Muhammad Irfan Yousuf Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi Conventions for Describing Networks
2-1. For the controlled (monitored) source shown in the figure, re!re ! lot simil!r to th!t given in Fig. 2-"(b).
v2 v1 # $b $b v1 # $! $!
i2 Fig. 2-" (b) %olution& 'en our book see the figure (*+) t is volt!ge controlled current source.
i2 /$e !0is
v2 -$e !0is gv1
i2
gv1
/ v2 current source -
Muhammad Irfan Yousuf Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 2-2. ee!t *rob. 2-1 for the controlled source given in the !ccom!ning figure. %olution& 'en our book see the figure (*+) t is current controlled volt!ge source. v2 ri1
i2
2-. 3he network of the !ccom!ning figure is ! model for ! b!tter of oen-circuit termin!l volt!ge $ !nd intern!l resist!nce b. For this network, lot i !s ! function v. dentif fe!tures of the lot such !s sloes, intercets, !nd so on. %olution& 'en our book see the figure (*+) 3ermin!l 3e rmin!l volt!ge v # $ - i b i b # $ - v i # ($ - v )+ b 4hen v # 5 i # ($ - v )+ b i # ($ - 5 )+ b i # $+ b !m 4hen v # $ i # ($ - $ )+ b i # (5 )+ b i # 5 !m v#5 i # $+ v#$ i#5 i $+ b
$ %loe&
v
Muhammad Irfan Yousuf Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 2-2. ee!t *rob. 2-1 for the controlled source given in the !ccom!ning figure. %olution& 'en our book see the figure (*+) t is current controlled volt!ge source. v2 ri1
i2
2-. 3he network of the !ccom!ning figure is ! model for ! b!tter of oen-circuit termin!l volt!ge $ !nd intern!l resist!nce b. For this network, lot i !s ! function v. dentif fe!tures of the lot such !s sloes, intercets, !nd so on. %olution& 'en our book see the figure (*+) 3ermin!l 3e rmin!l volt!ge v # $ - i b i b # $ - v i # ($ - v )+ b 4hen v # 5 i # ($ - v )+ b i # ($ - 5 )+ b i # $+ b !m 4hen v # $ i # ($ - $ )+ b i # (5 )+ b i # 5 !m v#5 i # $+ v#$ i#5 i $+ b
$ %loe&
v
Muhammad Irfan Yousuf Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi # m0 / c
(01, 1) # (5, $+ b)
(02, 2) # ($, 5) m # (2 6 1)+(02 6 01) # (5 6 $+ b)+($ - 5) # (-$+ b)+$ # (-$+b)(1+$) # -1+ b -intercet # $+ b 0-intercet # $ %loe -intercet 0-intercet -1+ b $+ b $ 2-. 3he m!gnetic sstem shown in the figure h!s three windings m!rked 1-17, 2-27, !nd -7. 8sing three different forms of dots, est!blish ol!rit m!rkings for these windings. %olution& 'en our book see the figure (*+) 9ets !ssume current in coil 1-17 h!s direction u !t 1 (incre!sing). t roduces flu0 (incre!sing) in th!t core in clockwise direction.
1
17 2
27
7
:ccording to the 9en;7s l!w current roduced in coil 2-27 is in such ! direction th!t it ooses the incre!sing flu0 . %o direction of current in 2-27 is down !t 27. )
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
1
17
2
27
7
27
7
(!)
1
17
2
(b) 2-. 3he figure shows four windings on ! m!gnetic flu0-conducting core. 8sing different sh!ed dots, est!blish ol!rit m!rkings for the windings. %olution& 'en our book see the figure (*+>) i1
i 2
i2 Coil 1
Coil
i
Coil 2 Coil
1
(Follow Fleming7s right h!nd rule) 2->. 3he !ccom!ning schem!tic shows the e?uiv!lent circuit of ! sstem with ol!rit m!rks on the three-couled coils. Dr!w ! tr!nsformer with ! core simil!r to th!t shown for *rob. 2- !nd l!ce windings on the legs of the core in such ! w! !s
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi to be e?uiv!lent to the schem!tic. %how connections between the elements in the s!me dr!wing. %olution& 'en our book see the figure (*+>) 2
i2 9
92
91
φ
1
2
2-". 3he !ccom!ning schem!tics e!ch show two inductors with couling but with different dot m!rkings. For e!ch of the two sstems, determine the e?uiv!lent induct!nce of the sstem !t termin!ls 1-17 b combining induct!nces. %olution& 'en our book see the figure (*+>) 9et ! b!tter be connected !cross it to c!use ! current i to flow. 3his is the c!se of !dditive flu0. M 91
92
$ i (!)
$ # self induced e.m.f. (1) / self induced e.m.f. (2) / mutu!ll induced e.m.f. (1) / mutu!ll induced e.m.f. (2) $ # 91di+dt / 92di+dt / @ di+dt / @ di+dt 9et 9e? be the e?uiv!lent induct!nce then $ # 9 e? di+dt 9e? di+dt # (91 / 92 / @ / @) di+dt
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 9e? # 91 / 92 / @ / @ 9e? # 91 / 92 / 2@ @ 91
92 i
$ (b)
3his is the c!se of subtr!ctive flu0. $ # 91di+dt / 92di+dt - @ di+dt - @ di+dt 9et 9e? be the e?uiv!lent induct!nce then $ # 9 e? di+dt 9e? di+dt # (91 / 92 - @ - @) di+dt 9e? # 91 / 92 - @ - @ 9e? # 91 / 92 - 2@ 2-A. : tr!nsformer h!s 155 turns on the rim!r (termin!ls 1-17) !nd 255 turns on the second!r (termin!ls 2-27). : current in the rim!r c!uses ! m!gnetic flu0, which links !ll turns of both the rim!r !nd the second!r. 3he flu0 decre!ses !ccording to the l!w # e-t 4eber, when t 5. Find& (!) the flu0 link!ges of the rim!r !nd second!r, (b) the volt!ge induced in the second!r. %olution& N1 # 155 N2 # 255 # e-t (t 5) *rim!r flu0 link!ge 1 # N1 # 155 e-t %econd!r flu0 link!ge 2 # N2 # 255 e-t @!gnitude of volt!ge induced in second!r v 2 # d 2+dt # d+dt(255 e -t) v2 # -255 e-t
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
b
2
1
!
c
=
d
d
c
=
R 1
b
2
!
2-11. 3hree gr!hs !re shown in figure. Cl!ssif e!ch of the gr!hs !s l!n!r or nonl!n!r. %olution& 'en our book see the figure (*+") :ll !re l!n!r. n th!t the m! be dr!wn on ! sheet of !er without crossing lines.
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 2-12. For the gr!h of figure, cl!ssif !s l!n!r or nonl!n!r, !nd determine the ?u!ntities secified in e?u!tions 2-1 2-1. %olution& 'en our book see the figure (*+") Cl!ssific!tion& Nonl!n!r Number of br!nches in tree # number of nodes 6 1 # = 6 1 # Number of chords # br!nches 6 nodes / 1 # 15 6 = / 1 # 15 6 # Chord me!ns B: str!ight line connecting two oints on ! curve7. 2-1. n (!) !nd (b) of the figure for *rob. 2-11 !re shown two gr!hs, which m! be e?uiv!lent. f the !re e?uiv!lent, wh!t must be the identific!tion of nodes !, b, c, d in terms of nodes 1, 2, , if ! is identic!l with 1 %olution& 'en our book see the figure (*+") (b) ! is identic!l with 1 b is identic!l with c is identic!l with 2 d is identic!l with 2-1. 3he figure shows ! network with elements !rr!nged !long the edges of ! cube. (!) Determine the number of nodes !nd br!nches in the network. (b) C!n the gr!h of this network be dr!wn !s ! l!n!r gr!h %olution& 'en our book see the figure (*+") Number of nodes # " Number of br!nches # 11 (b) es it c!n be dr!wn. 2-1=. 3he figure shows ! gr!h of si0 nodes !nd connecting br!nches. ou !re to !dd non!r!llel br!nches to this b!sic structure in order to !ccomlish the following different obEectives& (!) wh!t is the minimum number of br!nches th!t m! be !dded to m!ke the resulting structure nonl!n!r (b) 4h!t is the m!0imum number of br!nches ou m! !dd before the resulting structure becomes nonl!n!r %olution& 'en our book see the figure (*+A) @!ke the structure nonl!n!r @inimum number of br!nches # @!0imum number of br!nches # > 2-1. Disl! five different trees for the gr!h shown in the figure. %how br!nches with solid lines !nd chords with dotted lines. (b) ee!t (!) for the gr!h of (c) in *rob. 2-11. %olution& 'en our book see the figure (*+A)
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
1)
2)
)
)
=)
b)& 1)
2)
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
)
)
=)
2-1>. Determine !ll trees of the gr!hs shown in (!) of *rob. 2-11 !nd (b) of *rob. 215. 8se solid lines for tree br!nches !nd dotted lines for chords. %olution& 'en our book see the figure (*+A) :ll trees& 1) 2) ) )
=)
)
>)
")
A)
15)
11)
12)
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
1)
1)
1=)
1)
1>)
1")
1A)
25)
21)
22)
2)
2)
2=)
2)
2>)
2")
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 2A)
5)
:ll trees of
%olution& 1) 2)
) )
efore solving e0ercise following terms should be ket in mind& 1. Node 2. r!nch . 3ree . 3r!nsformer theor =. %loe . %tr!ight line e?u!tion >. ntercet ". %elf induction A. @utu!l induction 15. Current controlled volt!ge source
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 11. $olt!ge controlled current source 12. Coordin!te sstem
A""A# M$#AMMAD %P.&.$.#'
Network e?u!tions -1. 4h!t must be the rel!tionshi between C e? !nd C1 !nd C2 in (!) of the figure of the networks if (!) !nd (c) !re e?uiv!lent ee!t for the network shown in (b). %olution& 'en our book see the figure (*+">) / / + C1 C v(t) i
! kirchhoff7s volt!ge l!w&
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi v(t) # 1+C1 i dt / 1+C2 i dt v(t) # (1+C1 / 1+C2) i dt n second c!se / v(t)
Ce? i
v(t) # 1+Ce? i dt f (!) (c) !re e?uiv!lent 1+Ce? i dt # (1+C1 / 1+C2) i dt 1+Ce? # (1+C1 / 1+C2) (b)
/ i
-
!
C1
/
-
/
i2
C
-
C2
i1
b i # i1 / i2 i # C2dv!+dt / Cdv!+dt when v! is volt!ge !cross !b. 3he e?uiv!lent c!!cit!nce between ! b be C e?7 3hen i # Ce?7dv!+dt ∴ Ce?7dv!+dt # C2dv!+dt / Cdv!+dt Ce?7 # C2 / C Di!gr!m (b) reduces to / / C1
/
v Ce?7 From result obt!ined b (!) 1+Ce? # (1+C1 / 1+Ce?7)
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 1+Ce? # (1+C1 / 1+C2 / C) -2. 4h!t must be the rel!tionshi between 9 e? !nd 91, 92 !nd @ for the networks of (!) !nd of (b) to be e?uiv!lent to th!t of (c) %olution& 'en our book see the figure (*+">) n network (!) !ling G$9 v # 91di+dt / 92di+dt / @di+dt / @di+dt v # (91 / 92 / @ / @)di+dt v # (91 / 92 / 2@)di+dt n network (c) v # 9e?di+dt f (!) (c) !re e?uiv!lent (91 / 92 / 2@)di+dt # 9 e?di+dt (91 / 92 / 2@) # 9e? n network (b) !ling G$9 v # 91di+dt / 92di+dt - @di+dt - @di+dt v # (91 / 92 - @ 6 @)di+dt v # (91 / 92 - 2@)di+dt n network (c) v # 9e?di+dt f (b) (c) !re e?uiv!lent (91 / 92 - 2@)di+dt # 9e?di+dt (91 / 92 - 2@) # 9e? -. ee!t *rob. -2 for the three networks shown in the !ccom!ning figure. %olution& 'en our book see the figure (*+">)
/ @ v i1 loo 1 -
91
92 loo 2
i2
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi :ling G$9 in loo 1 v # 91d(i1 6 i2)+dt / @di2+dt v # 91di1+dt - 91di2+dt / @di2+dt v # 91di1+dt / @di2+dt - 91di2+dt v # 91di1+dt / (@ - 9 1)di2+dt :ling G$9 in loo 2 5 # 92di2+dt / 91d(i2 6 i1)+dt / H-@di2+dtI / H-@d(i2 6 i1)+dtI 5 # 92di2+dt / 91di2+dt - 91di1+dt - @di2+dt - @d(i2 6 i1)+dt 5 # 92di2+dt / 91di2+dt - 91di1+dt - @di2+dt - @di2+dt / @di1+dt 5 # 92di2+dt / 91di2+dt - 91di1+dt - 2@di2+dt / @di1+dt 5 # (@ 6 91) di1+dt / (91 / 92 6 2@) di2+dt 4riting in m!tri0 form 91
@ 6 91
di1+dt
v #
@ 6 91
di1+dt
91 / 92 6 2@
di2+dt
v
@ 6 91
5
91 / 92 6 2@
# 91
@ 6 91
@ 6 91
91 / 92 6 2@
v
@ 6 91
5
91 / 92 6 2@
# (v)( 91 / 92 6 2@) 6 5 # (v)(9 1 / 92 6 2@) 91
@ 6 91
@ 6 91
91 / 92 6 2@
# (91)(91 / 92 6 2@) 6 (@ 6 9 1)(@ 6 91) # (91)(91 / 92 6 2@) 6 (@ 6 9 1)2 # (912 / 9192 6 291@) 6 @2 6 912 / 2@91
5
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi # 912 / 9192 6 291@ 6 @2 6 912 / 2@91 # 9192 6 @2 di1+dt # (v)(91 / 92 6 2@)+9192 6 @2 di1+dt H(9192 6 @2)+(91 / 92 6 2@)I # v n network (c)
v
i1
9e?
