NETWORK ANALYSIS NOTES.doc

8/14/2019 NETWORK ANALYSIS NOTES.doc

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Network Analysis 10ES34

1

PART A

UNIT 1:

Basic Concepts: Practical sources, Source transformations, Network reduction using Star Delta

transformation, Loop and node analysis With linearly dependent and independent sources for DC

and C networks, Concepts of super node and super mesh

UNIT 2:

7 Hours

Network Topolog: !raph of a network, Concept of tree and co"tree, incidence matri#, tie"set,

tie"set and cut"set schedules, $ormulation of e%uili&rium e%uations in matri# form, Solution of

resisti'e networks, Principle of

UNIT !:

duality"

7 Hours

Network T#eore$s 1: Superposition, (eciprocity and )illman*s theorems

% Hours

UNIT &:

Network T#eore$s ' II:

+he'inin*s and Norton*s theorems )a#imum Power transfer theorem

% Hours

PART B

UNIT (: Resonant Circuits: Series and parallel resonance, fre%uency"response of series and

Parallel circuits, - factor, .andwidth/

%Hours

UNIT %:

Transient )e#a*ior an+ initial con+itions: .eha'ior of circuit elements under switching

condition and their (epresentation, e'aluation of initial and final conditions in (L, (C and (LCcircuits for C and DC e#citations/

UNIT 7:

,aplace Trans-or$ation . Applications : Solution of networks, step, ramp and

7 Hours

impulse

responses, wa'eform Synthesis


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7 Hours

UNIT /:

Two port network para$eters: Definition of 0, y, h and transmission parameters, modelingwith these parameters, relationship &etween parameters sets

T0T B34:

% Hours

1/ 5Network Analsis6 )/ 2/

3443/5an 5alken&urg, P67 8 Pearson 2ducation, 9

rd2dition/ (eprint

3/ 5Networks an+ sste$s6 (oy Choudhury, 3nd

edition, 344 re"print, New ge 7nternational

Pu&lications/

R080R0NC0 B34:

1/ 50ngineering Circuit Analsis6 6ayt, ;emmerly and Dur&in+)6 th

2dition, 3443

3/ 5Network analsis an+ 4nt#esis6

7nternational 2dition,$ranklin $/ ;uo, Wiley

9/ 5Analsis o- ,inear 4ste$s6, Da'id ;/ Cheng, Narosa Pu&lishing 6ouse, 11th

reprint, 3443


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I N DEX S HEET

SL.NO TOPIC PAGE NO.1 University syllabus 2-3

UNIT 1: Basic Cnce!ts 5-1401 Assignment !estions 15-1"

UNIT " #: Net$r% T!l&y 1#-3"01 Assignment !estions 3#-3$

UNIT " ': Net$r% T(ere)s 1 40-4"01 Assignment !estions 4#-4$

UNIT " *: Net$r% T(ere)s " II 50-5#01 Assignment !estions 5$-%0

UNIT " +: ,esnant Circuits %1-%$01 Assignment !estions "0-"1

UNIT " -: Transient be(avir an initial cnitins "2-#301 Assignment !estions #4-#5

UNIT /: La!lace Trans0r)atin A!!licatins #%-$501 Assignment !estions $%-$"

UNIT 2: T$ !rt net$r% !ara)eters $#-11001 Assignment !estions 111-112


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Unit : 1 :Basic

Cnce!ts Hr s: 07

Syllabus of unit 1:

&ra'ti'al so!r'es( So!r'e trans)ormations( Network re*!'tion !sing Star + Deltatrans)ormation( ,oo an* no*e analysis .it/ linearly *een*ent an* in*een*ent so!r'es)or D an* A networks on'ets o) s!er no*e an* s!er mes/

R ec omme nde d r e a ding s:

1 3Net$r% Analysis45 E an alken!rg( &HI &earson E*!'ation

#. 3Net$r%s an syste)s45 6oy /o!*/!ry( 2 e*ition( New Age International&!li'ations

3. Net$r% t(ery , 7anes/ 6ao.

4. 3Net$r% analysis , 6oy /o!*ry.


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1

B A4I C , A4:

1 8HS ,A. 9I: A IA;:i:?

< I 1 2 n

- I1 I2 In

n

Y YK Y1 Y2 Y3 Yn1

!rrent Di@isionII9>i?I


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I9:91:192:29-------------- 9I9I119I229-------------

P r o)le$s1al'!late t/e @oltages 12(23(34an*915-100

in t/e network s/own in Big( i) a91"32


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Practical surces:

Network is a system wit/ inter'onne'te* ele'tri'al elements Network an* 'ir'!it are t/e sameT/e only *i))eren'e eing a 'ir'!it s/all 'ontain at least one 'lose* at/

Ele'tri'al Elements

So!r'es &assi@e Elements

In*een*ent Deen*ant 6 , So!r'es So!r'es >Energy >Energy storing >Energy

N ons!ming element in a storingElement? magneti' )iel*? element in an

Ele'tri' )iel*?

oltage So!r'e>i*eal?

!rrent So!r'e>i*eal?

ki g@ ? >'? >*?

>a?!rrent 'ontrolle* '!rrent so!r'e>? oltage 'ontrolle* '!rrent so!r'e>'? oltage 'ontrolle* @oltage so!r'e

>*? !rrent 'ontrolle* @oltage so!r'e

>al!e o) so!r'eN

>So!r'e !antity is *etermine* y a @oltage!antity is not a))e'te* or '!rrent eisting at somein anyway y a'ti@itiesin t/e remin*er o) t/e'ir'!it?

ot/er ,o'ation in t/e 'ir'!it?T/ese aear in t/e e!i@alent mo*els )or manyele'troni' *e@i'es like transistors( 8&A&S an*integrate* 'ir'!its

&ra'ti'eoltageso!r'e

es/ >loo? ,oo &ra'ti'al '!rrentso!r'e

6e)eren'eno*e


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T6PES O7 NET8O,9S

L in e a r a n N n li n e a r N e t $ r % s:

A network is linear i) t/e rin'ile o) s!erosition /ol*s ie i) e1>t?( r1 >t? an* e2>t?(r2 >t? are e'itation< r2>t?

an* resonse airs t/en i) e'itation is e1 >t? < e2 >t? t/en t/e resonse is r1 >t?

T/e network not satis)ying t/is 'on*ition is nonlinearE- ,inear + 6esistors( In*!'tors( aa'itors

Nonlinear + Semi'on*!'tors *e@i'es like transistors( sat!rate* iron 'ore in*!'tor('aa'itan'e o) a -n )!n'tion

Pass iv e a n a c t iv e N e t $ r % s:

A ,inear network is assi@e i) >i? t/e energy *eli@ere* to t/e network is nonnegati@e

)or any e'itation >ii?e'itation is alie* no @oltages an* '!rrents aear etween any two terminals e)ore anyEamle- 6(, an*

A'ti@e network- Networks 'ontaining *e@i'es /a@ing internal energy +7enerators(amli)iers an* os'illators

U n il a t er a l B il a t e r a l:

T/e 'ir'!it( in w/i'/ @oltage '!rrent relations/i remains !naltere* wit/ t/e re@ersalo) olarities o) t/e so!r'e( is sai* to eilateral

E- 6( , J I) -I relations/is are *i))erent wit/ t/e re@ersal o) olarities o) t/e so!r'e( t/e

'ir'!it is sai* to e !nilateral E- semi'on*!'tor *io*es


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elements.

Lu m p e d & D is t r ib u t e d :Elements of a circuit, which are separated physically, are known aslumped

Ex:- ! ".

elements.

Elements, which are not separa#le for analytical purposes, areknown as distri#uted

Ex:- transmission lines ha$in% &, , " all alon% their len%th.

'n the former care (irchhoff)s laws hold %ood #ut in the latter case *axwell)s lawsare re+uired for ri%orous solution.

R ec ip r o c a l:

network is said to #e reciprocal if when the locations of excitation and responseare interchan%ed, the relationship #etween them remains the same.

4our ce Trans-or $ati on:

'n network analysis it may #e re+uired to transform a practical $olta%e source intoits e+ui$alentpractical current source and $ice$ersa .

hese are done as explained #elow

/ a aE/ '/ 0

fi% 1

# # fi% 2

"onsider a $olta%e source and a current source as shown ini%ure 1 and

2. or the same

load across the terminals a ! # in #oth the circuits,the currents

are

'4 E/ in fi% 1 and ' 4 '/ . 0 in fi% 2

: s


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Trans)ormation )rom a ra'ti'al '!rrent so!r'e to a '!rrent so!r'e eliminates a mes/A ra'ti'al '!rrent so!r'e is in arallel wit/ an ime*an'e : is e!i@alent to a @oltage so!r'e

Es9Is : in series wit/ :

A ra'ti'al @oltage so!r'e Es in series wit/ a ime*an'e :sEs:s in arallel wit/ :s

is e!i@alent to a '!rrent so!r'e


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10

:1

:4

:3

:1

:4

:3

4URC0 4HI 8TI N=:

So!r'e s/i)ting is o''asionally !se* to simli)y a network T/is sit!ation arises e'a!se o) t/e )a'tt/an an i*eal @oltage so!r'e 'annot e rela'e* y a '!rrent so!r'e ,ike wise i*eal '!rrent so!r'e'annot e rela'e* y a @oltage so!r'e ;!t s!'/ a so!r'e trans)ormation is still ossile i) t/e)ollowing te'/ni!es are )allowe*

' '

:3 :3

< a :1

:2 E

a :1 ? I s/i)t oeration


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:

-

4our ces w it# e;ui*al ent ter $i nal c#ar acter isti cs

i@?Series '!rrent so!r'es>i*eal?

