Electrical Fundmentals

8/12/2019 Electrical Fundmentals

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Chapter 1

Electrical Fundamentals

asic Electricit!

ELECR'CAL C'RC'$, no matter how comple.,

&ollow certain /asic principles" %nowlege o& these principles is

essential to uner stan how an electrical circuit operates an to

trou/leshoot e&&ecti!ely" An electrical circuit is an arrangement

or con &iguration o& components that &orm a close ioop" here

are three parameters in all electrical circuits0 current, !oltage,

an resistance"

Current

Current is a measure o& the electron &low in a circuit" his

may /e compare to the &low o& &lui through a pipe, which is

measure in gallons per minute (gpm)" he current &low o&

electrons in an electrical circuit is measure in amperes (A)"

1hen 2"343 5 +6+7electrons mo!e past a point in + secon (s),

the current &low is sai to /e ' ampere" he letter ' is use to

represent current" $cienti&ic notation is o&ten use to inicate the

amount o& current" here&ore" small amounts o& current may /e

measure in milliamperes (mA) (6"66+ ampere), an largeamounts o& current may /e measure in 8iloamperes (8A) (+666

amperes)"

"#lta$e

Voltage is the i&&erence in potential (charge) /etween two

points" E.presse in another way, !oltage is the amount o&

ri!ing &orce or pressure applie to a circuit" he !oltage in an

(3)


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%HM&S LA'( P%'ER( AND ENER)*

%hm&s La+ as Applied t# a Resisti,e Circuit

his section presents what is pro/a/ly the most

&unamental electric circuit relationship - the relationship

/etween !oltage an current in a circuit consisting o& a !oltage

source an a resistor" he circuit shown in *igure l"A"l illustrates

the /asic concept" 'n this case, the !oltage source is a /attery"

he /attery in *igure l"A"l pro!ies the &orce which causeselectrons to &low through the connecte circuit" his &orce,

terme the Electro


5/59

he rate at which electrons &low through a circuit is terme

?current>, an is measure in amperes" Current is enote /y the

sym/ol, '" One ampere represents the amount o& electrical

charge (i"e", electrons) mo!ing past a gi!en point in one secon"

'n this circuit, the connecte loa, perhaps an electric

heater, pro!ies resistance to the &low o& current, an is

measure in ohms" he sym/ol &or resistance is shown in *igure

+" + ari is assigne the letter, R@ ohms is a//re!iate /y the

ree8 letter, capital =omega> - " he larger the ohmic !alue o&

the resistor, the greater the resistance presente to the &low o&

current" it is important to note that resistance is a physical

property o& materials, li8e harness or tensile strength, an is,&or

our purposes, totally inepenent o& the !oltage applie to the

circuit"

A linear relationship e.ists /etween the applie !oltage

an the resulting current, with an increase in !oltage proucing a

proportional increase in current" hat is, &or any gi!en !alue o&

resistance, ou/ling the !oltage applie will ou/le the current"

his relationship is 8nown as Ohms Law, an is gi!en /y the

&ollowing e.pression0

' VR

where V !oltage, ' current, an R resistance

'&, in *igure +"A"l, a 2!olt /attery an a ; res n are use,

the current &lowing in the circuit woul /e ' VR 2; 3

amperes" ou/ling the !oltage to +3 !olts ou/les the current

&low to 4 amperes"

(D)


6/59

'& we a another ;c3 resistor in series with the e.isting

; resistor, the total circuit resistance increases to 2 1ith the

+3 !olt /attery in the circuit, the current &lowing in this new

circuit will /e +32 3 amperes" here&ore, ou/ling the

resistance, while 8eeping the applie !oltage the same, cuts the

current in hal&"

A goo way to remem/er the relationship /etween !oltage,

current, an resistance is to 8eep in min *igure +"A"3"

*or e.ample, the e.pression &or !oltage is o/taine /yco!ering up the V@ similarly, the e.pression &or current is

o/taine /y co!ering up the ', an resistance is o/taine /y

co!ering the R"

P#+er and Ener$! in Resisti,e Circuits

'n aition to the /asic relationship inicate /y Ohms

Law, it is important to 8now the amount o& energy consume /y

circuit elements an the rate at which this energy is consume"

%HM&S LA'

FI)URE 1A./

(2)


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he sym/ol use to enote energy (i"e", the a/ility o o

wor8) is E" he /asic unit o& energy is the =oule>"

he rate at which energy is consume in a circuit is power,enote /y the letter # an measure in watts"

'& the resistor o& *igure l"A"+ consumes + oule o& energy,

it is consuming + watt o& power in + secon"

'n e9uation &orm0

E #t or, e9ui!alently

# Et where E is energy (oules)

# is power (watts)

t is time (secons)

*rom the relationship /etween power an energy, we can

see that,

+ oule + watt-secon

+ watt + oulesecon

he power consume in the circuit o& *igure l"A"l is

relate to the applie !oltage an resulting current /y the

&ollowing e.pression0

# V'

*rom Ohms Law, (V 'R) e9ui!alent e.pressions &or

power can /e eri!e0

# '3R

# V3R

sing the e.ample o& *igure l"A"i, a +3-!olt /atteryconnecte to a 2 resistance prouce a current o& 3 amperes" he

(F)


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a o& power supplie to the resistor is gi!en /y the prouct o& the

!oltage an current@ that is, 3 5 +3 34 watts" o etermine the

total energy supplie to the resistor o!er a perio o& time, the

rate at which energy is consume (in other wors, the power) is

multiplie /y the esire time perio" '&, &or e.ample, we wish

to etermine the total energy supplie to the 2-ohm resistance

o!er a perio o& +3 hours, with energy supplie at the rate o&

34 watts, we simply multiply 34 watts /y +3 hours an get 377

watthours"

Al ternati!e+y, we can calculate the amount o& energy in

oules rather than watthours" $ince one watt is one oule per

secon 34 watts e9uals 34 oules per secon" wel!e hours 's

4;,366 secons (+3GR . 26


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Summar! #- Imp#rtant P#ints

!oltage, current an resistance in a resisti!e circuit are relate

/y Ohms Law0

' VR

V 'R

R V'

#ower an energy in a resisti!e circuit can L e.presse as

&ollows0

# V'

# V3R

# '3R

#ower is the time rate o& consuming energy, while the total

amount o& energy use o!er a perio o& time represents theactual wor8 one /y the circuit"