v # 9e?di1+dt For (!) (c) to be e?u!l di1+dt H(9192 6 @2)+(91 / 92 6 2@)I # 9e?di1+dt (9192 6 @2)+(91 / 92 6 2@) # 9e? (b) /
@ v i1
91
:ling G$9 in loo 1 v # 91d(i1 6 i2)+dt - @di2+dt v # 91di1+dt - 91di2+dt - @di2+dt v # 91di1+dt / @di2+dt - 91di2+dt v # 91di1+dt - (91 / @)di2+dt :ling G$9 in loo 2 5 # 92di2+dt / 91d(i2 6 i1)+dt / @di2+dt / @d(i2 6 i1)+dt 5 # 92di2+dt / 91di2+dt - 91di1+dt / @di2+dt / @d(i2 6 i1)+dt 5 # 92di2+dt / 91di2+dt - 91di1+dt / @di2+dt / @di2+dt - @di1+dt 5 # 92di2+dt / 91di2+dt - 91di1+dt / 2@di2+dt - @di1+dt 5 # - (91 / @) di1+dt / (91 / 92 / 2@) di2+dt
i2
92
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 4riting in m!tri0 form 91
- (91 / @)
di1+dt
v #
- (91 / @)
di1+dt
91 / 92 / 2@
di2+dt
v
- (91 / @)
5
91 / 92 / 2@
# 91
- (91 / @)
- (91 / @)
91 / 92 / 2@
v
- (91 / @)
5
91 / 92 / 2@
# (v)( 91 / 92 / 2@) 6 5 # (v)(9 1 / 92 / 2@) 91
- (91 / @)
- (91 / @)
91 / 92 / 2@
# (91)(91 / 92 / 2@) - (9 1 / @)(91 / @) # (91)(91 / 92 / 2@) - (9 1 / @)2 # (912 / 9192 / 291@) - @2 - 912 - 2@91 # 912 / 9192 / 291@ - @2 - 912 - 2@91 # 9192 6 @2 di1+dt # (v)(91 / 92 / 2@)+9192 6 @2 di1+dt H(9192 6 @2)+(91 / 92 / 2@)I # v n network (c)
v
i1
9e?
5
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
v # 9e?di1+dt For (!) (c) to be e?u!l di1+dt H(9192 6 @2)+(91 / 92 / 2@)I # 9e?di1+dt (9192 6 @2)+(91 / 92 / 2@) # 9e? -. 3he network of inductors shown in the figure is comosed of ! 1-< inductor on e!ch edge of ! cube with the inductors connected to the vertices of the cube !s shown. %how th!t, with resect to vertices ! !nd b, the network is e?uiv!lent to th!t in (b) of the figure when 9e? # =+ <. @!ke use of smmetr in working this roblem, r!ther th!n writing kirchhoff l!ws. %olution& 1-< 'en our book see the figure (*+"") 1-< 1-< 1-< 1-< 1-<
1-<
1
17 1-<
1-< 1-<
1-<
i+ i+
i+ i+
i
i+
i
i+
i+2
i+2 i+2
1
i+
i+ i+
i+
i
i+
i+2
i 17
i+
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 1+-<
1+-<
1+-<
9e? # 1+-< / 1+-< / 1+-< # =+-< -=. n the networks of *rob. -, e!ch 1-< inductor is rel!ced b ! 1-F c!!citor, !nd 9e? is rel!ced b C e?. 4h!t must be the v!lue of C e? for the two networks to be e?uiv!lent %olution& 'en our book see the figure (*+"")
1!(
1!(
1!(
1!(
1
1) 1!(
1!(
Ce? # 1+ / 1+ / 1+ # 1.2 F -. 3his roblem m! be solved using the two kirchoff l!ws !nd volt!ge current rel!tionshis for the elements. :t time t 5 !fter the switch k w!s closed, it is found th!t v2 # /= $. ou !re re?uired to determine the v!lue of i 2(t5) !nd di2(t5)+dt. node 1 / 15$
G
/ 1
-
2
1
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
i2 v2
1+2h
8sing kirchhoff7s current l!w !t node 1 v2 6 15+1 / v 2+2 / i2 # 5 v2 6 15 / v2+2 / i2 # 5 v2+2 / i2 # 15 i2 # 15 6 v2+2 !t t # t5 i2(t5) # 15 6 v2(t5)+2 i2(t5) # 15 6 (=)+2 # 2.= !m. :lso v2 v2 # i2(1) / 9di2+dt v2 # i2(1) / (1+2)di2+dt di2+dt # (v2 6 i2)(2) di2(t5)+dt # (v2(t5) 6 i2(t5))(2) # (= 6 2.=)(2) # (2.=)(2) # = !m+sec.
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
->. 3his roblem is simil!r to *rob. -. n the network given in the figure, it is given th!t v 2(t5) # 2 $, !nd (dv2+dt)(t 5) # -15 $+sec, where t 5 is the time !fter the switch G w!s closed. Determine the v!lue of C. %olution&
/ $
v2 /
2
-
1
C
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
! 8sing kirchhoff7s current l!w !t node v2 6 +2 / v2+1 / ic # 5 v2+2 / ic # +2 :t t # t5 v2(t5)+2 / ic(t5) # +2 (2)+2 / ic(t5) # +2 ic(t5) # - +2 !lso !t t # t5 ic(t5) # cdv2(t5)+dt - +2 # c(- 15) c # +25 5.1=-F 3he series of roblems described in the following t!ble !ll ert!in to the network of (g) of the figure with the network in : !nd secified in the t!ble.
-" (!) %olution&
/ v1
2 v2 J-h
'en our book see (*+"A) v2(t) v2(t) v2(t) v2(t) :ling G$9
5 1 5 2
5KtK1 1KtK2 2KtK KtK
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi v1 # 2(i) / (1+2)di+dt # 2(i) / v 2 v2 # (1+2)di+dt t i # 2 v2dt t 5 t i # 2 v2dt # 2 v2dt / 2 v2dt 5 t i(t) # 5 / 5dt # 5 !m. 5
5KtK1
t 1 t i # 2 v2dt # 2 v2dt / 2 v2dt 1 t t i(t) # i(1) / 2 (1)dt # 5 / 2 t 1 1 i(t) # 2(t - 1) !m.
1KtK2
2KtK
KtK
t 2 t i # 2 v2dt # 2 v2dt / 2 v2dt 2 t i(t) # i(2) / 5dt # 2 / 5 # 2 !m. 2
t t i # 2 v2dt # 2 v2dt / 2 v2dt t t i(t) # i() / 2 (2)dt # 2 / t i(t) # 2 / (t - ) !m. i() # 2 / ( - ) !m.
:t t #5 i(5) # 5 :t t #1 i(1) # 5 :t t #1 i(1) # 5 :t t #2 i(2) # 2
:t t # 2 i(2) # 2 :t t # i() # 2
:t t # i() # :t t # i() #
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi # !m.
v2(t) # 5
v1(t) # 2(i(t)) / v 2(t) :t t # 5 v1(5) # 2(i(5)) / v 2(5) v1(5) # 2(5) / 5 # 5 :t t # 1 v1(1) # 2(i(1)) / v 2(1) v1(1) # 2(5) / 5 # 5
1
v1(t) # 2(i(t)) / v 2(t) :t t # 2 v1(2) # 2(i(2)) / v 2(2) v1(2) # 2(2) / 1 # =
5
v1(t) # 2(i(t)) / v 2(t) :t t # v1() # 2(i()) / v 2() v1() # 2(2) / 5 #
2
v1(t) # 2(i(t)) / v 2(t) :t t # v1() # 2(i()) / v 2() v1() # 2() / 2 # 1
5KtK1
1KtK2
2KtK
KtK
v1(5) v1(1) v1(2) v1() v1()
5 5 = 1
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
4.5 4
4
3.5 3
e g a t l o v
3
2.5 Series2 2
2
1.5 1
1
0.5 0
Series2
0 1
2
3
4
5
0
1
2
3
4
time
-" (b)
v2 2
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 5
1
2
nterv!l 5KtK1 1KtK2 2KtK KtK tL
time
v2(t) 2t -2(t 6 2) 2(t 6 2) -2(t 6 ) 5
5KtK1 (01, 1) # (1, 2)
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
(05, 5) # (5, 5) %tr!ight-line e?u!tion # m0 / c m # (1 6 5)+(01 6 05) # (2 - 5)+(1 - 5) # 2+1 # 2 %loe # 2 -intercet # 5 # m0 / c v2(t) # 2t / 5 # 2t $olts
1KtK2 (02, 2) # (1, 2)
(01, 1) # (2, 5) %tr!ight-line e?u!tion # m0 / c m # (2 6 1)+(02 6 01) # (2 - 5)+(1 - 2) # 2+(-1) # - 2 %loe # - 2 -intercet # # m0 / c v2(t) # - 2t / # - 2(t 6 2) $olts
2KtK (0, ) # (, 2)
(02, 2) # (2, 5)
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi %tr!ight-line e?u!tion # m0 / c m # ( 6 2)+(0 6 01) # (2 - 5)+( - 2) # 2+1 # 2 %loe # 2 -intercet # - # m0 / c v2(t) # 2t / (-) # 2t 6 # 2(t - 2) $olts
1KtK2 (0, ) # (, 2)
(0, ) # (, 5) %tr!ight-line e?u!tion # m0 / c m # ( 6 )+(0 6 0) # (2 - 5)+( - ) # 2+(-1) # - 2 %loe # - 2 -intercet # " # m0 / c v2(t) # - 2t / " # - 2(t 6 ) $olts v1 # v2 / 2i v2 # (1+2)di+dt t i # 2 v2dt -
5KtK1
t 5 t i # 2 v2dt # 2 v2dt / 2 v2dt 5 t t i(t) # 5 / 2 2tdt # tdt 5 5 t 2 # t +2 5 2 2 i(t) # Mt +2 - 5 # Mt +2 # 2t2 !m.
:t t # 5 i(5) # 5 :t t # 1 i(1) # 2
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
1KtK2
2KtK
t 1 t i # 2 v2dt # 2 v2dt / 2 v2dt 1 t i(t) # i(1) / 2 -2(t - 2)dt 1 t i(t) # 2 / (-) (t - 2)dt 1 t i(t) # 2 - (t - 2)dt 1 t 2 i(t) # 2 - t +2 6 2t 1 2 i(t) # 2 - M(t +2 6 2t) 6 (1+2 - 2) i(t) # 2 - M(t 2+2 6 2t) 6 (- +2) i(t) # 2 - Mt 2+2 6 2t / +2) i(t) # 2 6 2t 2 / "t 6 i(t) # 6 2t2 / "t 6
t 2 t i # 2 v2dt # 2 v2dt / 2 v2dt 2 t i(t) # i(2) / 2 2(t - 2)dt 2 t i(t) # / (t - 2)dt 2
:t t # 2 i(2) # !m.
:t t # i() # !m.
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi t i(t) # / (t - 2)dt 1 t i(t) # / t +2 6 2t 2 2 i(t) # / M(t +2 6 2t) 6 (+2 - ) i(t) # / M(t 2+2 6 2t) 6 (- 2) i(t) # / Mt 2+2 6 2t / 2) i(t) # / 2t 2 - "t / " i(t) # 2t2 - "t / 12 2
KtK
t t i # 2 v2dt # 2 v2dt / 2 v2dt t i(t) # i() / 2 -2(t - )dt t i(t) # - (t - )dt t i(t) # - (t - )dt t i(t) # - t2+2 6 t 2 i(t) # - M(t +2 6 t) 6 (.= - 12) i(t) # - M(t 2+2 6 t) 6 (- >.=) i(t) # - Mt 2+2 6 t / >.=) i(t) # - 2t 2 / 1t - 5 i(t) # - 2t2 / 1t 6 2
v2(t) # 2t
:t t # i() # " !m.
v1(t) # 2(i(t)) / v 2(t) :t t # 5 v1(5) # 2(i(5)) / 2t
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
v1(5) # 2(5) / 5 # 5 :t t # 1 v1(1) # 2(i(1)) / 2t v1(1) # 2(5) / 2(1) # 2
5KtK1
1
v1(t) # 2(i(t)) / v 2(t) :t t # 2 v1(2) # 2(i(2)) 6 2(t - 2) v1(2) # 2() - 5 # "
5
v1(t) # 2(i(t)) / v 2(t) :t t # v1() # 2(i()) / 2(t - 2) v1() # 2() / 2 # 1
2
v1(t) # 2(i(t)) / v 2(t) :t t # v1() # 2(i()) 6 2(t - ) v1() # 2(") - 5 # 1
1KtK2
2KtK
KtK
v1(5) v1(1) v1(2) v1() v1()
5 2 " 1 1
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 4.5 4
4
3.5 3
e g a t l o v
3
2.5 Series2 2
2
1.5 1
1
0.5 0
Series2
0 1
2
3
4
5
0
1
2
3
4
time
5KtK1 1KtK2 2KtK tL :ling G$9 v1 # 2(i) / (1+2)di+dt # 2(i) / v 2 v2 # (1+2)di+dt t i # 2 v2dt -
$2 # 5 2 - 5
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
5KtK1
1KtK2
2KtK
t 5 t i # 2 v2dt # 2 v2dt / 2 v2dt 5 t i(t) # 5 / 5dt # 5 !m. 5
t 1 t i # 2 v2dt # 2 v2dt / 2 v2dt 1 t t i(t) # i(1) / 2 (2)dt # 5 / t 1 1 i(t) # (t - 1) !m.
t 2 t i # 2 v2dt # 2 v2dt / 2 v2dt 2 t t i(t) # i(2) / 2 (-)dt # - t 2 2 # 6 (t - 2) !m.
