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Course 2222 Electrical Circuits Theory I

1. GENERAL CONCEPTS

1.1 System of Units

The system of units employed is the international system whih is normally referred to as the S!

standard system. This system" whih is omposed of the #asi units

meter $m% kilogram $kg% second $s% mole $mol% ampere $A% degree kelvin $K% candela $cd%"

is defined in all modern physis te&t#oo's and therefore will not #e defined here. (owe)er" we will

disuss the units as we enounter them.

The standard prefi&es that are employed in S! are shown in the ne&t fi*ure. Note the deimal

relationship #etween these prefi&es. These standard prefi&es are employed throu*hout our study of

eletri iruits.

1+,1-

1+,1

1+,1/

1+,0

1+,

1+,2

1 1+2 1+

1+

0 1+

1/ 1+

1 1+

1-

a f p n m k M G T P E

atto femto pico nano micro mili kilo mega giga tera peta exa

1.2 Basic Quantities

3efore the #e*innin* of iruit analysis" the terms in)ol)ed must #e defined.

An electric circuitis an interonnetion of eletrial omponents" eah of these #ein* desri#ed throu*h

a mathematial model. The elementary 4uantity in analysis of eletri iruits is the eletri har*e.

Sine har*e in motion results in an ener*y transfer" we are partiularly interested of those situations in

whih the motion is onfined to a preise losed path. An eletri iruit is li'e a pipeline that failitates

the transfer of har*e from one point to another. The time rate of han*e in har*e means an electric

current" e&pressed as

5A6d

d

t

qi= " and measured in amperes" shortly amps" where s1

C1A1 = .


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Course 2222 Electrical Circuits Theory I2222

So" urrent flow in a metalli ondutor results from eletron motion" and the on)entional urrent flow

represents the mo)ement of positi)e har*es $no matter what really happens%. 7or e&ample" in the

fi*ure shown #elow !8,2 A indiates that at any point in the wire 2C of har*e pass from left to ri*ht eah

seond. Therefore" not only the ma*nitude of the urrent it is important" #ut also its diretion.

9ost often enountered types of urrent are alternating current $ac% and direct current $dc%" li'e in

household onsumers runnin*" respeti)ely in #atteries. !n addition to these ommon types" there are a

lot of other types" some of these #ein* e&amined later.

Let:s re)iew; har*e in motion means urrent" #ut also the motion of har*es yields an ener*y transfer.

Therefore" further is defined the voltage#etween two points in a iruit $usually alled potential% as the

differene in ener*y le)el of a unit of har*e loated at eah of the two points.


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Energyis another important 4uantity in iruits. Let:s ta'e the flashli*ht iruit. The #asi elements of a

flashli*ht are a #attery" a swith" a #ul# and onnetin* wires. !f the swith is losed" a urrent will flow

in the iruit" whih means that har*es flow out of the positi)e terminal of the #attery throu*h the

swith and li*ht #ul# and #a' into the ne*ati)e terminal. Char*es passin* throu*h the #ul# lose ener*y#eause the #ul# needs ener*y so as to li*ht. The same har*es a4uire ener*y as they pass throu*h the

#attery. An ener*y on)ersion proess is ourrin* in the flashli*ht; the hemial ener*y in the #attery is

on)erted to eletrial ener*y" whih is then on)erted to thermal ener*y in the #ul#.

Sine it is 'nown that the #ul# uses ener*y" the har*es omin* out of the #ul# ha)e less ener*y than

those enterin* in the #ul#. !n other words" if the har*es e&pend ener*y as they mo)e throu*h the #ul#"

the #ul# enterin* point has a hi*her potential than the output one" therefore" the )olta*e aross the

#ul# has a positi)e )alue in this diretion" from the enterin* point to e&it point. The har*es *ain ener*y

as they pass throu*h the #attery" this meanin* that the har*es enterin* point is at a lower potential to

the har*es e&it point. Thus" the positi)e )olta*e aross the #attery has the sense from the e&it point to

enterin* one. Now we may point out the followin*;

#ul# ? a#sor#s ener*y

? urrent and )olta*e ha)e the same diretion

#attery ? supply ener*y

? urrent and )olta*e ha)e opposite diretions

This leads us to the onlusion that a onsumer has positi)e )alues for urrent and )olta*e si*nals in the

same sense" and an ener*y supplier has positi)e )alues for urrent and )olta*e si*nals in opposite

diretions.