E #t

# #Et

Series Resist#rs

1hen resistors are connecte in series they ha!e one point

in common" he total resistance is e9ual to the sum o& the

ini!iual resistors" Alge/raically, this can /e written

Rtot R+I R3I R;IJ"" I Rn

he current in a series circuit is the same in each

component o& the circuit /ecause the current must &low through

(K)


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each resistor in the series to get to the ne.t resistor" his &act can

/e e.presse as &ollows

'tot '+ '3 ';J"" 'n

he applie !oltage i!ies across each component in a

series circuit in proportion to the resistance o& the component"

he greater the amount o& resistance, the greater the !oltage

rop across that resistor" 't is customary to use the letter E to

sym/olie a !oltage applie to a circuit an the letter V to

represent the !oltage rop in a circuit" 'n a series circuit,

E V+I V3I V;IJ"" I Vn

he !oltage i!ier rule is use to calculate the !oltage

across each resistor as &ollows0

tot

xx R

VRV =

E0AMPLE 1.1

R

EI=

3+

+66

RR + 4 amperes

V1totRVR+

3D

)+66(D 36 !olts

V2totRVR3

3D

)+66(36 76 !olts

o chec80 he !oltage applie to a circuit is roppe in the

circuit, that is, E V0

E V+I V3 V

Parallel Resist#rs

1hen resistors are connecte in parallel, they ha!e two

points in common" he total resistance o& parallel resistors is

(+6)


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e9ual to the reciprocal o& the sum o& the reciprocals o& the

ini!iual resistors" Re9 o& a parallel circuit is calle the

e9ui!alent resistance, Re9$tate alge/raically"

n

eq

RRRR

R+

"""""++++

;3+

++++

=

he current in a /ranch o& a parallel circuit is e9ual to the

e9ui!alent resistance o& the circuit i!ie /y the resistance o&

that /ranch multiplie /y the total current o& the circuit0 that is"

tot

x

q

x IR

I eR=

E0AMPLE 1./

+66 !olts is applie to each resistor"

===

D

+66

+

+

R

EI 36 amperes

===

36

+66

3

3

R

EI D amperes

he current eli!ere to the circuit /y the !oltage source is

the sum o& the /ranch currents, '+I '3, an e9uals 3D amperes"

he parallel con&iguration o&&ers less resistance to current

&low since each /ranch pro!ies a path &or current &low" he

parallel circuit may /e sai to o&&er more conuctance"

Conuctance is the reciprocal o& resistance an is represente /y

the letter " he unit o& conuctance is the mho which has the

sym/ol " *or the parallel circuit o& E.ample +"3,

D

+

36

+

+

++

++

3+

+

=

=

RR

GR

tot

eq

4 ohms

(++)


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SeriesParallel Circuits

A seriesparallel circuit contains some elements that are

connecte in se an others that are conucte in parallel

E.ample +"; il+ustrates a series-parallel circuit"

E0AMPLE 1.2

Rtot R+I R3R;

R+ +6 ohms

36

+

36

++

+++B

;3

;3

+

=

=

RR

RR +6 ohms

1hen resistors o& e9ual !alue are connecte in parallel, the

e9ui!alent resistance o& R3R;may /e &oun /y i!iing the

num/er o& parallel resistors into the !alue o& the resistance" 'n

this case" 36 i!ie /y 3 e9uals +6 ohms" O/!iously, this

particular metho is more con!enient to use than the reciprocal

metho, /ut remem/er that it only applies when the parallel

resistors are e9ual in !alue"

C#l#r C#din$ and Standard Resistance "alues

A wie !ariety resistors are physically large enough to

ha!e their resistance !alue printe on them" Gowe!er, car/on

composition 0 are too small &or this metho o& ienti&ication so

(+3)


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color coing system is use is use" *our color /ans are printe

on one en o& the resistor an are rea &rom the /an closest to

the en o& the resistor towar the center" Each color represents a

numerical !alue as inicate in a/le +"+" he &irst an secon

/ans represent the &irst an secon igits, respecti!ely" he

thir /an represents the multiplier, or the num/er o& eros that

&ollow the secon igit" he &ourth /an inicates the

manu&acturers tolerance" '& there is no &ourth /an the tolerance

is M36N o& the rate !alue"

a/le +"+Color Coing

6 Blac8 F Violet

+ Brown 7 ray

3 Re K 1hite

; Orange 6"+ ol

4 ellow 6"6+ $il!er

oleranceD reen DN ol

2 Blue +6N $il!er

E0AMPLE 1.3

$uppose the color /ans o& a resistor are yellow, !iolet,

re, an gol" he resistance !alue is etermine as &ollows0

(+;)


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he actual resistance !alue shoul /e /etween 442D an

4K;D ohms"

Occasionally a &i&th /an is use to inicate the &ailure rateo& the resistor0

ellow 6"66+N per +666 hours

Orange 6"6+N per +666 hours

Re 6"+N per +666 hours

Brown +"6N per +666 hours

4irchh#--&s "#lta$e La+

%irchho&&s !oltage law states that =the alge/raic sum o&

the potential rises an rops aroun a close loop is ero" A

close loop is any continuous circuit through which current can

&low &rom a point in one irection an return to that point &rom

another irection" $tate alge/raically, oV 6"

E0AMPLE 1.5

a) *in Rtot

$olution0

Rtot R+I R3I R; +D I D I +6 ;6 ohms

/) *in '"

$olution0 ' ;6

+D6=

totR

E D amperes

c) *inV+, V3an V;"

$olution0

V+ 'R+ (D) (+D) FD !olts

(+4)


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V3 'R3 (D) (D) 3D !olts

V; 'R; (D) (+6) D6 !olts

) Veri&y %irchho&&s !oltage law"

oV E P V+P V3P V; 6

E V+I V3I V;

+D6 FD I 3D I D6

+D6 +D6 (Chec8s)

4irchh#--&s Current La+

%irchho&&s current law states that =the alge/raic sum o&

the currents entering an lea!ing a noe is ero"> (A noe is a

unction o& two or more /ranches") $tate another way, the sum

o& the currents lea!ing a unction must e9ual the sum o& the

currents entering a unction"

E0AMPLE 1.6

*un ';"

$olution0

'3I ';P '+P '4 6

'3I '; '+I '4

'; '+I '4P '3

'; 7 I D P F

'; 2 amperes

C#nduct#rs and Insulat#rs

'n aition to resistors there are other components that

a&&ect the cu in a circuit, &or e.ample, conuctors insu+ators

(+D)