:t t # 5 i(5) # 5 :t t # 1 i(1) # 5
:t t # 2 i(2) #
:t t # i() # -2
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
v2(t) # 5
v1(t) # 2(i(t)) / v 2(t) :t t # 5 v1(5) # 2(i(5)) / v 2(5) v1(5) # 2(5) / 5 # 5 :t t # 1 v1(1) # 2(i(1)) / v 2(1) v1(1) # 2(5) / 5 # 5
2
v1(t) # 2(i(t)) / v 2(t) :t t # 2 v1(2) # 2(i(2)) / v 2(2) v1(2) # 2() / 2 # 15
-
v1(t) # 2(i(t)) / v 2(t) :t t # v1() # 2(i()) / v 2() v1() # 2(-2) - # ->
5KtK1
1KtK2
2KtK
v1(5) v1(1) v1(2) v1()
5 5 15 ->
15
10
e g a t l o v
10
5
0
1 0
0
3
2
Series2
-5 -7 -10 1
2
3
4
Series1
0
0
10
-7
Series2
0
1
2
3
time
Series1
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
-" (d) 5KtK , v2 # sint :ling G$9 v1 # 2(i) / (1+2)di+dt # 2(i) / v 2 v2 # (1+2)di+dt t i # 2 v2dt t 5 t i # 2 v2dt # 2 v2dt / 2 v2dt 5 t i(t) # 5 / 2 sintdt 5 t i(t) # - 2 cost # - 2(cost - 1) !m. 5 ec!use cos5 # 1
5KtK
v1 # 2(i) / v 2 v1(t) # 2(i(t)) / sint :t t # 5 v1(5) # 2(i(5)) / sin5 v1(5) # 2(5) / 5 # 5 $olt :t t # v1(t) # 2(i(t)) / sint v1( ) # 2(i( )) / sin v1( ) # 2() / 5 # " $olt v1(5) v1( )
5 "
:t t # 5 i(5) # 5 :t t # 1 i(1) # 5
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 10
8
e g a t l o v
8
6
Series1 Series2
4
2 1 0
0
0
1
2
Series1
0
0
Series2
1
8 time
-" (f) 5KtK1 1KtK KtK
2t 2 -2(t - )
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi v2
2
5
1
2
5KtK1 (01, 1) # (1, 2)
(05, 5) # (5, 5) %tr!ight-line e?u!tion # m0 / c m # (1 6 5)+(01 6 05) # (2 - 5)+(1 - 5) # 2+1 # 2 %loe # 2 -intercet # 5 # m0 / c
time
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi v2(t) # 2t / 5 # 2t $olts 1KtK v2(t) # 2 $olts
1KtK2 (0, ) # (, 2)
(0, ) # (, 5) %tr!ight-line e?u!tion # m0 / c m # ( 6 )+(0 6 0) # (2 - 5)+( - ) # 2+(-1) # - 2 %loe # - 2 -intercet # " # m0 / c v2(t) # - 2t / " # - 2(t 6 ) $olts v1 # v2 / 2i v2 # (1+2)di+dt t i # 2 v2dt -
5KtK1
1KtK
t 5 t i # 2 v2dt # 2 v2dt / 2 v2dt 5 t t i(t) # 5 / 2 2tdt # tdt 5 5 t 2 # t +2 5 2 2 i(t) # Mt +2 - 5 # Mt +2 # 2t2 !m. t 1 t i # 2 v2dt # 2 v2dt / 2 v2dt 1 t
:t t # 5 i(5) # 5 :t t # 1 i(1) # 2
:t t # i() # 15 !m.
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi i(t) # i(1) / 2 2dt 1 t i(t) # 2 / dt 1 t i(t) # 2 / t 1 i(t) # 2 / (t 6 1) # 2 / t 6 # -2 / t
KtK
t t i # 2 v2dt # 2 v2dt / 2 v2dt t i(t) # i() / 2 -2(t - )dt t i(t) # 15 - (t - )dt t i(t) # 15 - (t - )dt t 2 i(t) # 15 - t +2 6 t 2 i(t) # 15 - M(t +2 6 t) 6 (.= - 12) i(t) # 15 - M(t2+2 6 t) 6 (- >.=) i(t) # 15 - Mt 2+2 6 t / >.=) i(t) # 15 - 2t2 / 1t - 5 i(t) # - 2t2 / 1t 6 25
:t t # i() # 12 !m.
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
v2(t) # 2t
5KtK1
2
v1(t) # 2(i(t)) / v 2(t) :t t # v1() # 2(i()) / 2 v1(2) # 2(15) / 2 # 22
-2(t - )
v1(t) # 2(i(t)) / v 2(t) :t t # v1() # 2(i()) - 2(t - ) v1() # 2(12) - 5 # 2 $olts
1KtK
KtK
v1(5) v1(1) v1() v1()
v1(t) # 2(i(t)) / v 2(t) :t t # 5 v1(5) # 2(i(5)) / 2t v1(5) # 2(5) / 5 # 5 :t t # 1 v1(1) # 2(i(1)) / 2t v1(1) # 2(2) / 2(1) # $olts
5 22 2
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 30
25
24 22
20 s t l o v
15
Series2
10 6
5
0
Series2
0 1
2
3
4
0
6
22
24
time
-" (e)
v2 /1
time -1
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi nterv!l 5KtK1 1KtK2 2KtK KtK
v2(t) 5 1 5 -1
:ling G$9 v1 # 2(i) / (1+2)di+dt # 2(i) / v 2 v2 # (1+2)di+dt t i # 2 v2dt -
5KtK1
1KtK2
t 5 t i # 2 v2dt # 2 v2dt / 2 v2dt 5 t i(t) # 5 / 5dt # 5 !m. 5
t 1 t i # 2 v2dt # 2 v2dt / 2 v2dt 1 t t i(t) # i(1) / 2 (1)dt # 5 / 2 t 2 1 i(t) # 2(t - 1) !m.
:t t # 5 i(5) # 5 :t t # 1 i(1) # 5
:t t # 2 i(2) # 2
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
2KtK
KtK
t 2 t i # 2 v2dt # 2 v2dt / 2 v2dt 2 t i(t) # i(2) / 2 5dt # 2 2 # 2 !m.
t t i # 2 v2dt # 2 v2dt / 2 v2dt t i(t) # i() / 2 (-1)dt t t i(t) # 2 - 2 (1)dt # 2 - 2 t # 2 6 2(t 6 ) !m.
:t t # i() # 2
:t t # i() # 5
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi v2(t) # 5
v1(t) # 2(i(t)) / v 2(t) :t t # 5 v1(5) # 2(i(5)) / v 2(5) v1(5) # 2(5) / 5 # 5 :t t # 1 v1(1) # 2(i(1)) / v 2(1) v1(1) # 2(5) / 5 # 5
1
v1(t) # 2(i(t)) / v 2(t) :t t # 2 v1(2) # 2(i(2)) / v 2(2) v1(2) # 2(2) / 1 # =
5
v1(t) # 2(i(t)) / v 2(t) :t t # v1() # 2(i()) / v 2() v1() # 2(2) / 5 #
-1
v1(t) # 2(i(t)) / v 2(t) :t t # v1() # 2(i()) / v 2() v1() # 2(5) - 1 # -1
5KtK1
1KtK2
2KtK
KtK
v1(5) v1(1) v1(2) v1() v1()
5 5 = -1
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 6 5
5
4
4
3 e g a t l o v
2
Series2
1 0
0
0
-1
-1
-2
Series2
1
2
3
4
5
0
0
5
4
-1
time
-A (!) %olution& / J-F
/
v1 -
v2 1-h
-
v1 # vc / v2 t vc # (1+c) i(t)dt v2(t) v2(t) v2(t) v2(t) i(5) i(1) i(2) ()
5 1 5 2
5KtK1 1KtK2 2KtK KtK
5KtK1 5KtK1 1KtK2 2KtK
5 5 2 2
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi ()
KtK
5KtK1
1KtK2
2KtK
KtK
t 5 t vc # 2 idt # 2 idt / 2 idt 5 t vc # 5 / 2 5dt # 5 $olts. 5
:t t # 5 vc(5) # 5 :t t # 1 vc(1) # 5
t 1 t vc # 2 idt # 2 idt / 2 idt 1 t t vc(t) # vc(1) / 2 (2)dt # 5 / t 1 1 vc(t) # (t - 1) $olts.
:t t # 2 vc(2) #
t 2 t vc # 2 idt # 2 idt / 2 idt 2 t t vc(t) # vc(2) / 2 (2)dt # / t 2 2 vc(t) # / (t - 2) $olts.
:t t # vc() # "
t t vc # 2 idt # 2 idt / 2 idt t t vc(t) # vc() / 2 ()dt # " / 12 t vc(t) # " / 12(t - ) $olts.
:t t # vc() # 25
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
t#5 1 2
vc(t)
#5 5 " 25
v2(t) # 5
v1(t) # vc(t) / v2(t) :t t # 5 v1(5) # vc(5) / v2(5) v1(5) # (5) / 5 # 5 $olts :t t # 1 v1(1) # vc(1) / v2(1) v1(5) # (5) / 5 # 5 $olts
1
:t t # 2 v1(2) # vc(2) / v2(2) v1(5) # () / 1 # = $olts
5
:t t # v1() # vc() / v2() v1(5) # (") / 5 # " $olts
2
:t t # v1() # vc() / v2() v1(5) # (25) / 2 # 22 $olts
5KtK1
1KtK2
2KtK
KtK
v1(5) v1(1) v1(2) v1() v1()
5 5 = " 22
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 25 22 20
15
e g a t l o v
Series2 10 8 5
5
0
Series2
0
0
1
2
3
4
5
0
0
5
8
22
time
-A (b) v1 # vc / v2 t vc # (1+c) i(t)dt v2(t) v2(t) v2(t) v2(t) i(5) i(1) i(2) i() i()
2t -2(t 6 2) 2(t - 2) -2(t - ) 5KtK1 5KtK1 1KtK2 2KtK KtK
5KtK1
t 5 t vc # 2 idt # 2 idt / 2 idt 5 t vc # 5 / 2 5dt # 5 $olts. 5 t 5 t vc # 2 idt # 2 idt / 2 idt 5 t t vc # 5 / 2 2dt # t # t 5 5
5KtK1 1KtK2 2KtK KtK 5 2 " :t t # 5 vc(5) # 5 :t t # 1 vc(1) #
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
1KtK2
2KtK
KtK
t 1 t vc # 2 idt # 2 idt / 2 idt 1 t t vc(t) # vc(1) / 2 ()dt # / " t 1 1 vc(t) # / "(t - 1) $olts.
:t t # 2 vc(2) # 12
t 2 t :t t # vc() # 2 vc # 2 idt # 2 idt / 2 idt 2 t t vc(t) # vc(2) / 2 ()dt # 12 / 12 t 2 2 vc(t) # 12 / 12(t - 2) $olts.
t t vc # 2 idt # 2 idt / 2 idt t t vc(t) # vc() / 2 (")dt # 2 / 1 t vc(t) # 2 / 1(t - ) $olts.
:t t # vc() # 5
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi t#5 1 2
vc(t)
v2(t) # 2t
5KtK1
v1(t) # vc(t) / v2(t) v1(t) # 6 2(t 6 2) :t t # 2 v1(2) # $olts.
2(t 6 2)
:t t # v1() # vc() / v2() v1() # (") / 2(t - 2) # 15 $olts :t t # v1() # vc() / v2() v1(5) # (25) 6 2(t - ) # 25 $olts
2KtK -2(t - )
v1(5) v1(1) v1(2) v1() v1()
v1(t) # vc(t) / v2(t) v1(t) # 5 / 2t # 2t :t t # 5 v1(5) # 5 / 2t # 2(5) # 5 $olts. v1(t) # vc(t) / v2(t) v1(t) # 5 / 2t # 2t :t t # 1 v1(1) # 5 / 2t # 2(1) # 2 $olts.
-2(t 6 2)
1KtK2
KtK
#5 5 " 25
5 2 15 25
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
5
20
4
e m i t
10
3
4
2
Series2
2
1
0
0
Series2
5
10
15
20
25
1
2
3
4
5
0
2
4
10
20
voltage
-A (c) v1 # vc / v2 t vc # (1+c) i(t)dt v2(t) v2(t) v2(t) i(5) i(1) i(2) i()
5KtK1
5 2 -
5KtK1 1KtK2 2KtK
5KtK1 5KtK1 1KtK2 2KtK
5 5 2
t 5 t vc # 2 idt # 2 idt / 2 idt 5 t vc # 5 / 2 5dt # 5 $olts. 5 t 5 t vc # 2 idt # 2 idt / 2 idt 5 t vc # 5 / 2 5dt # 5 $olts 5
:t t # 5 vc(5) # 5 :t t # 1 vc(1) # 5
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
1KtK2
2KtK
t#5 1 2
t 1 t vc # 2 idt # 2 idt / 2 idt 1 t t vc(t) # vc(1) / 2 ()dt # 5 / " t 1 1 vc(t) # 5 / "(t - 1) $olts.
:t t # 2 vc(2) # "
t 2 t vc # 2 idt # 2 idt / 2 idt 2 t vc(t) # vc(2) / 2 (2)dt # " / t 2 vc(t) # " / (t - 2) $olts.
:t t # vc() # 12
vc(t)
#5 5 " 12
t 2
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi v2(t) # 5
v1(t) # vc(t) / v2(t) v1(t) # 5 / 5 # 5 :t t # 5 v1(5) # 5 $olts. v1(t) # vc(t) / v2(t) v1(t) # 5 / 5 # 5 :t t # 1 v1(1) # 5 $olts.