:< @?oltage so!r'e wit/ arallel : >@i?!rrent so!r'e wit/ series :

- @ii? an* I in &arallel >@iii? an* I in Series

>el ta'star tr ans-or$ati on:

A set o) star 'onne'te* > or T? immittan'es 'an e rela'e* y an e!i@alent set o) mes/>5 or M? 'onne'te* immittan'es or @i'e @ersa S!'/ a trans)ormation is o)ten ne'essary tosimli)yassi@e networks( t/!s a@oi*ing t/e nee* )or any mes/ or no*al analysis

Bor e!i@alen'e( t/e immittan'e meas!re* etween any two terminals !n*er se'i)ie*'on*itions m!st e t/e same in eit/er 'ase


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ELTA t 6 trans0r)atin:

onsi*er t/reeN-'onne'te*ime*an'es :A;( :; an* :A a'ross terminals A( ; an* Itis re!ire* to rela'e t/ese y an e!i@alent set :A( :; an* : 'onne'te* in star

A

:A;

;

:A

:;

A :A

:

;:;

In5(ime*an'e meas!re* etween A an* ; wit/ oen is

:A; >:; < :A?:A;< :; < :A

.it/ oen( in ( ime*an'e meas!re* etween A an* ; is :A :A< :A;?:A;< :; < :A

----------------------------->2?

Bor ime*an'e meas!re* etween an* A wit/ ; oen:A> :A; < :;?: < :A 9 :A;< :; < :A

-------------------------------->3?

A**ing >1?( >2? an* >3?

2 >:A< :; < :? 9 2 >:A; :; :; < :?

S!stit!ting )or :; < : )rom >2?

:A 9:A :A ; 9:A;< :; < :A

: A : A;

: A;

Similarly y symmetry :; 9

: 9

:A; :; :A;

:; :A

:A;

I) :A; 9 :; 9 :A 9 :O t/en :A 9 :; 9 : 9 : 9:O 3


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6 t ; trans0r)atin:

onsi*er t/ree 'onne'te* a*mittan'e a( an* ' a'ross t/e terminals A( ; an* It isre!ire* to rela'e t/em y a set o) e!i@alent O a*mittan'es a( ' an* 'a


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7 & 1

Assign me nt ue s t ions:

1 A Disting!is/ t/e )ollowing wit/ s!itale eamlesi? ,inear an* non-linear elementsii? Qnilateral an* trilateral elements

iii? In*een*ent an* *een*ent so!r'es

2? .rite t/e mes/ e!ation )or t/e 'ir'!it s/own in Big 1 an* *etermine mes/ '!rrents !singmes/ analysis

i .

3? Estalis/ star + *elta relations/i s!italy

4? Qsing so!r'e trans)ormation( )in* t/e ower *eli@ere* y t/eR50 in t/e 'ir'!it s/own in Big @oltage so!r'e

5? Bin* t/e ower *eli@ere* y t/e 5A '!rrent so!r'e in t/e 'ir'!it s/own in Big4 y !sing t/eno*al met/o*


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A Bor t/e network s/own in )ig 2( *etermine t/e @oltage !sing so!r'e s/i)t an* or so!r'etrans)ormation te'/ni!es only T/en @eri)y y no*e e!ations

BA Qse mes/ '!rrent met/o* to *etermine t/e '!rrent in t/e 'aa'itor o) %:o) t/e ri*ge networks/own in )ig 5

=A Qse no*e e!ations to *etermine w/at @al!e o) E will 'a!se to e Lero )or t/e networks/own in )ig %


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$? 8tain t/e *elta 'onne'te* e!iment o) t/e network s/own in )ig 1

10? '? Bin* t/e @oltage a'ross t/e 'aa'itor o) 20 : rea'tan'e o) t/e network s/own in )ig 2(y re*!'ing t/e network to 'ontain one so!r'e only( y so!r'e trans)ormation te'/ni!es

116 '? Qse 3 mes/ e!ations )or t/e network s/own in )ig 5 to *etermine 6 an* s!'/ t/att/e '!rrent in 3


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Unit : ! Net$r% T!l&y: Hr s: 07

Syllabus of unit :

7ra/ o) a Network( on'et o) t r ee an* o-tree(in'i*en'e

matri( tie-set( tie-set an*

!t-setresisti@e

s'/e*!les(networks(

Borm!lation 8)e!iliri!m

e!ations in matri )orm( Sol!tion o)

&rin'ile o) *!ality







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Net w o r k T o po l o g

e0 i n it i n:

T/e term 'ir'!it toology re)ers to t/e s'ien'e o) la'ement o) elements an* is a st!*y o) t/egeometri' 'on)ig!rations

Uir'!it toology is t/e st!*y o) geometri' roerties o) a 'ir'!it !se)!le/a@iorU )or *es'riing t/e 'ir'!it

Ter)s use in T!l&y:T/e )ollowing terms are o)ten !se* in network toology

Gra!(:

In t/e gi@en network i) all t/e ran'/es are reresente* y line segments t/en t/e res!lting )ig!reis 'alle* t/e gra/ o) a network >or linear gra/? T/e internal ime*an'e o) an i*eal @oltage so!r'eis Lero an* /en'e it is rela'e* y a s/ort 'ir'!it an* t/at o) an i*eal '!rrent so!r'e is in)inity an*/en'e it is reresente* y an oen 'ir'!it in t/e gra/

Eamle Network 7r a/

No*eIt is a oint in t/e network at w/i'/ two or more 'ir'!it elements are Coine*2( 3 an* 4 are no*es

In t/e gra/ s/own 1(

Branc(


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E E

Uncnnecte &ra!(I) t/ere is no at/ etween any two no*es(t/en t/e gra/ is 'alle* an !n'onne'te* gra/

1 2 4 1 2 4

3 5

3 5 3 5

Planar &ra!(A lanar gra/ is a gra/ *rawn on a two *imensional lane so t/at no two ran'/es interse't at aoint w/i'/ is not a no*e

; A ;

A E

D D

Nn !lanar &ra!(A gra/ on a two + *imensional lane s!'/ t/at two or moreran'/es interse't at a oint ot/er t/an no*e on a gra/

Tree o) a gra/Tree is a set o) ran'/es wit/ all no*es not )orming any loo or 'lose* at/

>? ontains all t/e no*es o) t/e gi@en network or all t/e no*es o) t/e gra/

>? No 'lose* at/>? N!mer o) ran'/es in a tree 9 n-1 ( w/ere n9n!mer o) no*esA 2 ; A ;

2 45 % 5 D

D

Two ossile trees

7ra/

C" treeA o- tree is a set o) ran'/es w/i'/ are remo@e* so as to )orm a tree or in ot/er wor*s( a 'o- treeis a set o) ran'/es w/i'/ w/en a**e* to t/e tree gi@es t/e 'omlete gra/ Ea'/ ran'/ soremo@e* is 'alle* a linkN!mer o) links 9 l 9 + >n-1? ./ere 9 Total n!mer o)ran'/es

n 9 N!mer o) no*es


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-1 1 1 0 0 00 -1 0 -1 1 00 0 -1 1 0 1

are aritrarily '/osen>? T/e in'i*en'e matri is )orme* y taking no*es as rows an* ran'/es as 'ol!mns

No*es ;ran'/es

1 2 3 4 5 %

A -1


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N!mer o) no*es 9 n 9 4 >say A ; an* D?N!mer o)ran'/es 9 9

%>say 1( 2( 3( 4( 5 an* %?