The 4uantity that desri#es the ener*y e&han*e in iruit is the power. Power is defined as the time

rate of han*e of ener*y and is measured in @oules per seond or watts. Also" power it is o#tained from

the produt #etween urrent and )olta*e.

56d

d

d

d

d

d

t

w

t

q

q

wtitvtp === )()()( where A1


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!t is also important to note that the eletrial iruits studied here satisfy the principle of conservation

of energy. 7or our purposes this means that in a *i)en iruit the power supplied is e&atly the same

with the power a#sor#edDonsumed.

1.3 Circuit Elements

!n *eneral" the elements defined here are terminal de)ies that are ompletely harateriBed #y the

relation #etween the urrent throu*h the element and the )olta*e aross it. These elements will #e

employed in onstrutin* eletri iruits. The elements are #roadly lassified as #ein* either ati)e or

passi)e. The distintion depends on one thin*" whether they supply or a#sor# ener*y; an activeelement

is apa#le of *eneratin* ener*y and a passiveelement annot *enerate ener*y. e will see later that

some passi)e elements are apa#le of storin* ener*y. Typial ati)e elements are batteries and

generators" usually denoted as sources. The three ommon passi)e elements are resistors" capacitors

and inductors. 7urther" our onern is to esta#lish mathematial models of these elements.

1.3.1 Active ElementsAs mentioned a#o)e" ati)e elements are soures. Generally spea'in* soures pro)ide eletrial ener*y"

#ut they do that in different manners. Soures pro)ide a 'ind of si*nal $urrent or )olta*e% either

independently or dependin* on other si*nal. So" we will study here two types of soures; independent

sourcesand dependent sources.

a) Independent sources

An independent soure is a two,terminal element that pro)ides a speified si*nal" urrent or )olta*e.

An independent voltage source maintains a speified )olta*e #etween its terminals re*ardless of the

urrent throu*h it. !n the ontrast" the independent current source is a two,terminal element that

maintains a speified urrent re*ardless of the )olta*e aross its terminals. Their models are

represented #elow.

element: voltage source current source

symbol:

operating equation: i(t)tetv %$" = )()( (t)titi g )%$" = )()(

e(t)

i(t)

v(t)+

ig(t)

i(t)

v(t)


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Those ones are ideal soures #eause pro)ide no ener*y in speial onditions $when )olta*e soure is

loaded #y an open iruit" and when urrent soure is loaded #y a short iruit% and also approah

infinite ener*y in opposite onditions $shorted )olta*e soure and opened urrent soure%. Of ourse"

these thin*s are physially impossi#le.

!n their normal mode of operation" independent soures supply power to the other iruit elements.

(owe)er" they may also #e onneted into a iruit in suh a way that they a#sor# power.

) !ependent sources

!n ontrast to the independent soures" whih produe a partiular )olta*e or urrent ompletely

unaffeted #y what is happenin* in the reminder of the iruit" dependent soures *enerate a )olta*e or

urrent that is determined #y a )olta*e or urrent at a speified loation in the iruit. The #eha)ior of

more other eletroni iruit elements is desri#ed throu*h the models of dependent soures.

The fi*ure #elow illustrates the four types of dependent soures. 7or eah sym#ol" the input terminals

on the left represent the si*nal $)olta*e or urrent% that ontrols the soure" and the output terminals

on the ri*ht represent the si*nal pro)ided #y the soure.

element: voltage controlled voltage source current controlled voltage source

symbol:

operating equation: )())(()( tvtvetv kkk == )())(()( tiRtietv kkk ==

element: voltage controlled current source current controlled current source

symbol:

operating equation: )())(()( tvGtviti kkkg == )())(()( titiiti kkkg ==

vk v(t)e(t)+

v(t)e(t)ik+

vk

i(t)

ig(t) ik

i(t)

ig(t)


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1.3.2 "assive ElementsA passi)e omponent may #e either a omponent that onsumes ener*y $#ut does not produe%" or a

omponent that is inapa#le of power *ain $they annot amplify si*nals%. Ciruit desi*ners will

sometimes refer to this lass of omponents as dissipati)e. At this point" we will onsider a passi)e

element as one that onsumes ener*y" #ut does not produe ener*y permanently $>>%. nder this

definition" passi)e omponents studied further on are; the resistor" the capacitor" the inductor.

a) #esistor

A resistor is a physial de)ie whose sym#ol and operatin* law are depited in ne&t fi*ure. The resistor

parameter is alled resistane and is measured in ohms 5F6.

element: resistor

parameter: resistane R 5F6

symbol:

operating equation:)()()( tvGi(t)tiRtv

== or

The power supplied to the terminals of a resistor is always dissipated" in other words is always positi)e;

0)()()()()( 22 === tvGtiRtitvtp .