5KtK1
2
v1(t) # vc(t) / v2(t) v1(t) # " / 2 # 15 :t t # 2 v1(2) # 15 $olts.
-
:t t # v1() # vc() / v2() v1() # (12) - # A $olts
1KtK2
2KtK
v1(5) v1(1) v1(2) v1()
5 5 15 A
12
10
10 9
8 e g a t l o v
6
Series2
4
2
0
Series2
0
0
1
2
3
4
0
0
10
9
time
-A (d) v1 # vc / v2 t vc # (1+c) i(t)dt v2(t)
Muhammad Irfan Yousuf Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
i(5)
5
5KtK 5KtK
i( )
t 5 t vc # 2 idt # 2 idt / 2 idt 5 t vc # 5 / 2 5dt # 5 $olts. 5 t 5 t vc # 2 idt # 2 idt / 2 dt 5 t t vc # 5 / " dt # " t # "t $olts 5 5
5KtK
t#5
vc(t)
v2(t) # sint
:t t # 5 vc(5) # 5 :t t # π vc(1) # " $olts
#5 "
v1(t) # vc(t) / v2(t) v1(t) # vc(t) / sint # 5 :t t # 5 v1(5) # 5 / sin5 # 5 $olts.
Muhammad Irfan Yousuf Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
v1(t) # vc(t) / v2(t) v1(t) # " / sint :t t # π v1( ) # " / sin # " $olts
5KtK
v1(5) v1( )
5 "
30
25.133
25
20 e g a t l o v
Series1 15 Series2 10
5 1 0
0
0
1
2
Series1
0
0
Series2
1
25.133 time
-A (e) v1 # vc / v2 t vc # (1+c) i(t)dt -
v2(t) v2(t) v2(t) v2(t)
5 1 5 -1
5KtK1 1KtK2 2KtK KtK
i(5)
5KtK1
5
Muhammad Irfan Yousuf Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi i(1) i(2) i() i()
5KtK1 1KtK2 2KtK KtK
5KtK1
1KtK2
2KtK
5 2 2 5 :t t # 5 vc(5) # 5 :t t # 1 vc(1) # 5
t 5 t vc # 2 idt # 2 idt / 2 idt 5 t vc # 5 / 2 5dt # 5 $olts. 5 t 5 t vc # 2 idt # 2 idt / 2 idt 5 t vc # 5 / 2 5dt # 5 $olts. 5
t 1 t vc # 2 idt # 2 idt / 2 idt 1 t t vc(t) # vc(1) / 2 (2)dt # 5 / t 1 1 vc(t) # 5 / (t - 1) $olts.
:t t # 2 vc(2) #
t 2 t vc # 2 idt # 2 idt / 2 idt 2 t
:t t # vc() # " t
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi vc(t) # vc(2) / 2 (2)dt # / t 2 vc(t) # / (t - 2) $olts.
t t vc # 2 idt # 2 idt / 2 idt t vc(t) # vc() / 2 (5)dt # " vc(t) # " $olts.
KtK
t#5 1 2
vc(t)
v2(t) # 5
5KtK1
1
1KtK2
2
:t t # vc() # " $olts
#5 5 " "
v1(t) # vc(t) / v2(t) v1(t) # 5 / 5 # 5 $olts :t t # 5 v1(5) # 5 $olts. v1(t) # vc(t) / v2(t) v1(t) # 5 / 5 # 5 :t t # 1 v1(1) # 5 $olts.
v1(t) # vc(t) / v2(t) v1(t) # / 1 # = :t t # 2 v1(2) # = $olts.
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
5
:t t # v1() # vc() / v2() v1() # (") / 5 # " $olts
-1
:t t # v1() # vc() / v2() v1() # " 6 1 # > $olts
2KtK
KtK
v1(5) v1(1) v1(2) v1() v1()
5 5 = " >
9
8
4
7
5
6
5
3
Series2 Series3
4
3
2
1
1
0
1
2
1
2
3
4
5
Series2
0
0
5
8
7
Series3
1 time
-1>. For e!ch of the four networks shown in the figure, determine the number of indeendent loo currents, !nd the number of indeendent node-to-node volt!ges th!t m! be used in writing e?uilibrium e?u!tions using the kirchhoff l!ws. %olution& 'en our book see (*+A5) (!) Number of indeendent loos # 2
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi Node-to-node volt!ges # (b) Number of indeendent loos # 2 Node-to-node volt!ges # (c) Number of indeendent loos # 2 Node-to-node volt!ges # (d) Number of indeendent loos # Node-to-node volt!ges # > -1". ee!t *rob. -1> for e!ch of the four networks shown in the figure on !ge A1. (e) Number of indeendent loos # > Node-to-node volt!ges # (f) Number of indeendent loos # Node-to-node volt!ges # = (g) Number of indeendent loos # Node-to-node volt!ges # = (h) Number of indeendent loos # = Node-to-node volt!ges # -1A. Demonstr!te the e?uiv!lence of the networks shown in figure -1> !nd so est!blish ! rule for converting ! volt!ge source in series with !n inductor into !n e?uiv!lent network cont!ining ! current source. %olution& 'en our book re!d !rticle source tr!nsform!tion (*+=>). -25. Demonstr!te th!t the two networks shown in figure -1" !re e?uiv!lent. %olution& 'en our book re!d (*+5). -21. 4rite ! set of e?u!tions using the kirchhoff volt!ge l!w in terms of !rori!te loo-current v!ri!bles for the four networks of *rob. -1>. (!) i1& 2i1 / 1+c (i1 6 i2) dt # 5 i2& v(t) # i2 1 / 1+c (i2 6 i1) dt / 9di 2+dt / i2 (b) i1& 1i1 / 9d(i1 6 i2)+dt # v(t) i2& 5 # i2 2 / 1+c i2 dt / 9d(i 2 6 i1)+dt (c) i1& (i1 6 i2) / 9di1+dt # v(t) i2& 5 # (i2 6 i1) / 1+c i2 dt (d) i1& 91d(i1 6 i)+dt / 1+c1 i1dt # 5 i2&
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
1i2 / 92d(i2 6 i)+dt / 1+c2 (i2 6 i)dt # 5 i& 91d(i 6 i1)+dt / 92d(i 6 i2)+dt / (i 6 i) # v(t) i& 2i / (i 6 i) / 1+c2 (i 6 i2)dt # 5 -22. @!ke use of the G$9 to write e?u!tions on the loo b!sis for the four networks of *rob. -1". %olution& 'en our book see (*+A1). (!) i1& 1i1 / 1+c (i1 6 i2)dt # - v(t) i2& 1+c (i2 6 i1)dt / 1(i2 6 i) # 5 i& 1+c1 idt / 1(i 6 i2) / (i 6 i) # 5 i& 1+c (i 6 i=)dt / 2(i 6 i) # 5 i=& 2i= / 1+c (i= 6 i)dt / 1+c2 i=dt # - v(t) i& 2(i 6 i=) / (i 6 i>) # - v(t) i>& 1+c= i>dt / (i> 6 i) # 5 (b) i1& 92di1+dt / 1+c1 (i1 6 i2)dt / 1+c (i1 6 i)dt / 9d(i1 6 i)+dt # v(t) i2& 91di2+dt / 1+c2 (i2 6 i)dt / 1+c1 (i2 6 i1)dt # 5 i& 9di+dt / 1+c2 (i 6 i2)dt / 1+c (i 6 i1)dt / 9d(i 6 i1)+dt / i# 5 (c) i1& 1+c (i1 6 i)dt / 1(i1 6 i2) # v(t) i2& 1+c (i2 6 i)dt / 1(i2 6 i1) / 9(i2 6 i) # 5 i& i / (i 6 i) / 1+c (i 6 i2)dt / 1+c (i 6 i1)dt # 5 i& 9(i 6 i2) / (i 6 i) / 1+c1 idt # 5 (d) i1& 1+c! (i1 6 i2)dt / 291di1+dt / 9bd(i1 6 i)+dt / 1+cb (i1 6 i)dt # v(t) i2&
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 9!d(i2 6 i)+dt / 1+c! (i2 6 i1)dt # 5 i& 292d(i 6 i)+dt / (i 6 i) / 1+c! (i 6 i=)dt / 9bd(i 6 i1)+dt / 1+cb (i 6 i1)dt # 5 i& 9!d(i 6 i2)+dt / 9bdi+dt / 1+cb idt / 292d(i 6 i)+dt / (i 6 i) # 5 (!)
/ -
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi i1
i2
i
i
i=
/ i
i>
(b)
i2
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
/ i1 i -
(c)
i
/
-
i
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
i1
i2
(d)
i2
i
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
/ i1 i
-
i=
-2. 4rite ! set of e?uilibrium e?u!tions on the loo b!sis to describe the network in the !ccom!ning figure. Note th!t the network cont!ins one controlled source. Collect terms in our formul!tion so th!t our e?u!tions h!ve the gener!l form of O?s. (>).
i2
/
-
! +
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi i1 i
i1& i1 / (i1 6 i2) / (i1 6 i) / 1 (i1 6 i)dt # v1(t) i2& 1(i2 6 i1) / 9di2+dt # 5 i& i / (i 6 i1) / 1 (i 6 i1)dt 6 k 1i1# 5 -2. For the couled network of the figure, write loo e?u!tions using the G$9. n our formul!tion, use the three loo currents, which !re identified. %olution& 'en our book see (*+A2). i1& 1i1 / (91 / 92)di1+dt / @di2+dt # v1 i2& 9di2+dt / @di1+dt / 1+c (i2 6 i)dt # v2 i& 2i / 1+c (i 6 i2)dt # 5 -2=. 8sing the secified currents, write the G$9 e?u!tions for this network. %olution& 'en our book see (*+A2). i1& 1(i1 / i2 / i) / 91di1+dt / @12di2+dt / 2i1 - @1di2+dt # v1(t) i2& 1(i1 / i2 / i) / 92di2+dt / @12di1+dt / @2di2+dt # v1(t) i& 1(i1 / i2 / i) / 9di+dt - @1di1+dt / @2di2+dt / 1+c idt# v1(t) -2. : network with m!gnetic couling is shown in figure. For the network, @ 12 # 5. Formul!te the loo e?u!tions for this network using the G$9. i1& 1i1 / 91di1+dt / @1d(i1 6 i2)+dt / 9d(i1 6 i2)+dt / @2d(-i2)+dt / @1di1+dt / 2(i1 6 i2) # v1(t) i2& i2 / 92di2+dt / @2d(i2 6 i1)+dt / 9d(i2 6 i1)+dt / @2d(i2)+dt / @1d(-i1)+dt / 2(i2 6 i1) #5 -2>. 4rite the loo-b!sis volt!ge e?u!tions for the m!gnetic!ll couled network with k closed. %olution& %!me !s .2. -2". 4rite e?u!tions using the GC9 in terms of node-to-d!tum volt!ge v!ri!bles for the four networks of *rob. -1>.
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi (!)
1
2
v1 v2 v 9
/ C -
v(t)
2
v1
v2
9 1 C v(t)+ 1
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
Node-v1 :ccording to GC9 %um of currents entering into the Eunction # %um of currents le!ving the Eunction v(t)+ 1 # v1+ 1 / (v1 6 v2)+ 2 / cd(v1 6 v2)+dt v(t)+ 1 # v1+ 1 / v1+ 2 6 v2+ 2 / cdv1+dt 6 cdv2+dt v(t)+ 1 # v1+ 1 / v1+ 2 / cdv1+dt 6 v2+ 2 6 cdv2+dt v(t)+ 1 # v1(1+ 1 / 1+ 2 / cd+dt) / (6 1+ 2 6 cd+dt)v2 v(t)+ 1 # v1(P1 / P2 / cd+dt) / (6 P2 6 cd+dt)v2
ec!use P # 1+
Node-v1 :ccording to GC9 %um of currents entering into the Eunction # %um of currents le!ving the Eunction 5 # (v2 6 v1)+ 2 / cd(v2 6 v1)+dt / 1+9 v2dt / v2+ 5 # (v2 6 v1)+ 2 / cd(v2 6 v1)+dt / Q v2dt / v2+ 5 # v2+ 2 6 v1+ 2 / cdv2+dt 6 cdv1+dt / Q v2dt / v2+ 5 # v2+ 2 / cdv2+dt / v2+ / Q v2dt 6 v1+ 2 6 cdv1+dt 5 # v2(1+ 2 / cd+dt / 1+ / Q dt) / v1(6 1+ 2 6 cd+dt) 5 # v2(P2 / cd+dt / P / Q dt) / v1(6 P2 6 cd+dt)
ec!use P # 1+, Q # 1+9
(b)
1 /
9
C
v(t) 2
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
v1
v(t)+ 1
9
C
1 2
Node-v1 :ccording to GC9 %um of currents entering into the Eunction # %um of currents le!ving the Eunction v(t)+ 1 # v1+ 1 / v1+ 2 / cdv1+dt / 1+9 v1dt v(t)+ 1 # v1+ 1 / v1+ 2 / cdv1+dt / Q v1dt Hec!use 1+9 # QI v(t)+ 1 # v1(1+ 1 / 1+ 2 / cd+dt / Q dt) (c)
9 / C v(t)
v1
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
9 C 1+9 v(t)dt
Node-v1 :ccording to GC9 %um of currents entering into the Eunction # %um of currents le!ving the Eunction 1+9 v1dt / v1+ / cdv1+dt # 1+9 v(t)dt v1(1+9 dt / 1+ / cd+dt) # 1+9 v(t)dt
(d)
/
v(t)
C1
92
91
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
1 C2
2
/
/
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
1+91 v(t)dt
91
c1dv(t)+dt C1 v1 v
v2
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
Node-v 1& c1dv(t)+dt / 1+91 v(t)dt # c1dv1+dt / 1+91 v1dt / 1+92 (v1 6 v)dt / (v1 6 v2)+ 1 (c1d+dt / 1+91 dt)v(t) # c 1dv1+dt / 1+91 v1dt / 1+92 v1dt - 1+92 vdt / v1+ 1 6 v2+ 1 (c1d+dt / 1+91 dt)v(t) # c 1dv1+dt / 1+91 v1dt / 1+92 v1dt / v1+ 1 6 v2+ 1 - 1+92 vdt (c1d+dt / 1+91 dt)v(t) # (c 1d+dt / 1+91 dt / 1+92 dt / 1+ 1) v1 6 v2+ 1 - 1+92 vdt Node-v 2& c2d(v2 6 v)+dt / (v2 6 v1)+ 1 / v2+ 2 # 5 c2dv2+dt - c2dv+dt / v2+ 1 6 v1+ 1/ v2+ 2 # 5 6 v1+ 1/ v2+ 2 / c2dv2+dt / v2+ 1 - c2dv+dt # 5 6 v1+ 1/ (1+ 2 / c2d+dt / 1+ 1)v2 - c2dv+dt # 5 Node-v & 1+92 (v 6 v1)dt / c2d(v 6 v2)+dt / v+ # 5 1+92 vdt - 1+92 v1dt / c2dv+dt - c2dv2+dt / v+ # 5 - 1+92 v1dt - c2dv2+dt / 1+92 vdt / v+ / c2dv+dt # 5 - 1+92 v1dt - c2dv2+dt / (1+92 dt / 1+ / c2d+dt)v # 5 -2A. @!king use of the GC9, write e?u!tions on the node b!sis for the four networks of *rob. -1". (!)