&reare a ta!lar 'ol!mn as s/ownNo*es ;ran'/es

1 2 3 4 5 %

A 1 0 0 0 1 -1; -1 1 1 0 0 0 0 -1 0 -1 0 1D 0 0 -1 1 -1 0

Brom t/e ta!lar 'ol!mn( t/e entries /a@e to e interrete* as )ollows


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Network Analysis 10ES34

Brom t/e )irst 'ol!mn t/e entries )or A an* ; are ones Hen'e ran'/ 1 is 'onne'te* etweenno*es A an* ; Sin'e )or no*e A entry is


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23

21

Tie s et Analy sis :

In or*er to )orm a tree )rom a network se@eral ran'/es nee* to e remo@e* so t/at t/e 'lose* loosoen ! All s!'/ remo@e* ran'/es are 'alle* links an* t/ey )orm a o-tree Alternati@ely w/en a

link is rela'e* in a tree( it )orms a 'lose* loo along wit/ )ew o) t/e treeran'/es A '!rrent 'an)low aro!n* t/is 'lose* loo T/e *ire'tion o) t/e loo '!rrent is ass!me* to e t/e same as t/at o)t/e '!rrent in t/e linkset

T/e tree + ran'/es an* t/e link t/at )orm a loo is sai* to 'onstit!te a tie +

De)inition

A tie + set is a set o)ran'/es 'ontaine* in a loo s!'/ t/at t/e loo /as at least one link an* t/eremain*er are twigs >treeran'/es?

% %

L4 ; 5 4 ; 5

A A

2

3 1 3

D D7ra/

tree >In t/i'k lines? o-tree >In *otte* lines?.e see t/at y rela'ing t/e links 1( 4 an* 5 t/ree loos are )orme* an* /en'e t/ree loo '!rrents( y an* L )low as s/own T/e relations/is otaine* etween loo '!rrents( tree ran'/es an*links 'an e s'/e*!le* as )ollows

Tie + set s'/e*!le

Tie + set Tree +ran'/es ,ink oop current1 2( 3 5 2 1( 4 2 y3 4( 5 % L

Tie + set matri >;)?

T/e Tie + set s'/e*!le s/own ao@e 'an e arrange* in t/e )orm o) a matri w/ere in t/e loo'!rrents 'onstit!te t/e rows an* ran'/es o) t/e network 'onstit!te t/e 'ol!mns Entries insi*e t/ematri are )ille* y t/e )ollowing ro'e*!re


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'!rrent 1 2 3 4 5 % 1 0 0 1 -1 0y 0 1 0 0 1 -1L 0 0 1 -1 0 1

./ere I; 9 ;ran'/ '!rrent matri; )TI,

9 Transose o) t/e tie- set matri9 ,oo '!rrent 'ol!mn matri

>ii?6ow wise a**ition )or ea'/ row gi@es t/e =, e!ations )or ea'/ )!n*amental loo

6ow - 1 6ow - 2

1 < 2 < 3 9 01 < 2 < 4 9 0

6ow - 3 4 - 5 - % 9 0

10 1 11 1 0

0 1 0 21 0 0 3

0 0 0 1 -1 -1 4 9 0 5

%

In 'oma't )orm ; ) ; 9 0 - - - >1?

./ere ; 9 ;ran'/ @oltage 'ol!mn matri an*; ) 9 Tie - set matri

A

Eamle Bor t/e network s/own in )ig!re( write aTie-set s'/e*!le an* t/en )in* all t/e ran'/ 50!rrents an* @oltages

5 510

V ;

10 5D

Sol!tion

1

T/e gra/ an* one ossile tree is s/own 5

A

25;

4 %

3 D

1 5 y

2

4 %

L3

,oo ;ran'/ N!mers


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Tie set atri ;) 9

1 0 0 1 -1 01 0 0 1 -10 1 -1 0 1

;ran'/ Ime*en'e matri :; 9 7 8 8 8 8 80 10 0 0 0 00 0 5 0 0 00 0 0 10 0 00 0 0 0 5 00 0 0 0 0 5

:, 9 1 0 0 1 -1 0 5 0 0 0 0 0 1 0 00 1 0 0 1 -1 0 10 0 0 0 0 0 1 00 0 1 -1 0 1 0 0 5 0 0 0 0 0 1

0 0 0 10 0 0 1 0 -10 0 0 0 5 0 -1 1 00 0 0 0 0 5 0 -1 1

:,

20 -5 -10

9 -5 20 -5-10 -5 20

,oo E!ations :, I, 9 - ;) s-50

20 -5 -10 1 0 0 1 - 1 0 0-5 20 -5-10 -5 20

y 9 -L

0 1 00 0 1

0 1 -1 0-1 0 1 0

00

20-5y-10L 950

-5


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Cut s et Analy sis

A '!t + set o) a gra/ is a set o) ran'/es w/ose remo@al ( '!ts t/e 'onne'te* gra/ in to two artss!'/A

t/at t/e rela'ement o) any one ran'/ o) t/e '!t set ren*ers t/e two arts 'onne'te*A

Eamle1 5

4 ; 3 D 4 ; D2 % 3

Dire'te* gra/ Two searate gra/s 'reate* y t/e '!t set >1( 2( 5( %?

7una)ental cut set is a '!t + set t/at 'ontains only one tree ran'/ an* t/e ot/ers are links

7r)ati n 0 7una )entalcut set

>?>?>?

Sele't a treeSele't a treeran'/Di@i*e t/e gra/ in to two sets o) no*es y *rawing a *otte* line t/ro!g/ t/e sele'te* tree

ran'/ an* aroriate links w/ile a@oi*ing interr!tion wit/ any ot/er tree +ran'/es

Eamle 1 A

Bor t/e gi@en gra/ write t/e '!t set s'/e*!le

1 5 A4 ; 3 D 1 5

2 4 ; 3 D% 2 %

T/e )!n*amental '!t +set o) t/eSele'te* tree is s/own in )ig!re

Note t/at BS - 1 yiel*s no*e A an* t/e set o) no*es > ;( ( D?T/e 8rientation o) t/e )!n*amental '!t + set is !s!ally ass!me* to e t/e same as t/e orientation o)t/e tree ran'/ in it( ./i'/ is s/own y an arrow ;y )ollowing t/e same ro'e*!re t/e BS- 2an* BS -3 are )orme* as s/ownelow


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1 0 0 -1 - 1 0

0 1 0 1 0 1

0 0 1 0 1 -1

A A1 5

1 54 ; 3 D ; 3

2 D2 4 %

%

BS -2 BS -3

It s/o!l* e note* t/at )or ea'/ tree ran'/ t/ere will e a )!n*amental '!t + set Bor a gra//a@ing n n!mer o) no*es t/e n!mer o) twigs is >n-1?T/ere)ore t/ere will e >n-1?>n-1? )!n*amental '!t-sets

8n'e t/e )!n*amental '!t sets are i*enti)ie* an* t/eir orientations are )ie*( it is ossile towrite a s'/e*!le( known as '!t + set s'/e*!le w/i'/ gi@es t/e relation etween tree +ran'/@oltages an* all ot/er ran'/ @oltages o) t/e gra/

,et t/e element o) a '!t + set s'/e*!le e *enote* y ik t/en(

ik 9 1 I) ran'/ = is in '!t + set I an* t/e *ire'tiono) t/e '!rrent in t/eran'/ = is same as '!t + set *ire'tion

-1 I)ran'/ k is in '!t + set I an* t/e *ire'tion o) t/e '!rrent in k is8osite to t/e '!t + set *ire'tion

!t set S'/e*!le8 I)ran'/ is k is not in '!t + set i

Tree

branc(

vlta&e

Branc( @lta&es

1 # ' * + -

e1 1 "1 "1

e2 1 1 1

e3 1 1 "1

T/e elements o) t/e '!t set s'/e*!le may e written in t/e )orm o) a matri known as t/e '!t setmatri

) 9


33/126

)ig!re =

0

Analysis 0 a ne t$ r % usin& c ut se t sc(e ule

>? ol!mn wise a**ition o) t/e '!t set s'/e*!le gi@es t/e relation etween tree ran'/@oltage an* t/e ran'/ @oltages )or t/e ao@e '!t t/e s'/e*!le

1 9 e12 9 e23 9 e34 9 - e1 < e25 9 - e1< e3

In matri )orm

% 9 e2 - e3

1 1 0 02 0 1 0 e1

3 0 0 1 e24 9 -1 1 0 e35 -1 0 1% 0 1 -1

In 'oma't )orm

@B T

@T WWW >1? ./ere ; 9;ran'/ @oltage atri

T) 9 Transose o) '!t set matri

an* T 9 Tree ran'/ @oltage matri

>?&ow wise addition %i$en (" at each nodeI1 - I4 - I5 9 0

I2 < I4 < I% 9 0I3 < I5 < I% 9 0

In matri )orm

1 0 0 -1 - 1 0 I1

0 1 0 1 0 1 I2

0 0 1 0 1 -1 I3 9 0

I4

I5I%

In 'oma't )orm ) I; 9 0 WWWWWWW>2? ./ere ) 9 '!t set matriI; 9 ;ran'/ '!rrent matri

>? ,et !s 'onsi*er a network /a@ing ran'/esI= =

Ea'/ o) t/e ran'/es /as a reresentation as s/own in


34/126

30

1 0 0 - - 0 ;ran'/ a*mittan'e0 2 0 - - 0 9 matri o) or*er

- - - - - -0 0 - - -

6e)erring to t/e )ig!re Ik 9 k k < IskIS=

Sin'e t/e network /as ran'/es( one s!'/ e!ation 'o!l* e written )or e@ery ran'/ o) t/e

network ie I1 9 1 1 < Is1I2 9 2 2 < Is2------------------------------

WWWWWWWWWW

I 9 < Is

&!tting t/e ao@e set o) e!ations in a matri )orm we get

IB 6B @B D Is WWWWWWWWW >3?