16/59

Qust as resistors oppose the &low o& current in a circuit,

conuct ors pro!ie a nearly resistance-&ree path &or current

&low" 'nsulators restrict the &low o& current to the intene path

/y isolating the circuit &rom aacent material"

he a/ility o& metals to conuct electricity, as well as their

physical strength, ma8es them goo conuctors" Gowe!er, not

all metals conuct the same" *or e.ample, gol an sil!er are

among the /est conuctors, /ut their high cost prohi/it their

e.tensi!e use" Copper is the most commonly use conuctor

/ecause o& its goo conucti!ity an relati!ely low cost"

Aluminum, which is less e.pensi!e than copper, is another

metal commonly use as a conuctor" Gowe!er, aluminum oes

not conuct as well as copper (it is only a/out 26N as goo)"

As the a/ility o& a material to conuct electricity (i"e",

conucti!ity) ecreases, its resistance increases" '& the material

oes not conuct current, it is calle an insulator" 'n actual

practice, no insulator is per&ect" here are i&&erent egrees o&

insulating properties, an insulators are 8nown to &ail at certain

!oltage le!els" he term that is use to rate an insulator is calle

the ielectric strength" he ielectric strength is the /rea8own

point o& an insulator" $ome common materials use as insulators

are mica, ru//er, paper, #yre., glass" an air"

A semiconuctor /elongs to the class o& materials /etween

the insulators an conuctors" ermanium, selenium, silicon,

an other similar compouns are semiconuctors" hese

materials are !ital ingreients &or the prouction o& transistors

an other soli-state e!ices"

(+2)


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Capacit#rs7

A capacitor is mae up o& two

or more plates, which are separate/y a ielectric (insulating) material"

1hen !oltage is applie across the

capacitor plates, current &lows out o&

one plate an into the other through

the !oltage source" 'eally, no

current &lows through the ielectric

separating the capacitor plates" he

charge on the capacitor is store in the electric &iel set up

/etween the oppositely charge plates" '& the applie !oltage is

c, the capacitor will charge to the !alue o& the applie !oltage

an will retain this charge until the applie !oltage changes" '&

the applie !oltage is ac, the capacitor will essentially &ollow the

polarity changes o& the applie !oltage, alternately charging an

ischarging in opposite irections each hal& cycle@ thus

e&&ecti!ely passing alternating current"

Capacitance (C) is the measure o& how much electrical

charge a capacitor can store" he unit o& capacitance is thefarad

(*), name in honor o&


18/59

C!"series#""""";B+3B++B+

+

CCC ++=

C!"parallel# C1 $ C2 $ C%$ &

*or two capacitors in series the total capacitance is e9ual to0

C!"series#3+

3+

CC

CC

+

=

Capacitors i!ie into two general categories@ electrostatic

an electrolytic" Electrostatic capacitors are generally small in

capacitance, are not polarie an theoretically can store a

charge ine&initely" his type o& capacitor is usually ienti&ie

/y the 8in o& ielectric material, such as air, paper, mica,

ceramic, mylar an polystyrene, an whether they are &i.e or

!aria/le in !alue" Varia/le capacitors use air, ceramic, glass, or

poly styrene as a ielectric" Electrolytic capacitors, on the other

han, use an electrolyte /etween the plates much li8e a /attery,

an thus are polarie" Electrolytic pro!ie much larger !alues

o& capacitance@ howe!er they o not retain a charge in e&initely

ue to lea8age through the electrolyte" 'n reality, all capacitors

ha!e lea8age /ut electro static capacitors might ta8e months to

ischarge /ecause o& lea8age"

Capacitors are one of the basic components in electronics.

!he' are used for signal coupling between stages( as b')pass

capacitors( decoupling capacitors( bloc*ing capacitors( tuning

capacitors and filter capacitors. !he' are used in electric power

applications for power factor compensation and for starting

certain t'pes of electric motors. Electrol'tic capacitors are used

to filter the dc output oltage of power supplies.

(+7)


19/59

Capacitanceis an unesira/le property in some instances"

At high &re9uencies, &or e.ample, stray capacitance /etween

lea wires an components prouces unesira/le coupling

/etween unrelate circuits" hese e&&ects must /e minimie /y

care&ul circuit esign an component layout"

*inally, it is help&ul to remem/er these important notes

a/out the capacitor 0

+" A Capacitor /loc8s the &low o& irect current"

3" A capacitor e&&ecti!ely passes alternating current"

;" he capacitance o& a capacitor is etermine /y the

spacing /etween the plates, the sur&ace area o& the

plates, an the type o& ielectric material,

4" Capacitors in series gi!e reuce total capacitance,

capacitors in parallel a irectly"

D" A capacitor temporarily stores energy when charge@

this energy is returne to the circuit when the capacitor

ischarges"

Induct#rs7

'nuctors use the a/ility o& electrical current to create a

magnetic &iel" '& a !oltage is applie to a coil o& wire, thecurrent &lowing in the coil will cause a magnetic &iel to "

e!elop" he more times the wire is celle an the more current

there is in the coil, the strength o& the strength o& the magnetic

&iel"

(+K)


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he property o& a coil that opposes a change in the current

&low is calle inuctance" he inuctance o& a coil epens on

&our &actors0

+" he num/er o& turns (winings) in the coil" 'nuctance

is proportional to the s9uare o& the num/er o& turns in

the coil"

3" he iameter o& the coil" he larger the iameter o& the

coil, the higher the inuctance"

;" he permea/ility (a/ility to /ecome magnetie) o& the

core material"

4" he length o& the coil" he shorter the coil, the higher

the inuctance"

Capacit#rs in Direct current circuits7 C#nsider the char$in$

circuit in -i$. 8 9

1hen the switch is close, charges &rom the source will

istri/ute themsel!es on the plates, that is, a current will &low"

'nitially, this current i is large, /ut as more charge is

accumulate, an hence more !oltage e!elope across the

plates, the ac cumulate charge tens to oppose the &urther &low

o& charge" *inally, when enough charge has /een trans&erre

&rom plate to the other, a !oltage + E will ha!e /een

e!elope across the plates" he plates are then charge to a

ma.imum, an since the !oltage across the plates e9uals the

source !oltage, the current imust /e ero" 'n the ieal situation,

the trans&er o& charge occurs in ero time, /ut in the practical

situation the charging process re9uires a !ery short /ut &inite

time"

