDUCED ORCE ON A 2IRE


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+he direction of the force is 'i&en )y the

ri'ht0hand rule: If the inde1 fin'er of the ri'ht

hand points in the direction of the &ector *

and the middle fin'er points in the direction

of the flu1 density &ector +, then the thum)points in the direction of the resultant force

on the #ire.

7I";RE 909

A current0carryin' #ire in the presence of a

ma'netic field.

+he ma'nitude of the force is 'i&en )y the euation

+he ma'nitude of the force is 'i&en )y the euation

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+he induction of a force in a #ire )y a current in the presence of a

ma'netic field is the )asis of motor action. Almost e&ery type of motor

depends on this )asic principle for the forces and torues #hich ma*e itmo&e.

507% INDU ED *'A E N A


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7% INDU ED *'A E N ACONDUC'OR MOIN; IN A

MA;NE'IC IE*D+here is a third ma4or #ay in #hich a ma'netic field interacts

#ith its surroundin's. If a #ire #ith the proper orientation

mo&es throu'h a ma'netic field, a &olta'e is induced in it. +his

idea is sho#n in 7i'ure 909F. +he &olta'e induced in the #ire

is 'i&en )y

Bector * points alon' the direction of the #ire to#ard the end

ma*in' the smallest an'le #ith respect to the &ector , - +.

+he &olta'e in the #ire #ill )e )uilt up so that the positi&e end

is in the direction of the &ector , - +. +he follo#in' e1amples

illustrate this concept.

51INDUCED O*'A;E ON A CONDUC'OR MOIN; IN A


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MA;NE'IC IE*D

7I";RE 909F

A conductor mo&in' in the presence of a

ma'netic field.



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MA;NE'IC IE*D



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MA;NE'IC IE*D

7I";RE 909

+he conductor of E1ample 900H.



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MA;NE'IC IE*D

+he induction of &olta'es in a #ire mo&in' in a ma'netic field is

fundamental to the operation of all types of 'enerators. 7or this reason, itis called generator action.

55%


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% A SIM"*EEAM"*E

A linear dc machine is a)out the simplest and easiest to

understand &ersion of a dc machine, yet it operates accordin'

to the same principles and e1hi)its the same )eha&ior as real

'enerators and motors. It thus ser&es as a 'ood startin' point

in the study of machines.

7I";RE 909H

A linear dc machine. +he ma'netic field points into the pa'e.

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'HE *INEAR DC MACHINE A SIM"*E EAM"*E


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A linear dc machine is sho#n in 7i'ure 909H. It consists of a )attery and a

resistance connected throu'h a s#itch to a pair of smooth, frictionless

rails. Alon' the )ed of this 8railroad trac*8 is a constant, uniform0densityma'netic field directed into the pa'e. A )ar of conductin' metal is lyin'

across the trac*s.

$o# does such a stran'e de&ice )eha&e/ Its )eha&ior can )e determined

from an application of four )asic euations to the machine. +hese

euations are


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(e #ill no# e1plore the fundamental )eha&ior of this simple dc machine

usin' these four euations as tools.

7I";RE 90?

Startin' a linear dc machine.

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Startn. the *near DC Machne

7i'ure 90? sho#s the linear dc machine under startin' conditions. +o start

this machine, simply close the s#itch. No# a current flo#s in the )ar,#hich is 'i&en )y irchhoff3s &olta'e la#:

Since the )ar is initially at rest, ein! < ?, so i . V+ /R. +he current flo#s do#n

throu'h the )ar across the trac*s. 2ut from Euation 590%>6, a currentflo#in' throu'h a #ire in the presence of a ma'netic field induces a force

on the #ire. 2ecause of the 'eometry of the machine, this force is

+herefore, the )ar #ill accelerate to the ri'ht 5)y Ne#ton3s la#6. $o#e&er,

#hen the &elocity of the )ar )e'ins to increase, a &olta'e appears across

the )ar.

+he &olta'e is 'i&en )y Euation 590%=6, #hich reduces for this 'eometry

to

+he &olta'e no# reduces the current flo#in' in the )ar, since )y irchhoff3s

&olta'e la#

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7I";RE 909

+he linear dc machine on startin'.5a6 Belocity &5t6 as a function of timeC

5)6 induced &olta'e eind 5t6C 5c6 current 5t6C

5d6 induced force 7ind 5t6.

As ein! increases, the current i

decreases.

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+he result of this action is that e&entually the )ar #ill reach a constant

steady0state speed #here the net force on the )ar is -ero. +his #ill occur#hen ein! has risen all the #ay up to eual the &olta'e V+. At that time, the

)ar #ill )e mo&in' at a speed 'i&en )y

+he )ar #ill continue to coast alon' at this no0load speed fore&er unless

some e1ternal force distur)s it. (hen the motor is started, the &elocity v ,

induced &olta'e eind , current i , and induced force Fin! are as s*etched in

7i'ure 909.

+o summari-e, at startin', the linear dc machine )eha&es as follo#s:9.losin' the s#itch produces a current flo# i = V B /R .

.+he current flo# produces a force on the )ar 'i&en by F = ilB.

>.+he )ar accelerates to the ri'ht, producin' an induced &olta'e ein! as it

speeds up.

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7I";RE 90

+he linear dc machine as a motor.

63



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'he *near DC Machne a$ a Motor

Assume that the linear machine is initially runnin' at the no0load steady0

state conditions descri)ed a)o&e. (hat #ill happen to this machine if an

e1ternal load is applied to it/ +o find out, let3s e1amine 7i'ure 90. $ere, a

force Floa! is applied to the )ar opposite the direction of motion. Since the

)ar #as initially at steady state, application of the force Floa! #ill result in a

net force on the )ar in the direction opposite the direction of motion 5Fnet .

Floa! Fin!6. +he effect of this force #ill )e to slo# the )ar. 2ut 4ust as soonas the )ar )e'ins to slo# do#n, the induced &olta'e on the )ar drops 5eind

= v↓BI 6. As the induced &olta'e decreases, the current flo# in the )ar

rises:

+herefore, the induced force rises too 5Fin! . i0*+6. +he o&erall result of thischain of e&ents is that the induced force rises until it is eual and opposite

to the load force, and the )ar a'ain tra&els in steady state, )ut at a lo#er

speed. (hen a load is attached to the )ar, the &elocity ,, induced &olta'e

ein!, current i, and induced force Fin! are as s*etched in 7i'ure 90 >.

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66



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68



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'he *near DC Machne a$ a ;enerator

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