I; 9 ;ran'/ '!rrent matri o) or*er 1

; 9

; 9 ;ran'/ oltage 'ol!mn matri o) or*er1 an*I; 9 So!r'e '!rrent matri o) or*er1

S!stit!ting >3? in >2? we get ) ; ; < IS 9 0

;!t )rom >1? we /a@e ; 9

) ; ;

)T T

< ) IS 9 0

Hen'e e!ation >4?e'omes

) ; T

) T < T 5?

./ere 9 ) ; )T

is 'alle* '!t - set a*mittan'e matri

Actin Plan 0 r c ut set analysis.

>? Borm t/e '!t- set matri )>? onstr!'t t/e ran'/ a*mittan'e matri ;>? 8tain t/e '!t + set a*mittan'e matri !sing t/e e!ation 9 ) ; )

T

>? Borm t/e =, or e!iliri!m e!ations !sing t/e relation T 9 - ) IS


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T/e elements o) t/e so!r'e '!rrent matri are ositi@e i) t/e *ire'tions o) t/e ran'/ '!rrentan* t/e so!r'e 'onne't atta'/e* to t/at ran'/ are same ot/erwise negati@e

>?>?

T/e ran'/ @oltages are )o!n* !sing t/e matri e!ation ; 9 )T T

Binally t/e ran'/ '!rrents are )o!n* !sing t/e matri e!ation I; 9 ; ; < IS

E?a)!le #A

51

D4

Bor t/e *ire'te* gra/ otain t/e '!t set matri

%

E ;2

# 3 "

Slutin T/e tree >marke* y t/i'k lines?an* t/e link >marke* y *o))e* lines?are as s/own

T/e )!n*amental '!t sets are )orme* at no*es A ; an* D keeing E as re)eren'e no*e

B's - 1 -- > 1( 5( %? A BS-1

B's - 2 -- >2( %( " ? 1

B's

B's

- 3

- 4

--

--

> 3(

>4(

"(

5(

#?

#?

DBS4

4 E 2 ; BS2

BS3 Hen'e t/e '!t + set s'/e*!le is as )ollows

Tree

branc(

vlta&e

Branc(

1 # ' * + - / #

e1 1 "1 1

e2 1 "1 1

e3 1 "1 1

e4 1 1 "1

Hen'e t/e re!ire* '!t +set matri ) 9 1 "1 1 1 "1 1

1 "1 1

1 1 "1


36/126

Tree

branc(

vlta&e

Branc(es

1 # ' * + -

e3 "1 1 "1

e4 1 1 1

e% 1 "1 1

-1 0 1 0 -1 00 1 0 1 1 01 -1 0 0 0 1

E?a)!le ': Bin* t/e ran'/ @oltages !sing t/e 'on'et o) '!t-sets

11 1

1 1

1@ V 1

Sluti n : T/e @oltage so!r'e is trans)orme* in to an e!i@alent '!rrent so!r'e It s/o!l* e note*t/at all t/e 'ir'!it &assi@e elements m!st e a*mittan'es an* t/e net work s/o!l* 'ontain only'!rrent so!r'es

T/e gra/ )or t/e network is s/own>S/own y *otte* lines? are s/own

A ossile tree >s/own wit/ t/i'k lines? an* 'o tree

1m/o

BS 1 9 3( 1( 5 1m/o 1 m/o

BS 2 9 4( 2( 51m/o

1m/o1m/o

BS 3 9 %( 1( 21A

!t set s'/e*!le

5 BS2

3 4A #

BS1 %1 2

BS3

) 9


37/126

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

; 9

-1 0 10 1 -1

-1 0 1 0 -1 0 1 0 0 0 0 0 1 0 00 1 0 0 0 0 0 1 0

9 0 1 0 1 1 0 0 0 1 0 0 0 -1 1 01 -1 0 0 0 1 0 0 0 1 0 0 0 0 1

0 0 0 0 1 00 0 0 0 0 1

3 -1 -1

9 -1 3 -1

-1 -1 3

E Fu ili b r iu ) EF u a t i n s T 9 - ) IS

3 -1 -1 e3 -1 0 1 0 -1 0 -1-1 3 -1 e4 9 - 0 1 0 1 1 0 0

-1 -1 3 e% 1 -1 0 0 0 1 0000

3 e3+ e4+ e% 9 - 1

-e3 < 3 e4+ e% 9 0- e3+ e4 < 3e% 9 1

Sol@ing we get e39-025 @olte490

e%9025 @olt


38/126

UALI T6 CONCEPT

Two ele'tri'al networks are *!als i) t/e mes/ e!ations t/at '/ara'teriLe one /a@e t/e samemat/emati'al )orm as t/e no*al e!ations o) t/e ot/er

Eamle 1

onsi*er an 6-,- series network e'ite* y a@oltage so so!r'e as s/own in t/e )ig!re T/ee!ation generatingt/e 'ir'!it e/a@ior is 6i2? w/en t/e)ollowing e'/anges are ma*e

6 Y 7( ,Y ( Y, an* YiHen'e networks o) )ig!re >1? an* >2? are *!al to ea'/ ot/er

Tale o) *!al !antities

1oltage So!r'e2,oo '!rrents3I*!'tan'es46esistan'es5aa'itan'es% 8n =,asis"lose o) swit'/

!rrent so!r'eNo*e @oltagesaa'itan'eson*!'tan'esIn*!'tan'es8n =,asis8ening o) swit'/

Note8nly lanar networks /a@e *!als

&ro'e*!re )or *rawing *!al networkT/e *!als o) lanar networks 'o!l* e otaine* y a gra/i'al te'/ni!e known as t/e *otmet/o* T/e *ot met/o* /as t/e )ollowing ro'e*!re

&!t a *ot in ea'/ in*een*ent loo o) t/e network T/ese *ots 'orreson* to in*een*ent no*esin t/e *!al network


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&!t a *ot o!tsi*e t/e network T/is *ot 'orreson*s to t/e re)eren'e no*e in t/e *!al networkonne't all internal *ots in t/e neig/oring loos y *otte* lines '!tting t/e 'ommon ran'/esT/ese ran'/es t/at are '!t y *as/e* lines will )orm t/e ran'/es 'onne'ting t/e'orreson*ing in*een*ent no*es in t/e *!al networkFoin all internal *ots to t/e eternal *ot y *as/e* lines '!tting all eternal ran'/es D!als o)t/ese

no*e

ran'/es will )orm t/e ran'/es 'onne'ting t/e in*een*ent no*es an* t/e re)eren'e

Eamle 1Draw t/e ea't *!al o) t/e ele'tri'al'ir'!it s/own in t/e )ig!re

Sol!tion ark two in*een*ent no*es 1 an* 2an* a re)eren'e no*e 0as s/own in t/e )ig!re

3o/m 4B Foin no*e 1 an* 2 y a *otte* line assing2sin%t

1

4o/mV

%H 2

t/ro!g/ t/e in*!'tan'e o) %Ht/is element willaear as 'aa'itor o) % B etween no*e 1 an*2 in t/e *!alFoin no*e 1 an * re)eren'e no*e t/ro!g/ a*otte* line assing t/e @oltage so!r'e o) 2sin%t@olts T/is will aear as a '!rrent so!r'e o)

2sin%t ameres etween no*e1 an* re)eren'e8 no*e

Foin no*e 1 an* re)eren'e no*e t/ro!g/ a *otte* line assing t/ro!g/ 3 o/ms resistor T/iselement aears as 3m/o 'on*!'tan'e etween no*e1 an* re)eren'e no*e in t/e *!alFoin no*e 2 an* re)eren'e no*e t/ro!g/ a *otte* line assing t/ro!g/ t/e 'aa'itor o) 4 Bara*sT/is element will aear as 4 Henry in*!'tor etween no*e 2 an* re)eren'e no*e in t/e *!alFoin no*e 2 an* re)eren'e no*e t/ro!g/ a *otte* line assing t/ro!g/ t/e resistor o) 4 o/ms

T/is element will aear as 4 m/o 'on*!'tan'e etween no*e 2 an* re)eren'e no*e

T/e D!al network *rawn !sing t/ese ro'e*!ral stes is s/own


40/126

3"