(36)


21/59

Fi$ure :;// The Char$in$ circuit

he charging transient o& *ig" F-33 /egins when the switch

is close, at a time calle t s" i& the switch is close, the

!oltage e9uation is

E R$ C

iRI C (F-33)

At the instant the switch is close, the capacitor, which is

consiere initially un charge, has ero !oltage (C 6) so that

E9" F-33 /ecomes

E iR $,

Or i '6E-R (F-3;)

he initial charging current + is thus limite /y the

resistance o& the circuit an is etermine /y the simple

application o& Ohms law" On the other han, a&ter su&&icient

time the capacitor is &ully charge an no current &lows" hen,

&rom E9" F-33,

E +,$ C

Capacitor charging current an !oltage are shown in &ig" ( )a

an /"

(3+)


22/59

ischarging is the proc o& remo!ing charge &rom a

pre!iously charge capacitor with a su/se9uent ecay in

capacitor !oltage" Consier *ig" F-3F"

'& the switch is initially place in position +, the capacitor

will charge towar the supply !oltage E an a&ter $t can

/e consiere &ully charge" '& the switch is then place in

position 3, the capacitor is irectly across the resistor so that the

charge lea8s through the resistor"

Fi$ure :;/: A char$e;dischar$e circuit

Capacitor ischarging !oltage an current are shown in *ig"( )a,

an /"

(33)


23/59

Time C#nstants 7

!ime constantscan he thought o& as the /uiling /loc8s o&

inustrial an commercial electronics" Very rarely will you

encounter an electrical or electronic circuit that oes not in!ol!e

time constants. Electric motor speed controls( light dimmers(

radio transmitters and receiers( practicall' eer'

circuit in a teleision set( burglar alarms( fire and smo*e

detectors( traffic light sequencers and the latest electronic

automotie ignition s'stems could not function without time

constants.'& you are to unerstan electronic circuits, you must

&irst unerstan the rather simple relationship /etween !oltage

an current in inuctors an capacitors"

'& you reuce the properties o& capacitors an inuctors to

/asic terms, you can say that a capacitor O##O$E$ a change in

!oltage while an inuctor opposes a change in current" A

resistor, o& course, o&&ers the same opposition to !oltage an

(3;)


24/59

current whether they are changing or not" hese three properties,

use singly or in com/ination, are the /asis o& operation &or all

electronic circuits, &rom the simplest to the most comple."

he time constanto& a circuit is the amount o& time it ta8es

&or current in an inucti!e circuit or !oltage in a capaciti!e

circuit to reach appro.imately 2;N o& ma.imum !alue" he

time uration (one time constant) is etermine /y the

resistance, inuctance an capacitance in the circuit" 'n a

resisti!e-inucti!e (RL) circuit, the uration o& One time

constant is calculate using the &ormula ! + -Rwhere !is the

time in secons, L is the inuctance in henrys an R is the

resistance in ohms" 'n the resisti!e-capaciti!e (RC) circuit, the

uration o& one time constant is calculate using the &ormula

RC where is the time in secons, R is the resistance in ohms

an C is the capacitance in &aras"

(34)


25/59

A.C. "%LTA)E )ENERATI%N

'n $egment A, a /attery was use as the !oltage source

ri!ing the current through the resistance" he !oltage isgenerate internally /y a chemical reaction which occurs within

the /attery cells" he energy o& the chemical reaction is then

con!erte to electrical energy" But a chemical reaction is only

one way o& proucing !oltage" $e!eral other techni9ues e.ist &or

the prouction o& !oltage, such as &riction, pressure, heat an

light" he most common metho &or the prouction o&electricity, (an the metho we will /e most intereste in) is

through motion o& a conuctor within a magnetic &iel, an,

e9ui!alently, through motion o& a magnetic &iel in the !icinity

o& a conuctor"

E.periments conucte in the +736s /y G" C" Oerste

showe that electric current &lowing through a wire generates amagnetic &iel aroun the wire" Later e.periments showe that

the opposite e&&ect is also true@ that is, a magnetic &iel can

generate a !oltage in the wire" 'n a close circuit, this will

prouce electric current"

Consier the e.perimental setup shown in *igure l"B"l" he

permanent magnet is shape so that the magnetic &iel isconcentrate /etween the two poles o& the magnet" he =lines o&

&orce> (i"e", the =&lu.>) are shown lea!ing the north pole an

entering the south pole" he two ens o& the coil are connecte

to a !oltmeter, an the coil is allowe to rotate in a

countercloc8wise irection, within the magnetic &iel" 't can /e

shown that a !oltage is prouce as the coil is rotate"*urthermore, i& a resistance is place /etween the two ens o&

(3D)


26/59

the coil, an an ammeter inserte in the circuit, a current will

&low through the resistance" his is 8nown as *araays Law@

an is state /elow0

1hen a conuctor passes through (=cuts>) a magnetic

&iel, a !oltage will /e inuce on the conuctor, an an

electric current will &low through the conuctor i& a

close path &or current conuction e.ists"

Actually, it oesnt matter whether the coil mo!es in the

magnetic &iel, or the &iel mo!es =through> the coil" All that is

re9uire &or !oltage generation, is that the coil must pass

through that is, =cut> the magnetic lines o& &orce"

(32)

C#il R#tatin$ in Ma$netic Filed

Fi$ure 1..1


27/59

As the

coil isrotate, the

!oltage

prouce across the +ea o& the coil will not remain constant"

he resulting wa!e shape is shown in *igure +"B"3" At the

position where the coil is mo!ing parallel to the lines o& &orce,

(as shown /y the position o& the coil in *igure l""B"l), no lines o&

&orce are c ari the !oltage inuce in the coil is ero" 1hen the

coil has rotate K66, it is mo!ing perpenicular to the lines o&

&lu., the ma.imum num/er o& &lu. lines per secon are /eing

cut, an the !oltage inuce in the coil will /e ma.imum"

O/!iously, as the coil continues to rotate an once again mo!es

perpenicular with the lines o& &orce, the !oltage inuce will

also /e a ma.imum /ut, since the coil is now =upsie own>

with respect to the &irst time it was mo!ing perpenicular to the

&lu. lines, the !oltage will /e in the opposite irection" hat is,

the =polarity> o& the !oltage will /e re!erse"

he horiontal a.is o& *igure l"B"3 inicates the num/er o&

egrees o& coil rotation, starting at ero egrees when the co is

mo!ing parallel to the &lu. lines" ;26 egrees correspons to one

(3F)