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
/ -
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
v(t)+ 1
v2
v
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
v v1(t)+ 2
:ccording to GC9 %um of currents entering into the Eunction # %um of currents le!ving the Eunction Node-v 2& v2+ 1/ v2+ 1 / cdv2+dt / c1d(v2 6 v)+dt # v(t)+ 1 v2+ 1/ v2+ 1 / cdv2+dt / c1dv2+dt 6 c1dv+dt # v(t)+ 1 v2(1+ 1/ 1+ 1 / cd+dt / c1d+dt) 6 c1dv+dt # v(t)+ 1 Node-v & $+ 2 / cdv+dt / c1d(v 6 v2)+dt / c2d(v 6 v)+dt # 5 $+ 2 / cdv+dt / c1dv+dt - c1dv2+dt / c2dv+dt - c2dv+dt # 5 - c1dv2+dt / $+ 2 / cdv+dt / c1dv+dt / c2dv+dt - c2dv+dt # 5 - c1dv2+dt / $(1+ 2 / cd+dt / c1d+dt / c2d+dt) - c2dv+dt # 5 Node-v & v+ / c2d(v 6 v)+dt / c=dv+dt / v+ 2 # v1(t)+ 2 v+ / c2dv +dt - c2dv +dt / c=dv+dt / v+ 2 # v1(t)+ 2 - c2dv +dt / (c=d+dt / 1+ 2 / 1+ / c2d +dt)v # v1(t)+ 2 (b)
/
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
91
v1
v2
v 9
C1
C2 C
92
" 1+92 v(t)dt
:ccording to GC9 %um of currents entering into the Eunction # %um of currents le!ving the Eunction Node-v 1& 1+92 v(t)dt # 1+92 v1d t / 1+91 (v1 6 v)d t / c1d(v1 6 v2)+dt 1+92 v(t)dt # 1+92 v1d t / 1+91 v1d t - 1+91 vd t / c1dv1+dt 6 c1dv2+dt 1+92 v(t)dt # 1+92 v1d t / 1+91 v1d t / c1dv1+dt 6 c1dv2+dt - 1+91 vd t
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 1+92 v(t)dt # v1(1+92 d t / 1+91 d t / c1d+dt) 6 c1dv2+dt - 1+91 vd t
:ccording to GC9 %um of currents entering into the Eunction # %um of currents le!ving the Eunction Node-v 2& c1d(v2 6 v1)+dt / c1d(v2 6 v)+dt / cdv2+dt / 1+92 v2dt # 5 c1dv2+dt - c1dv1+dt / c1dv2+dt - c1dv+dt / cdv2+dt / 1+92 v2dt # 5 - c1dv1+dt / c1dv2+dt / c1dv2+dt / cdv2+dt / 1+92 v2dt - c1dv+dt # 5 - c1dv1+dt / v2(c1d+dt / c1d+dt / cd+dt / 1+92 dt) - c1dv+dt # 5
:ccording to GC9 %um of currents entering into the Eunction # %um of currents le!ving the Eunction Node-v & 1+91 (v 6 v1)d t / c1d(v 6 v2)+dt / 1+9 vd t / v+ # 5 1+91 v d t - 1+91 v1d t / c1dv+dt - c1dv2+dt / 1+9 vd t / v+ # 5 - 1+91 v1d t - c1dv2+dt / c1dv+dt / 1+9 vd t / v+ / 1+91 v d t # 5 - 1+91 v1d t - c1dv2+dt / v(c1d+dt / 1+9 d t / 1+ / 1+91 d t) # 5
(c)
/
-
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
/
/
-
-
*1
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi v2
v C
1
%t',R
C1 9
C Cdv(t)+dt
:ccording to GC9 %um of currents entering into the Eunction # %um of currents le!ving the Eunction Node-v 1& v(t)+ # v1+ / (v1 6 v)+ / c1dv1+dt v(t)+ # v1+ / v1+ 6 v+/ c1dv1+dt v(t)+ # v1(1+ / 1+ / c1d+dt) - v+
:ccording to GC9 %um of currents entering into the Eunction # %um of currents le!ving the Eunction Node-v 2& cdv(t)+dt # cdv 2+dt / v2+ 1 / cd(v2 6 v)+dt cdv(t)+dt # cdv 2+dt / v2+ 1 / cdv2+dt - cdv +dt cdv(t)+dt # v 2(cd+dt / 1+ 1 / cd+dt) - cdv+dt
:ccording to GC9 %um of currents entering into the Eunction # %um of currents le!ving the Eunction Node-v & 5 # v+ 9 / (v 6 v1)+ / cd(v 6 v2)+dt 5 # v+ 9 / v+ 6 v1+ / cdv+dt - cdv2+dt 5 # 6 v1+ - cdv2+dt / cdv+dt / v+ 9 / v+ 5 # 6 v1+ - cdv2+dt / v(cd+dt / 1+ 9 / 1+)
(d)
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
/
-
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
/
-
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
291
/
-
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
v1
v2
291 1+291 v(t)dt
v
v
:ccording to GC9 %um of currents entering into the Eunction # %um of currents le!ving the Eunction Node-v 1& 1+291 v(t)dt # 1+291 (v1 6 v)dt / c!d(v1 6 v2)+dt / 1+9! (v1 6 v2)dt / 1+9b (v1 6 v)dt / cbd(v1 6 v)+dt
:ccording to GC9 %um of currents entering into the Eunction # %um of currents le!ving the Eunction Node-v 2&
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi c!d(v2 6 v1)+dt / cbd(v2 6 v)+dt / 1+9b
(v2 6 v)dt / 1+29 (v2 6 v)dt / (v2 6 v) # 5
:ccording to GC9 %um of currents entering into the Eunction # %um of currents le!ving the Eunction Node-v & c!d(v 6 v)+dt / 1+9b (v 6 v2)dt / cd(v 6 v2)+dt / 1+291 (v 6 v1)dt # 5 Node-v & 1+9! (v 6 v)dt / c!d(v 6 v)+dt / 1+292 (v 6 v2)dt / 1+9b (v 6 v1)dt / cbd(v 6 v1)+dt # 5 -5. For the given network, write the node-b!sis e?u!tions using the node-to-d!tum volt!ges !s v!ri!bles.
2
1
=
v2 v1
v
v=
v :ccording to GC9 %um of currents entering into the Eunction # %um of currents le!ving the Eunction Node-v 1& (v1 6 v2)+(1+2) / (1+2)d(v 1 6 v)+dt / (v1 6 v)+(1+2) # 5 (v1 6 v2)+(2) / (2)d(v 1 6 v)+dt / (v1 6 v)+(2) # 5 Node-v 2& i2 # (v2 6 v1)+(1+2) / (v2 6 5)+(1+2) i2 # (v2 6 v1)+2 / v2+2 Node-v & i2 # (1+2)d(v 6 v)+dt / (1+2)d(v 6 v1)+dt / (1+2)d(v 6 5)+dt
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi i2 # (2)d(v 6 v)+dt / (2)d(v 6 v1)+dt / (2)dv +dt Node-v & 5 # (1+2)d(v 6 v)+dt / (v 6 5)+(1+2) / (v 6 v1)+(1+2) 5 # (2)d(v 6 v)+dt / (v)+(2) / (v 6 v1)+(2) -1. 3he network in the figure cont!ins one indeendent volt!ge source !nd two controlled sources. 8sing the GC9, write node-b!sis e?u!tions.
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
/
*1
1
+ !
*-
/
:ccording to GC9
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi %um of currents entering into the Eunction # %um of currents le!ving the Eunction Node-$ 1& ($1 6 v1)+ 1 / C1d$1+dt / $1+ 2 # 5 Node-$ 2 vk & vk 6 $2 # (v1 - vk ) Node-$ 2& ($ 6 $2)+ / $+ / 1+9 vdt / ($ 6 $)+ = # 5 Node-$ & ($ 6 $)+ = / $+ #
i2 Hwhere i2 # $+ I
-2. 3he network of the figure is ! model suit!ble for Rmidb!ndS oer!tion of the Rc!scode-connectedS @'% tr!nsistor !mlifier. %olution& 'en our book see (*+A). %imlified di!gr!m&
-gm$ i $ $2 rd rd gm$1
9 i1
i2
9oo-b!sis& i2 # -gm$1 i # gm$ i1& (i1 6 i)rd / i1 9 - $ # 5
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi (i1 6 i)rd / i1 9 # $ i1rd 6 ird / i1 9 # $
i1 (rd / 9) 5 5
i2 5 -gm$1 5
i -rd 5 gm$
i1 i2 i
#
$ 5 5
Node-b!sis& Node-$ & gm$1 # $+rd - gm$ / ($ 6 $2)+rdT gm$1 / gm$ # $+rd / $+rd - $2+rd Node-$ 2& - gm$ # $2+ 9 / ($2 6 $)+rdT - gm$ # $2+ 9 / $2+rd 6 $+rd $2 (1+ 9 / 1+rd) -1+rd
$ -1+rd 2+rd
$2 $
- gm$ gm$1 / gm$
-. n the network of the figure, e!ch br!nch cont!ins ! 1-ohm resistor !nd four br!nches cont!in ! 1-$ volt!ge source. :n!l;e the network on the loo b!sis. %olution& / -
+ !
/ -
/ -
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
1
O?.
$olt!ge
>
0
2
i1
i2
i
3
i
i=
i
i>
i"
iA
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 1
1
-1
5
-1
5
5
5
5
5
2
-1
-1
-1
5
-1
5
5
5
5
5
5
-1
5
5
-1
5
5
5
5
-1
5
5
-1
5
-1
5
5
=
1
5
-1
5
-1
-1
5
-1
5
5
5
5
-1
5
-1
5
5
-1
>
1
5
5
5
-1
5
5
-1
5
"
-1
5
5
5
5
-1
5
-1
-1
A
-1
5
5
5
5
5
-1
5
-1
-. 4rite e?u!tions on the node b!sis.
ee!t *rob. - for the network.
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
1
2
=
/ -
O?.
Coefficients of $olt!ge di1+dt
di2+dt
di+dt
di+dt
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
/
1
5
-1
-1
5
2
1
-1
5
-1
5
-1
5
-1
5
5
-1
-1
/
-
-
/
/
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
2h
/
/
-
-
/
/
1h
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
h
1 h
1h
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi h 1h $2 $ 1+2 dt
1+ dt
h
1h
2h
Node-$ 1 1+2 dt # 1+2 $1dt / ($1 6 $)dt / 1+2 ($1 6 $2)dt 1+2 dt # 1+2 $1dt / $1dt - $dt / 1+2 $1dt - 1+2 $2dt
Node-$ 2 5 # 1+2 $2dt / ($2 6 $)dt / 1+2 ($2 6 $1)dt 5 # 1+2 $2dt / $2dt - $dt / 1+2 $2dt - 1+2 $1dt 1+ dt # 1+ $dt / $dt / ($ 6 $1)dt / ($ 6 $2)dt 1+ dt # 1+ $dt / $dt / $dt - $1dt / $dt - $2dt
O?.
Current
$1
$2
$
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 1
J dt
2 dt
-1+2 dt
- dt
2
5
-1+2 dt
2 dt
- dt
1+ dt
- dt
- dt
dt
$2
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi $ $1 $
Node-$ 1& $1+ / ($1 6 $)+1 / ($1 6 $2)+ # 5 $1+ / ($1 6 $) / ($1 6 $2)+ # 5 $1+ / $1 6 $ / $1+ - $2+ # 5 2$1+ / $1 6 $ - $2+ # 5 $1+2 / $1 6 $ - $2+ # 5 1.=$1 6 $ - $2+ # 5 Node-$ 2& ($2 6 $)+1 / ($2 6 $)+ / ($2 6 $1)+ # 5 $2 6 $ / $2+ - $+ / $2+ - $1+ # 5 1.=$2 6 $ - $+ - $1+ # 5 Node-$ & ($ 6 $2)+ / ($ 6 5)+ / ($ 6 $)+1 # 5 $+ - $2+ / $+ / $ 6 $ # 5 1.=$ - $2+ 6 $ # 5 Node-$ & ($ 6 $1)+1 / ($ 6 5)+1 / ($ 6 $)+1 / ($ 6 $2)+1 # $ 6 $1 / $ / $ 6 $ / $ 6 $2# $ 6 $1 6 $2 - $ #
Muhammad Irfan Yousuf Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi O?.