Resistors are normally ar#on omposition or wirewound" and also an #e fa#riated usin* thi' o&ide or

thin metal films" or they an #e diffused in semiondutor inte*ratin* iruits. 7urther" we assume that

resistor has a onstant parameter" and therefore its )olta*e,urrent e4uation is linear.

An interestin* definition for eletrial resistanceof an element states that this is the opposition to thepassa*e of an eletri urrent throu*h that element the in)erse 4uantity is eletrial conductance$G%"

whih means the ease at whih an eletri urrent passes throu*h the element.

A short circuitis a kind of resistor with next properties:

=

=

G

R

tv

0

0)(

.

In contrast, an open circuitis the oposite type, with next properties:

=

=

0

0)(

G

R

ti

.

v(t)

i(t)

R


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) Capacitor

A apaitor is a iruit element that onsists of two ondutin* surfaes separated #y a non,ondutin*

or dieletri material. The apaitor parameter is alled apaitane and is measured in farads 576. !ts

sym#ol and operatin* law are depited #elow.

element: capacitor

parameter: apaity C 576

symbol:

operating equation: == t

ttiCtvt

tvCi(t) dord

d

)(1

)(

)(

Capaitors are apa#le of storin* ener*y when a )olta*e is present aross the element. This ener*y is

stored in an eletri field. The power deli)ered to the apaitor is

t

tvtvCtitvtp

d

d )()()()()( == "

and hene the ener*y stored in its eletri field is

)(2

1)()()()( 2 tCvt

t

tvtvCttptW

tt=== dd

dd .

c) Inductor

An indutor is a iruit element that onsists of a ondutin* wire in the form of a oil. The indutor

parameter is alled indutane and is measured in henry 5(6. !ts sym#ol and operatin* law are;

element: inductor

parameter: indutane 5(6

symbol:

operating equation: == t

ttvL

tit

tiLtv dor

d

d)(

1)(

)()(

!ndutors are apa#le of storin* ener*y when a urrent is passin* throu*h the element" ausin* a

ma*neti field to form. The power deli)ered to indutor and the ener*y stored in its ma*neti field are;

t

titiLtitvtp

d

d )()()()()( == " )(

2

1)()()()( 2 tLit

t

titiLttptW

tt===

d

d

dd .

v(t)

i(t)

C

+q

-q

v(t)

i(t)

L


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1.$ %erms in Circuits %opolo&y

An eletri iruit is a onnetion of )arious iruit elements" onnetin* wires in a iruit #ein* assumed

to ha)e Bero resistane unless otherwise stated. Sine the elements of an eletri iruit an #einteronneted in se)eral ways" we need to understand some #asi onepts of networ' topolo*y. To

desri#e how it is omposed a iruit" we use the terms of node" !ranch" and loop.

"ode; is the point of connection between at least three circuit elements.se of the word point indefinition an #e misleadin*. A node is usually indiated #y a dot in a iruit. !f a short

iruit $a onnetin* wire% onnets two nodes" the two nodes ma'e a sin*le node.

node not a node a sin*le node

#ranch; is the path between two nodes; must contain at least one element, but no other node. !tfollows that #y a #ranh passes a sin*le urrent. 7or a networ' #ranh" the eletrial si*nals

$the urrent throu*h the #ranh and the )olta*e aross it% are always assoiated in the

same diretion.

#ranh not a #ranh two #ranhes

oop; is any closed path in a circuit.A loop is a losed path formed #y startin* at a node" passin*throu*h a set of nodes" and returnin* to the startin* node without passin* throu*h any

node more than one. A loop is said to #e independent if it ontains a #ranh whih is not in

any other loop. !ndependent loops result in independent sets of e4uations.

loop not a loop

A networ' with ##ranhes" "nodes" and independent loops will satisfy the fundamental theorem of

networ' topolo*y;

+ +

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