"#lta$e Induced in R#tatin$ c#il #-

Fi$ure 1../


28/59

&ull rotation (calle a =cycle>) o& the coil" At +76, the !oltage

changes polarity an /egins its =negati!e hal& cycle>"

he !oltage wa!e&orm shown in *ig" l"B"3 represents whatis 8nown as an alternating !oltage" 't prouces an alternating

current (ac) in any circuit connecte to it" Hote that the

wa!e&orm is not particularly well-shape" One o& the principal

as8s o& esigning an electrical generator is to prouce a

com/ination o& magnetic &iel an coil con&igurations that

prouces a smoother !oltage wa!e&orm, as shown in *igure

l"B";" his type o& wa!e &orm is calle a sine wa!e, or a

=sinusoi>"

Sine 'a,e #- ,#lta$e

Fi$ure 1..2

he rotation o& the coil through ;26 can also /e thought

o& in terms o& the passage o& time"

(37)


29/59

'& the coil completes one cycle (i"e", one &ull rotation) in

one secon, we say that !oltage is prouce at a &re9uency (&) o

one cycle per secon" '& the coil completes two cycles in one

secon, the &re9uency is two cycles per secon" he p () is the

time it ta8es to complete one cycle, an is the in!erse o&

&re9uency" hat is0

+&

'& &re9uency is 3 cycles per secon, the perio, is +3

secon" hat is, one &ull cycle is complete in +3 secon"


30/59

6< H= Sine +a,e #- ,#lta$e sh#+in$

an$ular and time scale

Fi$ure 1..3

(;6)


31/59


32/59

A.C. P%'ER IN RESISTI"E CIRCUITS

'n pre!ious iscussion it was shown that a !oltage is

inuce in a rotating coil when place in a stationary magnetic&iel" 'n power system generators, howe!er, the situation is

re!erse@ is the rotation o& a magnetic &iel that inuces a

!oltage on the stationary conuctors" (Remem/er, it oesnt

matter whether the coil rotates an the &iel remains stationary,

or the &iel rotates an the coil remains stationary")

he techni9ue &or proucing !oltage in power system

generators uses a rotating =electromagnet> as a source o&

magnetic lines o& &lu." n electromagnet, unli8e a permanent

magnet, sets up a magnetic &iel temporarily, as current is

passe through wires surrouning the core o& the electromagnet"

'n generators use on utility systems, copper /ars are im/ee

in the =rotor> (i"e", rotating part) o& the machine to &orm anelectromagnet" As current is passe through the copper /ars, a

magnetic &iel is prouce" he rotor an energie coils are

calle the =&iel>" he coils are calle the =&iel wining>"

he rotor coils are energie /y a c power source, calle

the =e.citer>, couple irectly to the rotor" 1hen c current (i"e",

=&iel current>) &lows in the rotor coils, a north an south poleare prouce on the rotor" Voltage is then inuce in the coil on

the stator o& the generator@ this stator completely surrouns the

rotor, an is =stationary>, as the name suggests" he stator is the

part o& a generator that supplies current an !oltage to the power

system"

(;3)


33/59

1hen an alternating !oltage source is connecte to a

resistor, the current that &lows through the resistor also

alternates" he current will /e =in phase> with the !oltage" his

means that the current will /e in its positi!e an negati!e hal&

cycles simultaneously with the !oltage" his is shown in *igure

l"C"l" he sym/olic representation o& a resistor is shown in the

circuit iagram an is assigne the letter R"

By Ohms Law, the pea8 current is gi!en /y pea8 !oltage

i!ie /y the resistance o& the circuit" Actually, the !alue o&

0 t any instant in time is o/taine /y i!iing the !oltage at

hat instant /y R" here&ore, Ohms Law (' VR) applies to ac s

well as c circuits"

(;;)


34/59

AC 6


35/59

' gro!es con!enient to characterie these sinusois /y the

=e&&ecti!e> !alues o& !oltage an current, as oppose to the pea8

!alues" he concept o& =e&&ecti!e> or =rms> !alue pro!ies thischaract he rms !alue o& a !oltage or current wa!e&orm is that

!alue o& !oltage or current that =e&&ects> the same amount o&

wor8 as a c !oltage or current o& the same magnitue" he term

=rms> stans &or =root mean s9uare> an can /e applie to any

perioic wa!e&orm" 't is calculate /y ta8ing the s9uare o& the

instantaneous !alues o& the wa!e&orm, calculating their a!erage

i"e", =mean>), then ta8ing the s9uare root o& this mean@ hence the

name, root mean s9uare *or any sine wa!e, the rms !alue wor8s"

out to /e the pea8 !alue i!ie /y the s9uare root o& 3" his is

the same as multiplying the pea8 !alue /y the &actor0 6"F6F" he

e9uations &or rms !oltage an current are shown /elow0

pea*pea*

rms VV

V F6F"6

3

==

pea*pea*

rms II

I F6F"63==

*or e.ample, suppose the ac !oltage applie in *igure l"C"l

is +66 !olts, pea8" hen the rms !oltage is F6"F !olts" his

means that , &or any gi!en perio o& time, the +66 !olt pea8 sine

wa!e applie to the resistor oes the same amount o& wor8 as a

c !oltage o& F6"F !olts applie to the same resistor"

't shoul /e note that the &actor 6"F6F applies only to

o/taining the rms !alue o& a sinusoial wa!e&orm, an not to

wa!e&orms o& any other shape"

'n the analysis o& power systems, almost all !oltage an

current 9uantities are gi!en as rms !alues rather than pea8

(;D)


36/59

!alues"


37/59

"#lta$e( Current and P#+er in a Resisti,e Circuit

Fi$ure 1.C./

he a!erage, or =e&&ecti!e> power o!er one cycle is gi!en

/y hal& the pea8 !alue o& power" As shown on the !ieotape

power is simply the prouct o& the rms !oltage an rms current"

hat is0

rmsrmsrmsrmspea**pea*

aerage IVIVIV

/ =

=

=

3

33

3

Hote0 )3433( ==

(;F)


38/59

Summar! #- Imp#rtant P#ints

#ower system generators prouce !oltage /y means o& a

rotating magnetic &iel, set up /y passing c current throughwinings on the rotor" Voltage is then inuce /y the rotating

magnetic &iel in the stator winings &or supply to the system"