Current
$1
$2
$
$
1
5
1.=
-5.2=
5
-1
2
5
-5.2=
1.=
-5.2=
-1
5
5
-5.2=
1.=
-1
-1
-1
-1
9oo-b!sis&
Muhammad Irfan Yousuf Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
+ !
O?.
$olt!ge
$1
$2
$
$
1
5
-1
-1
5
2
5
-1
5
-1
Muhammad Irfan Yousuf Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
-
-1
5
-1
5
-1
-1
-. For the network shown in the figure, determine the numeric!l v!lue of the br!nch current i 1. :ll sources in the network !re time inv!ri!nt.
$1
$2
2$
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
/ / -
($1 6 2)+(1+2) / ($1 6 $2)+1 / ($1 6 2)+2 # 5 ($1 6 2)2 / ($1 6 $2) / ($1 6 2)+2 # 5 2$1 6 / $1 6 $2 / $1+2 - 1 # 5 .=$1 6 $2 - = # 5
(i)
$2+(1+2) / ($2 6 $1)+1 / ($2 6 2)+1 # 1 2$2 / $2 6 $1 / $2 6 2 # 1 $2 6 $1 # $1 # - / $2
(ii)
*ut $1 in (i) .=$1 6 $2 - = # 5 .=(- / $2) 6 $2 - = # 5 -15.= / 1$2 6 $2 6 = # 5 1$2 6 1=.= # 5 1$2 # 1=.= $2 # 1=.=+1 # 1.1A2$olts *ut v!lue of $2 in (ii) $1 # - / $2 (ii) $1 # - / (1.1A2) $1 # - / (1.1A2) $1 # 1.>"$olts i1 # ($1 6 $2)+1 # $1 6 $2 # 1.>" $olts -1.1A2 $olts # 5.=> !meres. ->. n the network of the figure, !ll sources !re time inv!ri!nt. Determine the numeric!l v!lue of i 2.
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
1
:ccording to GC9 %um of currents entering into the Eunction # %um of currents le!ving the Eunction Node-v 1& + 1 4 1,1 + %1 5 ',1 + %1 5 ',1 # $1 / ($1 6 $2) / ($1 6 $) # $1 / $1 6 $2 / $1 6 $ # $1 6 $2 6 $
(i)
Node-v 2& ($2 6 $1)+1 / ($2 6 $)+2 / ($2 - 5)+1 # 1 $2 6 $1 / $2+2 6 $+2 / $2 # 1 $2 6 $1 / $2+2 6 $+2 / $2 # 1 2.=$2 6 $1 6 5.=$ # 1
(ii)
Node-v & ($ 6 $1)+1 / ($ 6 $2)+2 / $+1 # 1 $ 6 $1 / $+2 6 $2+2 / $ # 1 2.=$ 6 $1 6 $2+2 # 1
(iii)
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi # $1 6 $2 6 $ $1 # / $2 / $ $1 # ( / $2 / $)+ 2.=$ 6 $1 6 $2+2 # 1 2.=$ 6 (( / $2 / $)+) 6 $2+2 # 1 2.=$ 6 (+ / $2+ / $+) 6 $2+2 # 1 2.=$ 6 1 - $2+ - $+ 6 $2+2 # 1 2.=$ - $2+ - $+ 6 $2+2 # 2 2.=$ 6 5.$2 - 5.$ 6 5.=$2 # 2 2.1$ 6 5."$2 # 2 %ubtr!cting (ii) (iii) 2.=$2 6 $1 6 5.=$ # 1 (ii) 2.=$ 6 $1 6 $2+2 # 1 (iii) 2.=$2 6 2.=$ 6 5.=$ / $2+2 # 5 $2 6 $ # 5 $2 # $ $2 # $ 2.1$ 6 5."$2 # 2 utting $2 # $ 2.1$ 6 5."$ # 2 1.2$ # 2 $ # 2+1.2 # 1.=51 $ $ # 1.=51 $ i2 # (2 6 $)+2 # (2 6 1.=51)+2 # 5.2A= !meres. i2 # 5.2A= !meres. -". n the given network, !ll sources !re time inv!ri!nt. Determine the br!nch current in the 2 ohm resistor.
+ !
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
+ !
+ ! +
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
i1
+ !
+ !
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
i2
9oo-b!sis& i1& :ccording to kirchhoff7s volt!ge l!w %um of otenti!l rise # sum of otenti!l dro (+2)i 1 / 1(i1 6 i2) # 2 (+2)i 1 / i1 6 i2 # 2 (=+2)i 1 6 i2 # 2 i2& (i2 6 i1)1 / 2i2 / i2(1+2) # 2 i2 6 i1 / 2i2 / i2(1+2) # 2 .=i2 6 i1 # 2
=+2
-1
i1
2 #
-1
=+2
>+2
(/) (-)
i2
2
-1 # M(=+2)(>+2) - 1 # >.>=
-1
>+2
Determin!nt # >.>=
=+2
(/) (-)
2
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi # M(=+2)(2) / 2 # > -1
2
i2 # >+>.>= # 5.A5 !meres.
:ns.
-A. %olve for the four node-to-d!tum volt!ges.
$2 $1
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
/
Node-$ 1& ($1 6 $2)+(1+2) / ($1 6 $)+(1+2) / ($1 - 5)+(1+2) / 2 # " 2($1 6 $2) / 2($1 6 $) / 2$1 / 2 # " Node-$ 2& ($2 6 $1)+(1+2) / ($2 6 $)+(1+2) # 2($2 6 $1) / 2($2 6 $) # Node-$ & ($ 6 $)+(1+2) 6 ($ - 5)+(1+2) / ($ 6 $2)+(1+2) # 2 2($ 6 $) 6 2$ / 2($ 6 $2) # 2 Node-$ & ($ - 5)+(1+2) / ($ 6 $1)+(1+2) / ($ 6 $)+(1+2) # 2 2$ / 2($ 6 $1) / 2($ 6 $) # 2 -1 6 -", -= 6 -=> (Do ourself). -5. Find the e?uiv!lent induct!nce. %olution& %ee UV-2. for reference. -1. t is intended th!t the two networks of the figure be e?uiv!lent with resect to the !ir of termin!ls, which !re identified. 4h!t must be the v!lues for C 1, 92, !nd 9 %olution& Do ourself.
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi i
i2
i1
1
17
e e?uiv!lent with resect to the !ir of termin!ls
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
1
1
(!)
(b) O?u!ting (!) (b)
1
1
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
(!)
(b)
O?u!ting (!) (b)
1
(b) O?u!ting (!) (b) -2. %olution& %ee -1 for reference. efore solving e0ercise following terms should be ket in mind& 1. kirchhoffs current l!w 2. kirchhoffs volt!ge l!w . 9oo !n!lsis . Node !n!lsis =. Determin!nt . %t!te v!ri!ble !n!lsis >. %ource tr!nsform!tion
(!)
1
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
1-=. %olution& v # $5sin t C # C5(1 - cos t) U # t U # C$ i # d(?)+dt # d(Cv)+dt # Cdv+dt / vdC+dt i # Cdv+dt / vdC+dt i # C5(1 - cos t)d($5sin t)+dt / $5sin tdC5(1 - cos t)+dt i # C5(1 - cos t)
$5cos t / $5sin tH C5sin tI
1-15. t w # vi dt For !n inductor v9 # 9di+dt utting the v!lue of volt!ge t w # vi dt t w # (9di+dt)i dt t w # 9 idi t 2 w # 9 i +2 w # 9Mi2(t)+2 - i2(- )+2 w # 9Mi2(t)+2 6 (i(- ))2+2 w # 9Mi2(t)+2 6 (5) 2+2 w # 9Mi2(t)+2
Hec!use i(- ) # 5 for !n inductorI
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi :s we know # 9i 2 # 92i2 2 +9 # 9i2 w # 9Mi2(t)+2 w # 9i2+2 utting the v!lue of 9i 2 w # ( 2+9)+2 w#
2
+29
Hwhere
# flu0 link!geI
1-11. t w # vi dt For ! c!!citor i # Cdv+dt utting the v!lue of current t w # vi dt t w # (Cdv+dt)v dt t w # C vdv t 2 w # C v +2 w # CMv2(t)+2 - v2(- )+2 w # CMv2(t)+2 6 (v(- ))2+2 w # CMv2(t)+2 6 (5) 2+2 w # CMv2(t)+2 :s we know U # C$ $ # U+C w # CMv2(t)+2 w # CM(?+C)2+2 w # CM?2+2C2
Hec!use v(- ) # 5 for !n inductorI
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi w # ?2+2C w # ?2D+2 :ns. 1-12. w9 # (1+2)9i 2 * # vi * # dw9+dt utting v!lues of * w 9 vi # d((1+2)9i2)+dt vi # (1+2)d9i2+dt vi # (1+2)9di2+dt vi # (1+2)92iHdi+dtI v # 9Hdi+dtI 1-1. wc # (1+2)D?2 * # vi * # dw9+dt utting v!lues of * w 9 vi # d((1+2)D? 2)+dt vi # (1+2)dD?2+dt vi # (1+2)Dd?2+dt vi # (1+2)D2?Hd?+dtI vi # D?Hd?+dtI :s we know i # d?+dt vi # D?Hd?+dtI vi # D?HiI v # D? t ? # i dt v # D? t v#D
i dt -
1-1>. $ # 12 $ C#1 F w# w # (1+2)C$2 # (1+2)(1 15-)(12)2
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi w # >2 W 1-1". vc # 255 $ C#1 F m!ss # 155 lb # =. kg work done # Fd # mgd # (=.)(A.")d work done # energ # (1+2)C(v c)2 # (1+2)(1 15-)(255) 2 # 5.52 Eoule work done # (=.)(A.")d 5.52 # (=.)(A.")d d # 5.52+(=.)(A.") # 5.52+.A d # .=5= 15-=m
:ns.