3%


1? Elain in'i*en'e matri o) a network o) a network gra/Z 7i@e s!itale eamle2? De)ine t/e )ollowing wit/ s!itale eamles

i? &lanar an* non-lanar gra/ii? Twigs an* links

3? Bor t/e network s/own in Big # write t/e gra/ o) t/e network an* otain t/e tie-set

s'/e*!le 'onsi*ering F1( F2( F5 as tree ran'/es al'!late all ran'/es '!rrent

4? Elain rie)ly trees( 'otrees( an* loos in a gra/ o) network wit/ s!itale eamle5? Elain wit/ eamles t/e rin'ial o) *!ality%? Draw t/e oriente* gra/ o) t/e network s/own in Big 10 sele'ts a tree( write t/e set

s'/e*!le an* otain e!iliri!m e!ations


41/126


42/126

#? Bor t/e network s/own in Big 4( *raw its *!al .rite in intergo *i))erential )ormi? es/ e!ations )or t/e gi@en network

>t? 9 10 sin 40tii? No*e e!ations )or t/e *!al

H= ./at are *!al networksZ ./at is t/eir signi)i'an'e Z Draw t/e *!al o) t/e 'ir'!it

s/own in )ig #

10? Bor t/e network s/own in Big 5( er)orm so!r'e s/i)ts( *raw a gra/( sele't tree wit/ran'/es 1(2 an* 3 an* otain tie set an* matri'es


43/126

Unit : " Net$r%T(ere)s 1: Hr s: 0#

Syllabus of unit :S!erosition( 6e'iro'it y an* illmans t/eorems




3. Net$r% t(ery,

7anes/ 6ao.



44/126

40

NET8O,9 TEO,E>S

es/ '!rrent or no*e @oltage met/o*s are general met/o*s w/i'/ are ali'ale to any network An!mer o) sim!ltaneo!s e!ations are to e set ! Sol@ing t/ese e!ations( t/e resonse in all t/eran'/es o) t/e network may e attaine* ;!t in many 'ases( we re!ire t/e resonse in one ran'/or in a small art o) t/e network In s!'/ 'ases( we 'an !se network t/eorems( w/i'/ are t/e ai*esto simli)y t/e analysis To re*!'e t/e amo!nt o) work In@ol@e* y 'onsi*erale amo!nt( as'omare* to mes/ or no*al analysis ,et !s *is'!ss some o) t/em

S UPE,POS ITIONTEO,E >:

T/e resonse o) a linear network wit/ a n!mer o) e'itations alie* sim!ltaneo!sly ise!al to t/e s!m o) t/e resonses o) t/e network w/en ea'/ e'itation is alie* in*i@i*!ally

rela'ing all ot/er e'itations y t/eir internal ime*an'esHere t/e e'itation means an in*een*ent so!r'e Initial @oltage a'ross a 'aa'itor an* t/e

initial '!rrent in an in*!'tor are also treate* as in*een*ent so!r'es

T/is t/eorem is ali'ale only to linear resonses an* t/ere)ore ower is not s!Ce't tos!erosition

D!ring rela'ing o) so!r'es( *een*ent so!r'es are not to e rela'e* 6ela'ing an i*eal@oltage so!r'e is y s/ort 'ir'!it an* rela'ing an i*eal '!rrent so!r'e is y oen 'ir'!it

UIn any linear network 'ontaining a n!mer o) so!r'es( t/e resonse >'!rrent in or @oltagea'ross an element? may e 'al'!late* y s!erosing all t/e in*i@i*!al resonses 'a!se* y ea'/in*een*ent so!r'e a'ting alone( wit/ all ot/er in*een*ent @oltage so!r'es rela'e* y s/ort

'ir'!its an* all ot/er in*een*ent '!rrent so!r'es rela'e* y oen 'ir'!its[ Initial aa'itor@oltages an* initial in*!'tor '!rrents( i) any( are to e treate* as in*een*ent so!r'es

To ro@e t/is t/eorem 'onsi*er t/e network s/own in )ig

Ia

ISIS ES

Ia1

.e 'onsi*er only one-@oltage so!r'es an* only one '!rrent so!r'es )or simli'ity It isre!ire* to 'al'!late Ia wit/ Is a'ting alone t/e 'ir'!it e'omes

IS :1 :3Ia1 9 :1 < :2 < :3 :4 :3 < :4:3 < :4

9IS:1:3

>:1 < :2 < :3? :4 < >:1 < :2? :3------------------------------------>1?


45/126

S

Ia2

ES

wit/ ES a'ting alone-E

Ia1 9 :4 < >:1 < :2? :3:1 < :2 < :3

9-ES> :1< :2< :3?

>:1 < :2 < :3? :4 < >:1 < :2? :3---------------------------------------->2?

Net 'on@erting t/e '!rrent so!r'e to @oltage so!r'e( t/e loo e!ations

IS :1 I1 I2 ES

I2 9

:1


46/126

/imilarly if the sin%le currentsource 'x

#etween nodes 9 and 9) produces the$olta%e

response ;y #etween nodes and ) then the remo$al of the current sourcefrom 9 and 9) and its insertion #etween and ) will produce the $olta%eresponse ;y #etween the nodes 9 and 9).


47/126

J1 J#

J*A

J' D

I# EK

=ow the readin% of the mmeter is :

E :3I2 .

> :2

< :4? < :

1:

3:

1 :22?

rom >16 ! >26

I1 I#

't can #e similarly #e shown for a network with current sources #ywritin% node e+uations.

Trans -er I$pe+ an ce :

he transfer impedance #etween any two pairs of terminals of a linear passi$enetwork is the ratio of the $olta%e applied at one pair of terminals to theresultin% current at the other pair of terminals .

?ith this definition the reciprocity theorem can #e stated as :

@nly one $alue of transfer impedance is associated with two pairs ofterminals of a linear passi$e network .


48/126

E2

-

I2:1 a '

i ll )ans T(ere):

ertain simle 'ominations o) otential an* '!rrent so!r'e e!i@alents are o) !se e'a!se t/eyo))er simli)i'ation in sol!tions o) mor e etensi@e networks in w/i'/ 'ominations o''!rillmans T/eorem says t/at Ui) a n!mer o) @oltage so!r'es wit/ internal ime*an'es are'onne'te* in arallel a'ross two terminals( t/en t/e entire 'omination 'an e rela'e* y a single@oltage so!r'e in series wit/ single ime*an'e[

T/e single @oltage is t/e ratio

S!m o) t/e ro*!'t o) in*i@i*!al @oltage so!r'es an* t/eir series a*mittan'esS!m o) all series a*mittan'es

an* t/e single series ime*an'e is t/e re'iro'al o) s!m o) all series a*mittan'es

E1 :1

E2 :2

E3 :3

En :n


49/126

,et E1( E2WWWWEn e t/e @oltage so!r'es an* :1( :2WWWWWWW:n are t/eir rese'ti@eime*an'es All t/ese are 'onne'te* etween A J ; wit/ 91:( a''or*ing to illmansT/eorem( t/e single @oltage so!r'e t/at rela'es all t/ese etween A J ; is

n

EA; 9P E= ==91

nP ==91

An*

T/e single ime*an'e is : 9 1n

P ==91

&roo) Trans)orm ea'/ @oltage into its e!i@alent '!rrent so!r'eE1:1

:1

; E2:2 A

:2

T/en t/e 'ir'!it is as in Big

En:n

:n

.it/ 91: t/e 'ir'!it is simli)ie* as E1 1


50/126

1- E1

9

< E2? > :1 E1+ I S: S ? :1

:1: 2 < : S?

:1 E1 < E2? > :1 E1+ I S: S ? :1

9 :1>: 2 < : S? < :2 >:1 E1 < E2? :1

> E1 - IS :s? :1


58/126

I I

> E1 < E2? > :1 E1+ I S: S ? :19>: 1 < : ,? >:1 E1 < E2? > :1 E1+ I S: S ? :19:1>: 2 < : S? < :2 >:1


59/126

IN9Is'9E1< E2


60/126


61/126

>a?i )u ) Trans 0er T(ere):"

./en a linear network 'ontaining so!r'es an* assi@e elements is 'onne'te* at terminals Aan* ; to a assi@e linear network( maim!m ower is trans)erre* to t/e assi@e network w/en itsime*an'e e'omes t/e 'omle 'onC!gate o) t/e T/e@inins ime*an'e o) t/e so!r'e 'ontainingnetwork as @iewe* )rom t/e terminals A an* ;

Big reresents a network wit/ so!r'es rela'e* y its T/e@inins e!i@alent o) so!r'e o)ETH @olts an* ime*an'e :s( 'onne'te* to a assi@e network o) ime*an'e L at terminals A J ;.it/ :s 96s 6s


62/126


63/126

T/is means to transmit maim!m ower to t/e loa* 50d ower generate* is t/e loss S!'/ a lowe))i'ien'y 'annot e ermitte* in ower systems in@ol@ing large lo'ks o) ower w/ere 6, is @erylarge 'omare* to 6s T/ere)ore 'onstant @oltage ower systems are not *esigne* to oerate on t/easis o) maim!m ower trans)er