1hen an ac !oltage ?is applie to a purely resisti!e circuit,

the ac current that results is =in phase> with the !oltage@ that

is, it reaches its ma.imum, minimum, an ero !alues at the

same time as the !oltage wa!e"

An alternating !oltage or current wa!e can /e characterie

/y an rms or e&&ecti!e !alue, which has the same e&&ect as a

c !oltage or current o& the same magnitue" *or sinusois,

the rms !alue is e9ual to the pea8 !alue i!ie /y 3 "

he e&&ecti!e !alue o& power o!er one cycle is Vrms 5 'rms "

his is a+so e9ual to0 S (Vpea85 'pea8)"

All o& the e.pressions eri!e in $egment A &or Ohms Law,

power, an energy are applica/le to ac as well as c systems"

(;7)


39/59

EFFECTS %F INDUCTANCE AND

CAPACITANCE IN AC CIRCUITS

't was shown in $egment C that, in a purely resisti!ecircuit, the !oltage an current wa!e&orms are =in phase>@ that

is, /oth wa!e&orms are their positi!e an negati!e hal&cycles

the same time" Gowe!er, there are other circuit elements

that cause the current to /e isplace or =out o& phase> with the

applie !oltage" his segment iscusses the e&&ects o& these

elements - namely inuctance an capacitance

Inductance

'nuctance is the property o& an electric circuit that

opposes any change in current" 1hen a purely inucti!e circuit

is energie /y an ac !oltage source, the current will reach its

pea8 !alue K66 a&ter the !oltage has reache its pea8, as shown

/y the wa!e&orms o& *igure l""l" he sym/olic representation

o& an inuctor is shown in the circuit iagram ari is assigne

the letter =L> &or inuctance, or TL &or inucti!e reactance" ($ee

page ;7") 'n a circuit consisting o& pure inuctance, the cur rent

is sai to =lag> the !oltage /y K6"

(;K)


40/59

AC "#lta$e and Current in an inducti,e circuit

Fi$ure 1.D.1

(46)


41/59

Another way to e.press this phase relationship is to say

that the !oltage =leas> the current /y K66 in a purely inucti!e

circuit"A con!enient way o& picturing phase relationships /etween

sinusois (i"e", !oltages an currents) is through the concept o&

!ector"A !ector is really ust an arrow, ha!ing two attri/utes0

+) he length o& the !ector is proportional to the

magnitue (rms !alue) o& the 9uantity /eing

represente"

3) he irection o& the !ector inicates the phase angle

relationship /etween the 9uantity /eing represente an

some re&erence 9uantity"

he =!ector> iagram is also shown in *igure l""l, where

the source !oltage is consiere the re&erence 9uantity an is,

there&ore, rawn at an angle o& ero egrees" its length is opc"

ional to the rms !alue o& applie !oltage (i"e" 6"F6F 5 Vpea8)" he

current lags the !oltage /y K66, so is rawn at an angle o& -K6

(or I3F6) with respect to the !oltage" Hote that, & this were a

U Actually, a more correct term is =phasor>, since the phasor repre sents a time

!arying 9uantity, an the !ector oes not" Gowe!er, in the ac steay state, where

we assume all sinusois ha!e the same &re9uency, we can uni9uely represent a

phasor 9uantity with an arrow oriente in a twoimensional space (i"e", a

!ector) " 'n &act, the term !ector was coimr use to represent ac sinusoial

!oltages an currents until the term =phasor> was coine to a!oi am/iguity with

9uantities in other &iels o& stuy such as raiation an antenna theory" Another

term sometimes use is =sinor>" hrough cut this program we will use /oth the

terms =!ector> an =phasor>"

(4+)


42/59

resisti!e circuit, the current woul /e =in phase> with the

!oltage, so the current !ector woul /e rawn =on top o&> the

!oltage !ector, at an angle o& 6"

he length o& the current !ector is proportional to the rms

!alue o& current" his is o/taine /y i!iing rms !oltage /y the

inucti!e reactance TL, i"e"

'rms VrmsTL

his e9uation is simply another e.pression o& Ohms Law,

/ut &or a circuit consisting o& pure inuctance" 'n a resisti!e

circuit, the =TL> woul /e replace /y the sym/ol =R>, were R

is the resistance o& the circuit"

Qust as R represents the opposition to current &low in a

resisti!e circuit, TLrepresents the opposition to current &low in

an inucti!e circuit"

he term =inucti!e reactance> warrants some more

e.planation" 'nucti!e reactance epens on the inuctance o&

the circuit (i"e", a physical property o& materials) an the

&re9uency o& the applie !oltage0 he e9uation &or inucti!e

reactance, TL, is shown /elow0

TL 3&L (TLgi!en in ohms -)

where0

;"+4+DK (is actually a repeating, ecimal,

/ut this is close enough &or our purposes"

& &re9uency o& the applie !oltage, measure

in Gert, an

L the inuctance o& the circuit"

(43)


43/59

'nuctance is measure in henries, a//re!iate as =G>"

he 9uantity (3&) is calle the angular &re9uency an is

sym/olie /y the term ui (small omega) " his term shoul not/e con&use with =capital omega> (), use to enote =ohms>"

't can /e seen that, &or a c !oltage source, the &re9uency is

ero, hence" the inucti!e reactance, TL, is ero" his means that

inuctor acts as a short circuit (ero inucti!e reactance) &or c

!oltage application" As the source &re9uency increases, the

inucti!e reactance will increase in proportion to he &re9uency"

1e can summarie our iscussion on inuctance thus &ar

as &ollows0

+) 'n a purely inucti!e circuit, the current =lags> the

!oltage /y K6 egrees"

3) he relationship /etween !oltages an current in an

inucti!e circuit is gi!en /y Ohms Law, ' VTL,

where TL is terme the inucti!e reactance an is

measure in ohms" TLincreases with the &re9uency o&

the applie !oltage" TLcan /e consiere constant i& the

&re9uency o& the applie !oltage is constant"

*igure l""3 shows the power in an inucti!e circuit" he

power cur!e was o/taine in the same manner as &or a resisti!e

circuit@ i"e", /y multiplying !oltage an current at each instant o&

time" 1hene!er either the !oltage or current is ero, the power

eli!ere to the inuctor is ero" 1hene!er the !olt age ari

current wa!e&orms are /oth positi!e, their prouct will also /e

positi!e" he same is true i& /oth the !oltage an cur rent are

negati!e" uring these perios o& positi!e power, the inuctor is

(4;)