1-1A. %olution& $m v 5
1
2
-$m for 5 t 1
(01, 1) # (1, $m)
(05, 5) # (5, 5) %loe # m m # (1 6 5)+(01 6 05) # ($m 6 5)+(1 6 5) m # $m
# m0 / c # $m(t) / 5 # $mt
time
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
%tr!ight-line e?u!tion
-intercet # c # 5
for 1 t
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi (01, 1) # (1, $m)
%loe # m m # ( 6 1)+(0 6 01) # (-$m 6 $m)+( - 1) # -2$m+2 # -$m m # -$m
(0, ) # (, -$m)
%tr!ight-line e?u!tion # m0 / c # -$ m(t) / 2$m # -$mt / 2$m -intercet # c # 2$m for t
(0, ) # (, 5)
(0, ) # (, -$m) %loe # m m # ( 6 )+(0 6 0) # (5 6 (-$ m))+( 6 ) m # $m %tr!ight-line e?u!tion # m0 / c # $m(t) - $m # $mt 6 $m
-intercet # c # -$ m 9et c!!cit!nce be C
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi for 5 t 1 i # Cdv+dt # Cd($mt)+dt # C$m i # C$m for 1 t i # Cdv+dt # Cd(-$ mt / 2$m)+dt # -C$m i # -C$m for t i # Cdv+dt # Cd($mt 6 $m)+dt # C$m i # C$m
C$m
-C$m for 5 t 1 ? # C$ ? # C$mt for 1 t ? # C$
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi ? # C$m(2 6 t) for t ? # C$ ? # C$m(t - ) for 5 t 1 for 1 t for t
? # C$mt ? # C$m(2 6 t) ? # C$m(t - )
Ch!rge w!veform s!me !s volt!ge w!veform. (b)
i(t)
t # 5, ? # 5
t # 1, ? # C$m t # , ? # -C$m t # , ? # 5
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 5
1
2
for 5 t 1
(01, 1) # (1, m)
time
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi (05, 5) # (5, 5) %loe # m m # (1 6 5)+(01 6 05) # (m 6 5)+(1 6 5) m # m
# m0 / c # m(t) / 5 # mt %tr!ight-line e?u!tion -intercet # c # 5 for 1 t
(01, 1) # (1, m)
%loe # m m # ( 6 1)+(0 6 01) # (-m 6 m)+( - 1) # -2m+2 # -m m # -m
(0, ) # (, -m)
%tr!ight-line e?u!tion # m0 / c # - m(t) / 2m # -mt / 2m -intercet # c # 2 m for t
(0, ) # (, 5)
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
(0, ) # (, -m) %loe # m m # ( 6 )+(0 6 0) # (5 6 (- m))+( 6 ) m # m %tr!ight-line e?u!tion # m0 / c # m(t) - m # mt 6 m
-intercet # c # - m for 5 t 1 t v(t) # (1+C) id(t) / v(t1) t1 t v(t) # (1+C) mtd(t) / 5 5 t v(t) # (1+C) mtd(t) 5 t 2 v(t) # (1+C)m t +2 5 v(t) # (1+C)mM(t2+2) - ((5) 2+2) v(t) # (1+C)m(t2+2) v(1) # (1+C)m((1)2+2) # (1+C) m(1+2) # m+2C for 1 t t v(t) # (1+C) id(t) / v(t1) t1 t v(t) # (1+C) m(2 6 t)d(t) / m+2C 1
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi t v(t) # (1+C) 2t 6 t +2 / m+2C 1 2
v(t) # (1+C)M(2t 6 t 2+2) - (2(1) 6 12+2) / m+2C v(t) # (1+C)M(2t 6 t 2+2) - (2 6 1+2) / m+2C v(t) # (1+C)M(2t 6 t 2+2) - (+2) / m+2C !t time t # v() # (1+C)M(2() 6 () 2+2) - (+2) / m+2C v() # (1+C)M 6 .= 6 1.= / m+2C v() # m+2C for t t v(t) # (1+C) id(t) / v() t1 t v(t) # (1+C) m(t - )d(t) / m+2C 1 t 2 v(t) # (1+C)m t +2 6 t / m+2C 1 v(t) # (1+C)mM(t2+2 6 t) 6 (1+2 - ) / m+2C v(t) # (1+C)mM(t2+2 6 t) / 2.= / m+2C !t time t # v() # (1+C)mM(()2+2 6 ()) / 2.= / m+2C v() # (1+C)mM1+2 6 12 / 2.= / m+2C v() # (1+C)mM" 6 12 / 2.= / m+2C v() # (1+C)mM61.= / m+2C v() # -m+C v(5) v(1) v(2) v() v()
5 m+2C # 5.=(m+C) m2+C # 2(m+C) m+2C # 5.=(m+C) -m+C # -1(m+C)
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi v(t) # (1+C)m(t2+2) !t time t # 2 v(2) # (1+C)m((2)2+2) # m(2)+C
5
4
5
3
4
2
3
e g a t l o v
Series2 Series1 1
2
2
0
4
1
-1
5
-2 1
2
3
4
5
Series2
0
0.5
2
0.5
-1
Series1
0
1
2
3
4
time
for 5 t 1 ? # C$ ? # C(mt2+2C) # mt2+2 for 1 t ? # C$ ? # Cm(t 6 t2 - 2)+2C
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi for t ? # C$ ? # C(1+C)M mM(t2+2 6 t) / 2.= / m+2C # mM(t2+2 6 t) / 2.= / m+2C
:t time t # 5 ? # C(mt2+2C) # mt2+2 # m(5)2+2 # 5 C :t time t # 1 ? # C(mt2+2C) # mt2+2 # m12+2 # m+2 C :t time t # 2 ? # C(mt2+2C) # mt2+2 # m22+2 # 2m C :t time t # ? # Cm(t 6 t2 - 2)+2C # C m(() 6 2 - 2)+2C # (5.=m+C) C :t time t 4 / ? # C(1+C)M mM(t2+2 6 t) / 2.= / m+2C # mM(t2+2 6 t) / 2.= / m+2C ? # mM(2+2 6 ()) / 2.= / m+2C # -m+C Ch!rge w!veform s!me !s volt!ge w!veform. 1-25. %olution& # 1: 9#J< w9 # (1+2)92 # (1+2)(1+2)(1) 2 # 5.2= W
:s we know energ in !n inductor # (1+2)9 2 W
/ -
source 9
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
%hort circuit 9
Onerg will be lost !fter short-circuiting. 1-21. %olution& 9 # 1< (!) (flu0 link!ge) !t t # 1sec. (flu0 link!ge) # 9 # t Hec!use # m0 / cT m # 1 # sloeI (flu0 link!ge) # 9t !t t # 1sec. (flu0 link!ge) # 9t # (1)(1) # 1 <(henr):(!mere). (b) d +dt # 9d(t)+dt # 9 # 1 (c) t ? # idt t ? # tdt t 2 ? # t +2 # Mt2+2 6 (- )2+2 # :t time t # 1sec ? # t 2+2 # ? # (1) 2+2 # J Coulomb ? # t2+2 1-2. %olution& G is closed !t t # 5 i(t) # 1 6 e-t, tL5 i(t5) # 5. : 1 6 e-t5 # 5. : 6e-t5 # -1 / 5. : -e-t5 # -5.> e-t5 # 5.> 3!king log!rithm of both the sides
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi loge-t5 # log5.> -t5loge # -5.2 t5(5.) # 5.2 Hec!use e # 2.>1"I t5 # 5.2+5. # 5.AA= sec # 1 sec. t5 # 1 sec. (!) di(t5)+dt # di(t)+dt # d(1 6 e -t)+dt di(t)+dt # d(1)+dt - d(e -t)+dt di(t)+dt # 5 - e -tHd(-t)+dtI di(t)+dt # 5 - e -t(-1) di(t)+dt # e-t di(t5)+dt # e-t5 t5 # 1 sec. di(1)+dt # e-1 di(1)+dt # 1+e # J.>1" # 5.> :mere er second di(1)+dt # 1+e # J.>1" # 5.> :mere er second (b) # 9i i(t) # 1 6 e-t # 9i(t) # 9(1 6 e-t) (t5) # 9(1 6 e-t5) t5 # 1 sec (1) # 9(1 6 e -1) (1) # 9(1 6 1+e) :s 9 # 1< 1+e # 5.> (1) # (1)(1 6 5.>) # 5. weber (c) d +dt # # 9(1 6 e-t) # # (1 6 e-t) d +dt # d(1 6 e -t)+dt # d1+dt - de-t+dt # 5 / e-t # e-t d +dt # e-t d (t5)+dt # e-t5 t5 # 1 sec. d (1)+dt # e-1 # 1+e # 5.> weber er sec. (d) v(t) # 9di(t)+dt i(t) # 1 6 e-t 9 # 1 v(t) # (1)d(1 6 e -t)+dt
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi v(t) # d(1 6 e-t)+dt # e-t v(t5) # e-t5 t5 # 1 sec. v(1) # e-1 # 1+e # 5.> $ (e) w # (1+2)9i 2 # (1+2)(1)(1 6 e -t)2 w # (1+2)(1 6 e-t)2 w # (1+2)(1 / 2e-2t 6 2e-t) w(t5) # (1+2)(1 / 2e -2t5 6 2e-t5) t5 # 1 sec. w(1) # (1+2)(1 / 2e -2(1) 6 2e-1) w(1) # (1+2)(1 / 2e -2 6 2e-1) w(1) # (1+2)(1 / 2(1+e 2) 6 2(1+e)) H1+e # 5.>T 1+e2 # 5.1=I w(1) # (1+2)(1 / 2(5.1=) 6 2(5.>)) w(1) # (1+2)(1 / 5.2> 6 5.>) w(1) # 5.2= Woule (f) v # v # i # (1 6 e-t)(1) # (1 6 e -t) v # i # (1 6 e -t) v (t5) # (1 6 e-t5) !t time t5 # 1 sec. v (1) # (1 6 e-1) # 5. $ (g) w # (1+2)(1 / 2e-2t 6 2e-t) dw+dt # d((1+2)(1 / 2e -2t 6 2e-t))+dt dw+dt # (1+2)d(1 / 2e -2t 6 2e-t)+dt dw+dt # (1+2)Hd(1)+dt / d(2e -2t)+dt - d(2e -t)+dtI dw+dt # (1+2)H5 / 2e -2t)(-2) - 2e-t)(-1)I dw+dt # (1+2)H-e-2t / 2e-t)I dw(t5)+dt # (1+2)H-e -2t5 / 2e-t5)I dw(1)+dt # (1+2)H-e -2 / 2e-1)I dw(1)+dt # (1+2)H-(1+e 2) / 2(1+e)I dw(1)+dt # (1+2)H-(5.1=) / 2(5.>)IH1+e # 5.>T 1+e 2 # 5.1=I dw(1)+dt # (1+2)H-5.= / 5.>I # 5.1 w!tts (h) * # i2 # (1 / e-2t 6 2e-t)(1) * # i2 # (1 / e-2t 6 2e-t) * (t5) # i2 # (1 / e-2t5 6 2e-t5)
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi :t time t5 # 1 sec. * (1) # i2 # (1 / e-2(1) 6 2e-(1)) * (1) # i2 # (1 / e-2 6 2e-1) * (1) # i2 # (1 / 1+e 2 6 2(1+e)) * (1) # i2 # (1 / 5.1= 6 2(5.>)) * (1) # i2 # (1 / 5.1= 6 5.>) * (1) # i2 # (5.A=) w!tts (i) *tot!l # vi # (1)(1 6 e -t) # (1 6 e-t) :t time t5 # 1 sec. *tot!l(t5) # vi # (1)(1 6 e -t) # (1 6 e-t5) *tot!l(1) # (1 6 e -1) *tot!l(1) # (1 6 e -1) # 5. w!tts. 1-2=. $olt!ge !cross the c!!citor !t time t # 5 vc(5) # 1 $olt k is closed !t t # 5 i(t) # e-t, tL5 i(t5) # 5.> : 5.> # e-t5 3!king log!rithm of both the sides log5.> # loge -t5 -t5loge # -5.2 t5(5.) # 5.2 Hec!use e # 2.>1"I t5 # 1 sec. (!) dvc(t5)+dt # 8sing loo e?u!tion vc(t) # i # e -t(1) # e-t $olts dvc(t)+dt # -e-t $olts dvc(t5)+dt # -e-t5 $olts t5 # 1 sec. dvc(t5)+dt # -e-1 $olts dvc(t5)+dt # -5.> $+sec (b) Ch!rge on the c!!citor # ? # Cv # (1)(e -t) # e-t coulomb Ch!rge on the c!!citor # ?(t 5) # Cv # (1)(e -t) # e-t5 coulomb t5 # 1 sec.
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi Ch!rge on the c!!citor # ?(1) # Cv # (1)(e -t) # e-1 coulomb # 5.> coulomb (c) d(Cv)+dt # Cdv+dt # Cde -t+dt # -Ce-t d(Cv(t 5))+dt # -Ce-t5 t5 # 1 sec. d(Cv(t 5))+dt # -Ce-1 :s C # 1F d(Cv(t 5))+dt # -e-1 # -5.> coulomb+sec. (d) vc(t) # e-t t5 # 1 sec. vc(t5) # e-t5 vc(1) # e-1 # 5.> $olt (e) wc # wc # (1+2)Cv 2 # (1+2)(1)(e -t)2 # (1+2)e-2t wc(t5) # (1+2)Cv 2 # (1+2)(1)(e-t5)2 # (1+2)e-2t5 t5 # 1 sec. wc(1) # (1+2)e-2(1) wc(1) # (1+2)e-2 wc(1) # (1+2)(1+e 2) H1+e2 # 5.1=I wc(1) # (1+2)(5.1=) wc(1) # (1+2)(5.1=) # 5.5> Woules (f) v (t) # i # e -t(1) # e-t $olts v (t5) # i # e-t(1) # e-t5 $olts t5 # 1 sec. v (1) # i # e -t(1) # e-1 $olts # 5.> $olts (g) dwc+dt # wc # (1+2)e-2t dwc+dt # d(1+2)e -2t+dt dwc+dt # (1+2)e -2t(-2) # -e-2t dwc(t5)+dt # (1+2)e -2t(-2) # -e-2t5 t5 # 1 sec. dwc(1)+dt # (1+2)e -2t(-2) # -e-2(1) dwc(1)+dt # (1+2)e -2t(-2) # -e-2 dwc(1)+dt # (1+2)e -2t(-2) # -e-2 # - 5.1= w!tts. (h) * # i2 # (e-t)2(1) # e-2t *(t5) # i2 # (e-t)2(1) # e-2t5
Muhammad Irfan Yousuf Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi t5 # 1 sec. *(1) # i2 # (e-t)2(1) # e-2(1) *(1) # i2 # (e-t)2(1) # e-2 *(1) # i2 # (e-t)2(1) # e-2 # 5.1= w!tts. 1-2. %olution& (!) C # (1+)(?) # ?+(?+t) # t (b) 9+ $ # 9di+dt 9 # $+(di+dt) 9 # $dt+di # $+ 9+ # ($dt+di)+($+) # $ 2dt+di (c) 9C # ($dt+di)(?+$) # (dt+di)(?) (d) 2C # ($2+2)(?+$) # $?+ 2 # $( t)+2 # $ t+ (e) 9C # ($dt+di)+(?+$) # ($2dt)+(?di) # ($2+di)+(1+(?+t)) (f) 9+ 2 # # ($dt+di)+($ 2+2) # dt+$ # ?+$ # C
1-A.
1.5 vc(t)
sint 5.=
Muhammad Irfan Yousuf Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
5
+
+2
= +
for vc -5.=$olt C # (-1.5 / 5.=)+(-1.= / 5.=) # -5.=+-1 # 5.= F
for 65.= vc 5.= C # (5.= / 5.=)+(5.= / 5.=) # 1+1 # 1F for 5.= vc 1.= C # (1.5 - 5.=)+(1.= - 5.=) # 5.=+1 # 5.=F for 5 vc
5.=
for 5.= vc for 5.= vc
1 5
ic(t) # d(Cv)+dt # Cdv+dt / vdC+dt ic(t) # Cdv+dt / vdC+dt for 5 t + C # 1F $ # sint ic(t) # (1)dsint+dt / sintd(1)+dt ic(t) # cost ic(t) # d(Cv)+dt # Cdv+dt / vdC+dt ic(t) # Cdv+dt / vdC+dt for + t = + C # 5.=F v # sint ic(t) # (5.=)dsint+dt / sintd(5.=)+dt ic(t) # (5.=)cost ic(t) # d(Cv)+dt # Cdv+dt / vdC+dt ic(t) # Cdv+dt / vdC+dt for = + t
for 5 t for + t for = + t
+ = +
Muhammad Irfan Yousuf Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi C # 1F v # sint ic(t) # (1)dsint+dt / sintd(1)+dt ic(t) # cost
vc(t)
5
+
+2 time
ic(t)
= +
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
5
1-5.