Howe@er in 'omm!ni'ation systems t/e ower to e /an*le* is small as t/ese systems are low'!rrent 'ir'!its T/!s ime*an'e mat'/ing is 'onsi*erale )a'tor in 'omm!ni'ation networks

Howe@er etween 6 J X i) eit/er 6 or X is restri'te* an* etween : an* i) eit/er b:b or isrestri'te* t/e 'on*itions )or a & is state* as )ollows

ase >i? - 6 o) : is @arie* keeing X 'onstant wit/ 6 only ariale( 'on*itions )or maowertrans)er is >6s#?ase >i@?- I) b:b is 'onstant !t is @arie*

T/en )rom en >$? >:2:2


64/126


65/126


66/126

%0

Unit : *: ,esnant Circuits: Hr s: 0#

Syllabus of unit :Series an* arallel resonan'e( )re!en'y resonse o) series an* &arallel 'ir'!its( +)a'tor(;an*wi*t/



#. 3Net$r%s an Syste)s45 6oy /o!*/!ry( 2 e*ition( New Age International&!li'ations




67/126

,e snant C irc uits

6esonan'e is an imortant /enomenon w/i'/ may o''!r in 'ir'!its 'ontaining ot/ in*!'tors'aa'itors

an*

In a two terminal ele'tri'al network 'ontaining at least one in*!'tor an* one 'aa'itor( we*e)ine resonan'e as t/e 'on*ition( w/i'/ eists w/en t/e in!t ime*an'e o) t/e network is!relyresisti@e In ot/er wor*s a network is in resonan'e w/en t/e @oltage an* '!rrent at t/e network in!t terminals are in/ase

6esonan'e 'on*ition is a'/ie@e* eit/er y keeing in*!'tor an* 'aa'itor same an* @arying)re!en'y or y keeing t/e )re!en'y same an* @arying in*!'tor an* 'aa'itor St!*y o)resonan'e is @ery !se)!l in t/e area o) 'omm!ni'ation T/e aility o) a ra*io re'ei@er to sele't t/e'orre't )re!en'y transmitte* y a roa* 'asting station an* to eliminate )re!en'ies )rom ot/erstations is ase* on t/e rin'ile o) resonan'e

T/e resonan'e 'ir'!its 'an e 'lassi)ie* in to two 'ategories

Series + 6esonan'e ir'!its&arallel + 6esonan'e ir'!its

1Series 6esonan'e ir'!it6 ,

A series resonan'e 'ir'!it is one in w/i'/ a 'oil an*a 'aa'itan'e are 'onne'te* in series a'ross an alternating I@oltage o)

@arying )re!en'y as s/own in )ig!re

T/e resonse I o) t/e 'ir'!it is *een*ent on t/e ime*an'e o) t/e 'ir'!it(./ere :9 6 )r?

Eression )or 6esonant Bre!en'y >At resonan'e X, 9 X

)r ?


68/126

X

Salient 7eatures 0 ,esnant circuit

>? At resonan'e X, 9 X>? At resonan'e : 9 6 ie ime*an'e is minim!m an* /en'e I 9 is maim!m

:>?

>?

T/e '!rrent at resonan'e >Ir? is in /ase wit/ t/e @oltage

T/e 'ir'!it ower )a'tor is !nity>? oltage a'ross t/e 'aa'itor is e!al an* oosite to t/e @oltage a'ross t/e in*!'tor

Bre!en'y resonse o) a series resonan'e 'ir'!itBor a 6-,- Series ir'!it t/e '!rrent I is gi@en y

I96 < C >X,- X?

At resonan'e X, 9 X an* /en'e t/e '!rrent at resonan'e >Ir? is gi@en y Ir 9 6At o)) resonan'e )re!en'ies sin'e t/e ime*an'e o) t/e 'ir'!it in'reases t/e '!rrentin t/e 'ir'!it will re*!'e At )re!en'ies ) ./ere )f )r ( t/e ime*an'e is going to e morein*!'ti@e Similarly at )re!en'ies ) )r t/e 'ir'!it ime*an'e is going to e more 'aa'iti@eT/!s t/e resonan'e '!r@e will e as s/own in )ig!re

I

IrX' f Xl

l f X'

)r )

uali 0y 0actr


69/126


70/126


71/126

;!t 62


72/126

,

,

6

C

,

II A c il a n a P r a c t ic a l Ca ! a c it r in ! a r a lle l

onsi*er a arallel resonant 'ir'!it in w/i'/t/e resistan'e o) t/e 'aa'itan'e is also 'onsi*ere*

IL,L L

Ime*an'e o) t/e 'oil 9 :, 9 6, < C , IC, 9 1 :, 9 1 6,6

2+ ,?

0 1 < ,L# LMC=

#/ LC < , # LMC=

At 6esonan'e t/e a*mittan'e is !rely 'on*!'ti@e gi@en y


73/126

0 9 6, 610 200? 9 0

25 25 mH,C - ()s Answer

E?a)!le #: An Ime*an'e 'oil o) 25 o/ms6esistan'e an* 25 mH in*!'tan'eis 'onne'te* in arallel wit/ a @ariale'aa'itor Bor w/at @al!e o) aa'itor

will t/e 'ir'!it resonateI) $0 @olts(400HL so!r'e is !se*( w/at will e t/e line!rrent !n*er t/ese 'on*itions

$0 olts(400HL

Sol!tion 0 9 2/)0 9 2/> 400?

h02 9 1 - 62

, ,2


74/126

%31%10% 9 1 - 62


75/126

, , 2

1 9 h02 < 62

, ,2

9 %31% 10% 25 10-3?2

:0 9 ,6 9 >25 10-3? 25 54%" 10-%

9 1#2#$ o/ms

I0 9 0 :0 9 $0 1#2#$ 9 04$2 amere


76/126


1? Elain arallel resonan'e Z Deri@e t/e 'on*ition )or arallel resonan'e w/en 6, 'onne'te*arallel to 6

2? S/ow t/at resonant )re!en'y o) series resonan'e 'ir'!it is e!al to t/e geometri'mean o) two /al) ower )re!en'ies

3? In t/e 'ir'!it gi@en elow in Big 21( an in*!'tan'e o) 01 H /a@ing a o) 5 is inarallelwit/ a 'aa'itor Determine t/e @al!e o) 'aa'itan'e an* 'oil resistan'e at resonant )re!en'y o)500 ra*sa'

4? In t/e 'ir'!it s/own in Big 22( *etermine t/e 'omlete sol!tion )or t/e '!rrent w/ent/e swit'/ S is 'lose* at t 9 0 Alie* @oltage is >t? 9 400 'os 500or 'on*ition? o) t/e '!rrent )rom t/e instant o) swit'/ing to attainment o) stea*y state is 'alle*transient state or transient T/e '!rrent an* @oltage o) 'ir'!it elements *!ring transient erio* is'alle* transient resonseT/e transient may also o''!r *!e to @ariation in 'ir'!it elements Transient analysis is an !se)!ltool in ele'tri'al engineering )or analysis o) swit'/ing 'on*itions in ir'!it reakers( 6elays(7enerators et'It is also !se)!l )or t/e analysis o) )a!lty 'on*itions in ele'tri'al *e@i'es Transient analysis is also!se)!l )or analyLing swit'/ing on*itions in analog an* *igital Ele'troni' *e@i'es

,"L Series circuit transient:

onsi*er T/e 6-, series 'ir'!it s/own in t/e )ig Swit'/ = is'lose* at t90 6e)erring to t/e 'ir'!it( alan'e e!ation !sing

6 ,

=i>t?

5t ( tldi

t

t 9 0

=ir'/o))s law 'an e written asTaking ,ala'e Trans)orm we get

idt

5 s

s7su ( LHS7s i4I

Ass!ming t/ere is no store* energy in t/e in*!'torI>0?905 s

s(7s LS7s

J

5 s 5 sK 1 K

7 sS ( SL L

LM ( N

O

7s .