44/59

a/sor/ing energy &rom the connecte !oltage source" Gowe!er,

when either the !oltage or current is negati!e an the other is

positi!e, the power a/sor/e is negati!e, 'nicating that the

inuctor is supplying power to the !oltage source" Recall that

this is not the case with a purely resisti!e circuit, where power is

always positi!e"

"#lta$e( Current( and P#+er in An Inducti,e Circuit

Fi$ure 1.D/

(44)


45/59

*urthermore, *igure L""3 shows that the a!erage power

consume o!er one cycle is ero, since as much power is

a/sor/e uring one 9uarter cycle as is prouce uring the ne.t9uarter cycle" 'n the case o& an inuctor, the magnetic &iel set

up aroun the inuctor acts as an energy storage e!ice, with the

sa amount o& energy release uring one 9uarter cycle as was

store uring the pre!ious 9uarter cycle"

he power in a purely inucti!e circuit there&ore oscillates

/etween the generator an the magnetic &iel e!ery 9uartercycle, meaning that no acti!e power is consume /y the

inuctor" $ince the prouct o& !oltage an current in an inuctor

cannot /e terme acti!e power, we gi!e it the name reacti!e

power, or !oltamperes reacti!e" his is a//re!iate as !ar" Vars

in a purely inucti!e circuit are o/taine /y multiplying rtns

!oltage times rms current"

'nucti!e !ars are rawn at an angle o& IK66 with respect

to acti!e power (watts)" By con!ention, watts are assigne phase

angle o& ero egrees, an are consiere the re&erence !ector on

a !ector iagram in!ol!ing power" (Hote that there are no watts

prouce in the circuit o& *igure +"6"3" hey are rawn on the

!ector iagram only &or re&erence)"

'nucti!e !ars are consiere positi!e !ars, since the

generator must supply them" hey are gi!en a positi!e sign an

rawn as an arrow oriente in the IK6 egree position"

o summarie the main points concerning power in a

purely inucti!e circuit0

(4D)


46/59

+) he a!erage power consume in an inuctor o!er one

cycle is ero"

3) he power oscillates /etween the inuctor an thesource, an is gi!en the name reacti!e power, or

!oltamperes reacti!e, to istinguish it &rom watts, or

acti!e power" Reacti!e power is a//re!iate !ars"

;) Reacti!e power can /e illustrate /y means o& a !ector

iagram" '& the acti!e power !ector is at an angle o& ero

egrees, reacti!e power in an inucti!e circuit can /erawn at an angle o& IK6 egrees" Vars in an inucti!e

circuit are gi!en a positi!e sign"

(42)


47/59

Capacitance

*igure +""; shows a purely capaciti!e circuit" he

capacitor is shown sym/olically in the circuit iagram, an isassigne the letter C (&or capacitance) , or TC (&or capaciti!e

reactance)"

Capacitance is the property o& an electric circuit that

opposes any change in !oltage here&ore, i& an ac !oltage is

applie to a purely capaciti!e circuit, the resulting current will

reach its pea8 !alue K66 ahea o& the !oltage, as shown /y the

wa!e&orms o& *igure l"";" he current =leas> the !oltage /y

K6, as inicate in the !ector iagram"

he amount o& current &lowing in a capaciti!e circuit is

once again gi!en /y Ohms Law0

'VTC

where TC is terme the capaciti!e reactance Capaciti!e

reactance represents the opposition to current &low in a

capaciti!e circuit" Capacitance, li8e resistance an inuctance, is

a physical property o& materials" he capaciti!e reactance TC, is

a &unction o& the capacitance o& the circuit element, an the

&re9uency o& the applie !oltage" 't is gi!en /y the &ollowing

e9uation0

Tl(3&C) (TCe.presse in )

where0

;"+4+DK

(4F)


48/59

"#lta$e( Current( and P#+er In A Capaciti,e Circuit

Fi$ure 1.D.3

(47)


49/59

$ince no acti!e power (watts) is consume, the prouct o&

r !oltage an rms current is calle again, !oltamperes reacti!e,

or !ars" Gowe!er, the !ars prouce /y a capaciti!e circuit arealways gi!en a negati!e sign, inicating that capacitors are

consiere a source o& !ars"

'llustrate on the !ector iagram, o& *igure l""4,

capaciti!e !ars are always rawn at an angle o& minus K6

egrees with respect to the acti!e power" Again, acti!e power is

shown here only &or re&erence"

'& /oth capacitance an inuctance e.ist in a circuit, some

or all o& the !ars re9uire /y the inuctor may actually /e

supplie /y the capacitor (i"e", the capaciti!e !ars =compensate>

&or the inucti!e !ars re9uire) his will /e illustrate in the

ne.t segment"

Capaciti,e circuits can >e summari=ed as -#ll#+s7

+) 'n a purely capaciti!e circuit, the current &lowing through the

capacitor always =leas> the applie !oltage /y K6 egrees"

3) he relationship /etween current an !oltage in a capaciti!e

circuit is gi!en /y Ohms Law, ' VTC, where TCis terme

the capaciti!e reactance, an is measure in ohms" TC is

in!ersely proportional to the &re9uency o& the applie !oltage"

;) Ho acti!e power is consume /y capaciti!e circuits" Li8e

inucti!e circuits, the prouct o& !oltage an current is

terme !ars /ut is gi!en a negati!e sign (opposite to inucti!e

!ars), inicating that capacitors act as =sources> o& !ars" (By

the same to8en, inuctors act as =sin8s> o& !ars)"

(4K)


50/59

P%'ER FACT%R

'n pre!ious iscussion the e&&ects o& resistance, inuctance

an capacitance were iscusse ini!iually" But a p electriccircuit e.hi/its mc than one o& these properties an perhaps all

three at the same time, with one /eing preominant" his

segment presents an e.ample o& a circuit containing all three

properties - resistance, inuctance an capacitance - an

emonstrates how their e&&ects are com/ine" 'n so oing, the

concept o& =power &actor> is introuce"

*igure l"E"l shows a !oltage source connecte to a pure

resistance" he !oltage source has a &re9uency o& 26 G, an

!oltage o& +66 !olts rms" he resistance has a !alue o& 3"D ohms"

he current through the resistor is gi!en /y Ohms Law,

' VR, an is e9ual to 46 amperes rms" his current, calle 'R,

is in phase with the !oltage, as shown on the !ector iagram"

he source !oltage is ta8en to /e the re&erence !ector"

(D6)


51/59

"#lta$e Current and P#+er;Resistance %nl!