KSPSKQ R

SK

L TK

U

S

M (N

S(

L

5sPSR

S .SL TM(N

L

5s


80/126


81/126

5s

(

M (N 5s

!t

. 5s

(

S(L

V

.P SR L T

L

W

7s X Y5 s X 1 1 Y

T/ere)ore

(X

S

ZS

(Y

L [

Taking in@erse ,ala'e we get

V P S Wit

5M ( N

tX1 e R

L T Y( X Y

Z [

T/e e!ation 'learly in*i'ates transient nat!re o) '!rrent( w/i'/ is also s/own in )ig!reL

./ere ( T!ne 'onstant o) t/e 'ir'!it( w/i'/ is *enote* y : gi@en in se'on*s

5 V W

Hen'e

it X1 e t Y( XZ Y[

5 5

&!tting t9L we get i>L? 9 0%32 ( ./ere ( 9 stea*y state '!rrent Hen'e Time 'onstant )or an

6-, series '!rrent 'ir'!it is *e)ine* as t/e time taken y t/e 'ir'!it to rea'/ %32d o) its )inalstea*y @al!e

6- series 'ir'!it Transient


82/126


83/126

c

P

s

S

onsi*er t/e 6 'ir'!it s/own ,et t/e swit'/ e 'lose* at t90

.riting t/e alan'e e!ation !sing =ir'/o))s @oltage law (

'ti(

1\idt c

Taking ,ala'e trans)orm( we get

5 s 1 V7s - 4 W7 s ( X Y

S Z s [

,et !s ass!me t/at t/ere is no store* energy in t/e 'ir'!it

Hen'e

-o 905s

s

7s(7sCS

7sL(Q

1 JO

CSU

M N

P S7s

5 s P 1 S(

P S1 S

R (C TTaking ,ala'e in@erse we get

M 1 N5 t P St(C

i t e RT

(

1

T/e sket'/ o) transient '!rrent is s/own in )ig!re ./ere (C?

t/e time 'onstant o) t/e 'ir'!it

?

&!tting

1

(C in t/e '!rrent e!ation we get5

i>L? 9 03%" (

Hen'e time 'onstant o) 6 series '!rrent 'an e *e)ine* as t/e time taken y '!rrent transient to)all to 3%"d o) its initial @al!e


84/126

Eamle 1

In t/e 'ir'!it s/own in )ig!re t/e swit'/ = is mo@e* )rom osition 1 to osition 2 at time t 9 0T/e stea*y state '!rrent /a@ing een re@io!sly estalis/e* in 6-, 'ir'!it Bin* t/e '!rrent i>t? a)terswit'/ingSol!tionBrom t/e gi@en *ata t/e 'ir'!it is !n*er stea*y state w/en swit'/ = is in osition 1 !n*er stea*y

14

state 'on*ition in*!'tan'e is a s/ort an* /en'e i>0? 9 14 9 1 Am

./en t/e 'ir'!it is swit'/e* to osition 2( t/is 1 Am '!rrent 'onstit!te* t/e store* energy in t/e'oil

.riting t/e alan'e e!ation )or osition 2 we get

Taking ,ala'e trans)ormation347s


85/126

PT

Eamle 2

A series 6- 'ir'!it is s/own in )ig!re T/e 'aa'itor /as an initial '/arge o) #00o!loms on itslates( at t/e time t/e swit'/ is 'lose* Bin* t/e res!lting '!rrent transient

=44 u 14Sol!tion Brom t/e *ata gi@en >0? 9 .riting t/e alan'e e!ation we get

1144 14i

t

\i>tAdt

tA

94e 3?444 t


86/126


87/126

Eamle3Bor t/e 'ir'!it s/own in )ig!re t/e relay 'oil is a*C!ste* to oerate at a '!rrent o) 5 Ams Swit'/ =is 'lose* at t 9 0 an* t/e relay is )o!n* to oerate at t 9 034" se'on*s Bin* t/e @al!e o) in*!'tan'e, o) t/e relaySoln .riting t/e alan'e e!ation )or t/e relay 'ir'!it

5>tA (i>tA Ldi

dtAlying ,ala'e trans)ormation

5 > s A

S(7>SA LS]7>SA i>4A^

Sin'e t/ere is no mention o) initial '!rrent in t/e 'oil i>0? 9014

Hen'e S7>sA 7>sALS

7>SAHSL 1I14

S

14

7>sA 14 L .

S1 SL M 1N

SPSS

R LT

S M 1 N

PS SR L T

14 PS 1 S .SM N

L R L TA910

7>SA

;9 -10

1414

S1

L

Taking In@erse ,ala'e we get

t

i>tA 14 14e L

T/e relay oerates at t 9 034" se'on*s w/en t/e '!rrent @al!e rea'/es 5A Hen'e4/9


88/126

Eamle 4

In )ig!re t/e swit'/ = is 'lose* Bin* t/e time w/en t/e '!rrent in t/e 'ir'!itry rea'/es to 500 mA

Soln ./en t/e swit'/ is 'lose* ' >0? 9 0./en t/e swit'/ is 'lose* at t 9 0

I1 >t?50 9 10

7 >tAu B41

143144 u

14

\

i3dt

Taking ,ala'e )or ot/ t/e e!ations

71>SA

14

?4S

4/ 3

>1A?

7 >SA B41 73 > s A 14

3 uc S ?

B473 >sA73 > S A 14

144 u 14

s

?

7 >SA 14 1 1

E1

>3A3B4S 14

tA 4/31

e 1


89/126

X( Y

3

6-,- Series Transient 'ir'!it

Ass!ming Lero initial 'on*itions w/en swit'/ = is 'lose* t/e alan'e* e!ation is gi@en y

5 i( Ldi

dt

1\idt

CTaking ,ala'e trans)ormation we get5 > s A

s1>sA( LS7>sA

7> s A

CS

7>sAV SL 1 WZCS[ 5>sA

7>sA 5 > s A L

S>( SL1

A S3 (

S1

CS L LCT/e time resonse o) t/e 'ir'!it *een*s on t/e oles or roots o) t/e '/ara'teristi' e!ation

S3

S( 1

4L LC

6oots o) t/e '/ara'teristi' e!ation are gi@en y

(r P ( S


90/126

&lease note t/at t 9 0 in*i'ates t/et 9 0- in*i'ates time imme*iately e)ore t/rowing t/e swit'/ an*


91/126

#0

t 90< in*i'ates time imme*iately a)ter t/rowing t/e swit'/.e are @ery m!'/ intereste* in t/e '/ange in '!rrents an* @oltages o) energy storage elements>in*!'tor an* 'aa'itor? a)ter t/e swit'/ is t/rown sin'e t/ese @ariales along wit/ t/e so!r'es will*i'tate t/e 'ir'!it e/a@ior )or t f 0

Initial 'on*itions in a network *een* on t/e ast /istory o) t/e 'ir'!it >e)ore t9 0-? an*

str!'t!re o) t/e network at t 9 0t? 9 1t

@ *t

0- t, -

9 1, @*t- < 1,t @*t0-9 i>0-? < 1,

0-@*t

&!tting t 9 0or stea*y state?,

'on*ition I

T/e )inal+'on*ition, *i *t

e!i@alent 'ir'!it o) an in*!'tor is *eri@e* )rom t/e asi' relation s/i 9

Qn*er stea*y state 'on*ition *i 9 0 w/i'/ means @ 9 o an* /en'e , a'ts as a s/ort

at t 9 > )inal or stea*y state?I0

S

I, S

At t9

T(e ca!acitr

T/e swit'/ is 'lose* at t 9 0 T/e eression At t90Bor @oltage a'ross t/e

t

@ 9 1 i *t-

'aa'itor is gi@enyi>t?

@

0- t@>t? 9 1 i *t < 1 i *t

-&!tting t9 00< ? 9 @>0-? < 1

0-

0< i *t

>0


93/126

T/e )inal-'on*ition e!i@alent network is *etermine* )rom t/e asi' relations/ii9 *@*t

Qn*er stea*y state 'on*ition *@ *t 9 0 w/i'/ means at t9 t/e aa'itor a'tsas a oen 'ir'!it

8

0- < 0

- -0


95/126

#? A arallel 6-, 'ir'!it is energiLe* y a '!rrent so!r'e o) 1 A t/e swit'/ a'ross t/e so!r'e isoene* at t90 Sol@e )or ( D@ an* D

2@ all at t90< i) 69100:an* ,91H$? Determine t/e T/e@enins e!i@alent a>S? an* :a>S? )or t/e network on Big12 )or Leroinitial 'on*itions

1= Bor t/e 'ir'!it s/own in Big 15 t/e swit'/ is oene* at t90 I) , 9 H(

791m/o( 91B an* 91@( )in* t/e no*e @oltages @1>t? an* @2>t?


96/126

Unit : 7 La!lace Trans0r)atin A!!licatins : Hr s: 07

Syllabus of unit :

Sol!tion o) networks( ste( ram an* im!lse resonses( wa@e)orm Synt/esis







97/126

LAPLACE T,ANS7O,>ATI ON:

,ala'e trans)orm is a @ery !se)!l an* ower)!l tool in 'ir'!it analysisIntegro-*i))erential e!ationsan e trans)orme* in to algerai' e!ations !sing t/e te'/ni!e o) ,ala'e trans)ormation an*'omlete sol!tion in@ol@iong ot/ nat!ral resonse an* )or'e* resonse is otaine* in one ste

e0initin 0 La!lace Trans0r):

,et )>t? e a )!n'tion o) time Ass!ming t/e @al!e o) )!n'tion to e Lero )or t0( t/e,ala'e trans)orm o) )>t? is gi@en as

, ) >t?j 9 B>S? 9 )>t? e-St *t

0)>t? is a )!n'tion in time-*omain an* B>S? is a )!n'tion in 'omle )re!en'y *omain omle)re!en'y S is gi@en y S9 t? `9 B1>S?

An* ,_ )2>t? `9B2>S?

T/en , _a1 )1>t? `< a2 )2>t?`9 a1 B1>S?< a2 B2>S?

&roo) , _a1 )1>t? `< a2 )2>t?`9 _a1 )1>t? `< a2 )2>t? `e-St *t0

9 a1 )1>t? e-St *t t? e-St *t

9 a1 B1>S? < a2 B2 >S?

Ti)e"S(i0tin& Pr!erty: u


98/126

.e get , >t-t0? !>t-to?j 9 >t- t0?e-St *tto

,et t9t0 < *t 9 *As tAs t

t0 0

Hen'e we get , _ >t-t0? !>t-to?`9 >? e-S > < t0? *0

9 e-Sto0

>? e-S *

9 X> S? e-Sto

7reFuency")ain s(i0tin& !r!erty

I) ,_>t?`9 X>S? t/en,_eSot >t? `9 X>S-So?

&roo) ,_eSot >t? `9 eSot >t? e-St *t

0

9 >t? e-> S-So?t0

9 X> S- S0?

*. Ti)e"Scalin& Pr!erty

I) , >t? j 9 X>S? t/en

, >at? j 9 1a X> Sa?

&roo) , >at? j 9

&!t at9 a *t 9 *

>at? e-St *t0

Hen'e , >at?j 9

01a >? e- S a *

Hen'e , >at?j 9 1a

0>? e- S a *

, >at?j 9 1 a X>S a?


99/126


100/126

/. Ti)e"Periicity Pr!erty

>t?

1>t?T 2T 3T

time4T

0 T

2>t?

T 2T

3>t?

2T 3T

,et !s 'onsi*er a B!n'tion >t? t/at is erio*i' as s/own in )ig!reT/e )!n'tion>t? 'an e reresente* as t/e s!m o) time-s/i)te* )!n'tions as s/own in )ig!re

Hen'e >t? 9 1>t? < 2>t? < 3>t? < WWWWWWW./ere 2>t? 9 1>t- T? !>t-T?

3>t? 9 1>t- 2T? !>t-2T?WWWWWWWWWWWWWWWWWWWWWW an* so on

Hen'e >t? 9 1>t? < 1>t- T? !>t-T?< 1>t- 2T? !>t-2T? < WWWWWW./ere 1>t? is t/e wa@e)orm *es'rie* o@er t/e )irst erio* o) >t?Taking ,ala'e trans)ormation on ot/ si*es o) t/e ao@e e!ation we get

X>s? 9 X1>s? < X1>s? e-TS < X1>s? e-2TS < X1>s? e-3TS < WWWWWW


101/126


102/126


103/126

to re*!'e it to roer )ra'tionii? >S? s/o!l* e in t/e )a'tore* )orm

on@ol!tion-integral met/o*

I) , _ >t? ` 9 X>S?

, _ />t? ` 9 H>S?

T/en , _ >t? />t? `9 X>s? H>S?./ere is t/e 'on@ol!tion o) two )!n'tions gi@en y

>t? />t? 9

- > ? /> t- ? *

T(ree i)!rtant Sin&ularity 0unctins :

T/e t/ree imortant sing!larity )!n'tions emloye* in 'ir'!it analysis aret/e !nit ste )!n'tiont/e *elta )!n'tion _>t?t/e ram )!n'tion r>t?

!>t?

T/ey are 'alle* Sing!larity )!n'tions e'a!se t/ey are eit/er not )inite or t/ey *o notosess )inite *eri@ati@e e@eryw/ere

Unit ste! 0unctin:! >t?

T/e Qnit-ste )!n'tion is *e)ine* as !>t? 9 0

9 1

t 0

t f 0

T/e ste )!n'tion 'an /a@e a *is'ontin!iyBor eamle in se!ential swit'/ing T/e !nitSte )!n'tion t/at o''!rs at t9a is eresse* as

!>t-a? w/i'/ is eresse* as

t

!>t-a?

!> t-a ? 9 09 1

>t-a ? 0>t-a?f 0

t

.e !se ste )!n'tion to reresent an ar!t '/ange in @oltage or '!rrent ( like t/e '/anges t/at8''!r in t/e 'ir'!it o) 'ontrol engineering an* *igital systems


104/126

,ala'e trans)ormation o) !nit ste )!n'tion is gi@eny , _ !> t? ` 9 e-St *t0

9 - 1 S e- - e0 j

Similarly ,_ ! >t-a? ` 9

! > t-a? e-St *ta

9 1S

9 -9 1S

e-St

a9 1

Se- a S

I)!ulse 0unctin:

T/e *eri@ati@e o) t/e !nit ste )!n'tion is t/e !nit im!lse )!n'tion _>t?ie _>t?9 **t _ !>t? ` 9 0

9 19 0

t 0

t 9 0t f 0

_>t?

T/e !nit im!lse may e @is!aliLe* as @ery s/ort *!ration !lse o) !nit areaT/is may e eresse* mat/emati'ally as 0t?*t 9 10-

./ere t9 0- in*i'ates t/e time C!st e)ore t90 an* t90< *enotes t/e timeF!st a)ter t90 Sin'e t/e area !n*er t/e !nit im!lse is !nity( it is ra'ti'e to write

1 esi*e t/e arrow ./en t/e im!lse /as a strengt/ ot/er t/an !nity t/e areao) t/e im!lse )!n'tion is e!al to its strengt/

Sin'e _>t?9 **t _ !>t? `

, _ _>t?` 9 , **t _ !>t? `j 9 S X 1 S 9 1

,a)! 0unctin:

Integrating t/e !nit ste )!n'tion res!lts in t/e !nit ram )!n'tion r>t?t

r >t? 9 ! > ? * 9 t ! > t? r>t?- 9 0 t 09 t t f 0

In general a ram is a )!n'tion t/at '/anges at aonstant rate 0 t

A *elaye* ram )!n'tion is s/own in )ig!reat/emati'ally it is *es'rie* as )ollows


105/126

r>t-t0 ? 9 0 t 0r >t- t0 ?

9 t + t0 t f 0

,ala'e trans)ormation o) a ram )!n'tion is gi@en y

t, _ r > t ? ` 9 , ! > t ? *t j

0t 0 t

9 1 S X 1 S 9 1 S2

, _ r > t + t 0 ? ` 9 1 S X 1 S e-t0S

9 1 e- toS


106/126

Assign me nt ue s t ions:1? 8tain t/e ,ala'e trans)orm o) a erio*i' )!n'tion wit/ a s!itale eamle wa@e )orm Also)in* t/e ,ala'e trans)orm o) t/e )ollowing wa@e )orm s/own in Big

$? = State an* ro@e 'on@ol!tion t/eorem Qsing t/e same )in* )>t? w/en B>s? 9

10? Qsing ,ala'e trans)orm *etermine t/e '!rrent in t/e 'ir'!it s/own in Big 25 w/en t/eswit'/ S is 'lose* at t90 Ass!me Lero initial 'on*ition

11? Bin* t/e ,ala'e trans)orm o) i? `>t? ii? t iii? e-at i@? sin w t @?!>t?12? 8tain t/e ,ala'e trans)orm o) sawtoot/ wa@e)orm in t/e Big

13? Bin* t/e ,ala'e in@erse o) !sing 'on@ol!tion integral

14? State an* ro@e >i? initial @al!e t/eorem an* >ii? )inal @al!e t/eorem as alie* to ,trans)orm ./at are t/e limitations o) ea'/ t/eoremZ

#? 8tain t/e ,ala'e trans)orm o) a )!ll wa@e re'ti)ie* sine wa@e o) amlit!*e 1 an* erio* 3

se's


107/126

age a' oss /e 'aa' o o p 0

$? T/e '!rrent )!n'tion i>t? s/own in )ig 10 is imresse* on a 'aa'itor w/at s/o!l* e t/estrengt/ A o) t/e im!lse so t/at t/e @oltage a'ross t/e e'omes Lero )or t f 5se's

1=

9 1In t/e 'ir'!it s/own in )ig 11( t/e swit'/ is oene* at t 9 0( wit/ 9 1( 9 1B , 9 q H( 7)in* t/e no*e @oltages 1>t? an* 2>t? y , trans)orm met/o*

11+ = T/e swit'/ in t/e network o) Big 1" oen at t90 !se ,ala'e trans)ormation analysis to

*etermine t/e @olt r t it r ) r t


108/126

Unit : , T$ !rt net$r% !ara)eters : Hr s: 0#

Syllabus of unit :

De)inition o) L( y( / an* transmission arameters( mo*eling wit/ t/ese arameters(relations/i etween arameters sets







109/126

Network

T P RT PAR A@0 T0R4:

i1 i1 i2< 1 < 0 CNN 0CTI N:'

< I1

1

< I1a

1a

I2a< I1 I2

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