Fi$ure 1.E.1

(D+)


52/59

he power a/sor/e /y the resistor is0 +66 !olts 5 46

amperes 4666 watts" his is also shown in *igure +"E"l, rawn

at an angle o& ero egrees"$uppose a 3"D ohm inuctor is connecte to the circuit, as

shown in *igure +"E"3" Hotice that the !oltage across the

inuctor is the same as that across the resistor-namely, +66 ring"

he resistor an inuctor are sai to /e connecte =in parallel>"

By Ohms Law, the current &lowing through the incuctor,

'Lis +66 !olts3"D ohms 46 amperes" Gowe!er, the current in

an inuctor lags the source !oltage /y K6 egrees" he acti!e

power issipate in the inuctor is ero, /ut the !ars in the

inucti!e circuit are0

!olts 5 46 amperes 4666 !ars

*igure l"E"3 shows the graphical representation o& the

source !oltage, the current in the resistor an the inuctor, an

acti!e an reacti!e power &or this com/ine circuit"

As a &inal step, a capacitor with a capaciti!e reactance o&

+6 is connecte across the source !oltage in para++e with the

resistor an inuctor, as shown in *igure +"E";" Once again, the

&ull source !oltage o& +66 !olts is applie across the capacitor"

he current &lowing through the capacitor, 'C, is gi!en /y +66

!olts+6 +6 amperes, at an angle o& IK6 with respect to the

source !oltage" Li8e an inuctor, no acti!e power is consume

/y a capacitor" Rather, the reacti!e power (!ars) are calculate

/y0 +66 !olts 5 +6 amperes +666 !ars" he three currents, 'R, 'L,

an 'Care shown graphically in *igure l"E";, as are acti!e power

(watts) , inucti!e !ars, an capaciti!e !ars"

(D3)


53/59

"#ta$e( Current and P#+er

Resistance and Inductance in Parallel

Fi$ure 1.E./

(D;)


54/59

"#lta$e( Current and P#+er

Resistance( Inductance( and Capacitance in Parallel

Fi$ure 1.E.2

(D4)


55/59

$ince capaciti!e !ars are negati!e, they are supplie to the

circuit" his reuces the !ars re9uire /y the inuctor@ that is,

the 4666 !ars neee /y the inuctor can /e partially met /y the+666 !ars generate /y the capacitor" he capacitor is sai to

=compensate> &or the !ar re9uirement o& the inuctor" he total

!ar re9uirement &or the circuit is thus 4666 !ars - +666 !ars

;666 !ars" his is shown graphically in *igure +"E"4(a), where

the reacti!e power !ectors &or inuctance an capacitance

(*igure l"E";) are simply ae !ectorially to gi!e net inucti!e

!ars e9ual to ;666 !ars"

Gowe!er, recall that only !ectors in the same irection or

in the opposite irection (+76 apart) can /e ae) or

su/tracte) irectly" o a !ectors o& watts an !ars (which are

K6apart) we must resort to a /asic theorem o& geometry (the

#ythagorean heorem), which states that the s9uare o& the

hypotenuse o& a right triangle is e9ual to the sum o& the s9uares

he two remaining sies o& the triangle"

'n *igure +"E"4(/), the reacti!e power !ector has /een

rerawn /y mo!ing it &rom the tail o& the acti!e power !ector to

the tip o& the acti!e power !ector" (Recall0 wo !ectors, K6

apart, can /e ae graphically /y rawing them tip to tail, then

completing the triangle)" he sum o& the two power !ectors is

then, the hypotenuse o& the right triangle, la/ele on the

iagram, =!oltamperes>, an a//re!iate =VA>"

(DD)


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to the hypotenuse (VA) e&ines the cosine o& the angle , hence,

power &actor can also /e etermine /y0

R/12ER/0C!1V0

20!!3

45/1!E673E

0890CE6!

Cos ===

'n this e.ample, the power &actor is calculate to /e0

#"*" 4666 wattsD666 !oltamperes 6"7" $ince the total

current, ', lags the source !oltage (i"e", this is an inucti!e

circuit)@ the circuit o& *igure l"E"4 has what is terme a 6"7

lagging power &actor" '& the !alues o& capacitance an

inuctance were such that negati!e !ars ominate (i"e",

capaciti!e circuit), the total current woul lea the !oltage, an

the circuit woul operate with what is terme a leaing power

&actor" he angle can /e calculate /y &ining that angle who

cosine is 6"7" *rom trigonometric ta/les or a calculator, this

turns out to /e ;F egrees"

A power &actor o& 6"7 means that 76"percent o& the

apparent power is acti!e power (watts) " 'n a circuit whose

power &actor is +"6 (unity power &actor) , +66 percent o& the

apparent power is acti!e power" his situation is characteristic

o& purely resisti!e circuits, an also o& circuits where the !ars

a/sor/e /y inuctors is e.actly matche /y the !ars generate

/y capacitors" A circuit ha!ing a power &actor o& 6"6 has no

resistance elements to a/sor/ acti!e power@ thus none o& the

apparent power is acti!e power the circuit consists only o&

inucti!e anor capaciti!e elements"

A power &actor can /e speci&ie at e!ery point in a power

system@ i"e", we can consier the power &actor at a generator, the

power &actor at the point o& loa, or anywhere else in the system"

(D7)


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'n practice, it is esira/le to 8eep the power &actor as to unity as

possi/le, since the acti!e power (watts) eli!ere to a loa

actually oes the wor8 in operating electrical e9uipment"

Vars, howe!er, are a/solutely essential to the operation o&

the power system" Vars prouce the ac magnetic &iels in

inucti!e components o& the system, such as trans&ormers,

transmission lines, inuction motors, an so on" hese e!ices

re9uire /oth acti!e power (to o the use&ul wor8) , an reacti!e

power (to set up the magnetic &iels which allow use&ul wor8 to

/e one)

Reacti!e power re9uire /y inucti!e loas must /e

supplie /y the power system generators, unless enough !ar

compensation is present (capacitors, synchronous conensers,

static compensators)" n&ortunately, this has the e&&ect o&

limiting the amount o& megawatts that can /e supplie /y the

generator" A simple e.ample will illustrate this point"

Electrical Fundmentals

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