+
+2
= +
time
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
5
1
5.2=
5.2=
2
5.2=
5.2=
5.2=
2
5.2=
5.2=
time
5.2=
2
vc(t) 2t -2t / 2t 6 -2t / " 5 vc(t) 2t 2t -2t / -2t / 2t 6
interv!l for 5 t 1 for 1 t 2 for 52 t for t for t interv!l for 5 t 5.2= for 5.2= t 1 for 1 t 1.>= for 1.>= t 2 for 2 t 2.2=
C!!citor(v!lue) 1F 5.=F 5.=F 1F 1F
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 2t 6 -2t / " -2t / " 5
5.=F 5.=F 1F 1F
for 2.2= t for t .>= for .>= t for t
For the rem!ining !rt see 1-A for reference. 1-2> 6 1-". (%ee ch!terV for reference) efore solving ch!terV1 following oints should be ket in mind& 1. $olt!ge !cross !n inductor 2. Current through the c!!citor . Pr!hic!l !n!lsis . *ower dissi!tion
-1. %olution& 1 *osition B17 /
switch $
9
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
: ste!d st!te current h!ving reviousl been est!blished in the 9 circuit. 4h!t does th!t me!n 1
$ i %hort circuit
n the ste!d st!te inductor beh!ves like ! short circuit i(5-) # $+ 1 (current in 9 circuit before switch Bk7 is closed) n !n inductor i(5-) # i(5/) # 5 t me!ns th!t i(5-) # i(5/) # $+ 1 G is moved from osition 1 to osition 2 !t t # 5.
1
9 2
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi for t 5 :ccording to kirchhoffs volt!ge l!w %um of volt!ge rise # sum of volt!ge dro Circuit simlific!tion&
( 2 / 1)
9
(!) 9di+dt / ( 1 / 2)i # 5 9di+dt # -( 1 / 2)i di+dt # -( 1 / 2)i+9 di+i # -( 1 / 2)dt+9 ntegr!ting both the sides, di+i # -( 1 / 2)dt+9 di+i # -( 1 / 2)+9 dt lni # -( 1 / 2)t+9 / C i # e-(1 / 2)t+9 / C i # e-(1 / 2)t+9eC i # ke-(1 / 2)t+9 :ling initi!l condition i(5/) # $+ 1 i(5/) # ke-(1 / 2)(5)+9 i(5/) # ke5 i(5/) # k i(5/) # $+ 1 i(5/) # k O?u!ting G # $+ 1
(b)
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi i # ke-(1 / 2)t+9 i # ($+ 1)e-(1 / 2)t+9 is the !rticul!r solution. -2. %olution& %witch is closed to b !t t # 5 niti!l conditions v 2(5/) # 5 i(5/) # $ 5+ 1 for t L 5 (1+C1) idt / (1+C2) idt / 1i # 5 Differenti!ting both sides with resect to Bt7 (1+C1)i / (1+C2)i / 1di+dt # 5 (1+C1 / 1+C2)i / 1di+dt # 5 i+Ce? / 1di+dt # 5 i+Ce? # - 1di+dt di+i # (-1+Ce? 1)dt ntegr!ting both the sides di+i # (-1+Ce? 1)dt di+i # (-1+Ce? 1) dt lni # (-1+Ce? 1)t / k 1 i # e(-1+Ce?1)t /k1 i # e(-1+Ce?1)tek1 i # ke(-1+Ce?1)t :ling initi!l condition i(5/) # ke(-1+Ce?1)(5) i(5/) # ke5 i(5/) # k(1) i(5/) # k i(5/) # $5+ 1 O?u!ting G # $5+ 1 3herefore i # ke(-1+Ce?1)t i # ($5+ 1)e(-1+Ce?1)t t v2(t) # (1+C2) idt 5 t v2(t) # (1+C2) idt / (1+C2) idt 5 t # v2(5/) / (1+C2) ($5+ 1)e(-1+Ce?1)tdt
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 5 t (-1+Ce?1)t
# 5 / (1+C2) ($5+ 1)(-Ce? 1) e
5 t # (1+C2) ($5)(-Ce?) e
(-1+Ce?1)t
5 # (1+C2) ($5)(-Ce?)Me - e(-1+Ce?1)(5) # (1+C2) ($5)(-Ce?)Me (-1+Ce?1)t 6 e5 # (1+C2) ($5)(-Ce?)Me (-1+Ce?1)t 6 1 (-1+Ce?1)t
v2(t) # (1+C2) ($5)(Ce?)M1 - e (-1+Ce?1)t t v1(t) # (1+C1) idt 5 t v1(t) # (1+C1) idt / (1+C1) idt 5 t # v1(5/) / (1+C1) ($5+ 1)e(-1+Ce?1)tdt 5 t # -$5 / (1+C1) ($5+ 1)(-Ce? 1) e
(-1+Ce?1)t
5
t # (1+C1) ($5)(-Ce?) e
(-1+Ce?1)t
5 # (1+C1) ($5)(-Ce?)Me - e(-1+Ce?1)(5) # (1+C1) ($5)(-Ce?)Me (-1+Ce?1)t 6 e5 # (1+C1) ($5)(-Ce?)Me (-1+Ce?1)t 6 1 (-1+Ce?1)t
v1(t) # (1+C1) ($5)(Ce?)M1 - e (-1+Ce?1)t $ (t) # i 1 # (($5+ 1)e(-1+Ce?1)t) 1 $ (t) # i 1 # ($5)e(-1+Ce?1)t !
b
/
/ $5
G
1
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi -
v2 C1 C2 -
!
b
1 C2
////////
------------
---------------
/////// C1
-. %olution& k is closed !t t # 5 niti!l condition i(5/) # ($ 1 6 $2)+ 1 for t L 5 C!!citor ch!rging (1+C (1+C 1) idt /is 2) idt / i # 5 $olt!ge !cross both the sides with resect to Bt7 Differenti!ting c!!citor # $5 / di+dt # 5 (1+C 1)i / (1+C2)i (1+C1 / 1+C2)i / di+dt # 5 i+Ce? / di+dt # 5 i+Ce? # -di+dt di+i # (-1+Ce?)dt ntegr!ting both the sides di+i # (-1+Ce?)dt di+i # (-1+Ce?) dt lni # (-1+Ce?)t / k 1 i # e(-1+Ce?)t /k1 i # e(-1+Ce?)tek1 i # ke(-1+Ce?)t :ling initi!l condition i(5/) # ke(-1+Ce?)(5) i(5/) # ke5 i(5/) # k(1) i(5/) # k i(5/) # ($1 6 $2)+
C!!citor is disch!rging $olt!ge !cross the c!!citor # - $ 5
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi O?u!ting G # ($1 6 $2)+ 3herefore i # ke(-1+Ce?1)t i # ($1 6 $2)+)e(-1+Ce?)t t v2(t) # (1+C2) idt 5 t v2(t) # (1+C2) idt / (1+C2) idt 5 t # v2(5/) / (1+C2) (( $1 6 $2)+)e(-1+Ce?)tdt 5 t (-1+Ce?)t
# $2 / (1+C2) (($1 6 $2)+)(-Ce?) e
5 t (-1+Ce?)t
# (1+C2) ($1 6 $2))(-Ce?) e
/ $2
5 (-1+Ce?)t
# (1+C2) ($1 6 $2)(-Ce?)Me - e(-1+Ce?)(5) / $2 # (1+C2) ($1 6 $2)(-Ce?)Me (-1+Ce?)t 6 e5 / $2 # (1+C2) ($1 6 $2)(-Ce?)Me (-1+Ce?)t 6 1 v2(t) # (1+C2) ($1 6 $2)(Ce?)M1 - e (-1+Ce?)t / $2 (i) t v1(t) # (1+C1) idt 5 t v1(t) # (1+C1) idt / (1+C1) idt 5 t # v1(5/) / (1+C1) (($1 6 $2)+)e(-1+Ce?)tdt 5 t (-1+Ce?)t
# -$1 / (1+C1) (($1 6 $2)+)(-Ce?) e
5
t
Muhammad Irfan Yousuf Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi # (1+C1) ($1 6 $2)(-Ce?) e(-1+Ce?)t - $1 5 (-1+Ce?)t # (1+C1) ($1 6 $2)(-Ce?)Me - e(-1+Ce?)(5) 6 $1 # (1+C1) ($1 6 $2)(-Ce?)Me (-1+Ce?)t 6 e5 6 $1 # (1+C1) ($1 6 $2)(-Ce?)Me (-1+Ce?)t 6 1 6 $1 v1(t) # (1+C1) ($1 6 $2)(Ce?)M1 - e (-1+Ce?)t 6 $1
(ii)
from (i) v2(t) # (1+C2) ($1 6 $2)(Ce?)M1 - e (-1+Ce?)t / $2 v2( ) # (1+C2) ($1 6 $2)(Ce?)M1 - e (-1+Ce?)( ) / $2 v2( ) # (1+C2) ($1 6 $2)(Ce?)M1 - e -( ) / $2 v2( ) # (1+C2) ($1 6 $2)(Ce?)M1 - 5 / $2 v2( ) # (1+C2) ($1 6 $2)(Ce?) / $2 v2( ) # (1+C2) ($1 6 $2)(Ce?) / $2 from (ii) v1(t) # (1+C1) ($1 6 $2)(Ce?)M1 - e (-1+Ce?)t 6 $1 v1( ) # (1+C1) ($1 6 $2)(Ce?)M1 - e (-1+Ce?)( ) 6 $1 v1( ) # (1+C1) ($1 6 $2)(Ce?)M1 - e -( ) 6 $1 v1( ) # (1+C1) ($1 6 $2)(Ce?)M1 - 5 6 $1 He-( ) # 5I v1( ) # (1+C1) ($1 6 $2)(Ce?) 6 $1 v1( ) # (1+C1) ($1 6 $2)(Ce?) 6 $1
Muhammad Irfan Yousuf Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 2.5
2
2
1.5 t n e r r u c
Series1 Series2 1
1
1
0.5
0
0.05
0
0.002
1
2
3
Series1
0
1
2
Series2
1
0.05
0.002
time
v2(t) # (1+C2)($1 6 $2)(Ce?)M1 - e (-1+Ce?)t / $2 v2(t) # (1+(1+2))(2 6 (1+(1+2))(2 6 1)(Ce?)M1 1)(Ce?)M1 - e(-1+Ce?)t / 1 v2(t) # 2(1+)M1 - e (-1+(1+))t / 1 v2(t) # (2+)M1 - e -t / 1
Muhammad Irfan Yousuf Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 2.5
2
2
1.666 1.634
1.5
Seri es1 Seri es2
1
1
1
0.5
0
0 1
2
Series1
0
1
2
Series2
1
1.634
1.666
time
:t t # 5 switch is moved to osition b. niti!l condition i 91(5-) # i91(5/) # $+ # 1+1 # 1:. $2(5/) # (-1)(1+2) # -5.= volts for t 5, GC9 (1+1) v2dt / v2+(1+2) / (1+2) v2dt # 5
3
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi (1 / 1+2) v2dt / 2v2 # 5 (+2) v2dt / 2v2 # 5 Differenti!ting both sides with resect to Bt7 (+2)v 2 / 2dv2+dt # 5 Dividing both the sides b 2 H(+2)+2Iv 2 / (2+2)dv2+dt # 5 (+)v 2 / dv2+dt # 5 %olving b method of integr!ting f!ctor * # X, U # 5 v2(t) # e-*t e*t.Udt / ke-*t v2(t) # e-*t e*t.Udt / ke-*t v2(t) # e-(+)t e(+)t.(5)dt / ke-(+)t v2(t) # ke-(+)t :ling initi!l condition v2(5/) # ke-(+)(5/) v2(5/) # ke5 v2(5/) # k(1) v2(5/) # k -5.= # k v2(t) # -5.=e-(+)t efore switching
%hort circuit
-
!
b
n c!se of D.C. inductor beh!ves like ! short circuit
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 1
e0!nding
+
!
O?uiv!lent network !t t # 5/
Coll!sing -
/
-=. %olution& %witch is closed !t t # 5 niti!l condition&i(5-) # i(5/) # (25 / 15)+(5 / 25) # 5+=5 # += : for t 5, :ccording to G$9 %um of volt!ge rise # sum of volt!ge dro 25i / (1+2)di+dt # 15 @ultiling both the sides b B27 2(25i) / 2(1+2)di+dt # 15(2)
5i / di+dt # 25 di+dt / 5i # 25 %olving b the method of integr!ting f!ctor * # 5 -*t *t i(t) # e e .Udt / ke-*t
U # 25
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi i(t) # e-*t e*t.Udt / ke-*t i(t) # e-5t e5t.(25)dt / ke-5t i(t) # 25e-5t e5tdt / ke-5t i(t) # 25e-5t(e5t+5) / ke-5t i(t) # 1+2 / ke -5t :ling initi!l condition i(5/) # 1+2 / ke-5(5/) i(5/) # 1+2 / ke5 i(5/) # 1+2 / k(1) += # 1+2 / k += - 1+2 # k 5. 6 5.= # k 5.1 # k 3herefore i(t) # 1+2 / ke -5t i(t) # 1+2 / 5.1e -5t 3ime const!nt # C # 1+5 # 5.52= secs. i(t) # 1+2 / 5.1e -5t # 1+2 / 5.1e-t+C 1+C # 5
%te!d st!te lus tr!nsien t
C # 1+5 5. 5.=
s.s.# 5.=
5.1 5
time
6
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 16
efore switching
n c!se of D.C. inductor beh!ves like ! short circuit
:fter switching
16
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
-. %olution& %witch is 5ened !t t # 5 niti!l condition&i(5-) # i(5/) # 15+25 # 1+2 : for t 5, :ccording to G$9 %um of volt!ge rise # sum of volt!ge dro (25 / 5)i / (1+2)di+dt # 5 =5i / (1+2)di+dt # 5 @ultiling both the sides b B27 2(=5i) / 2(1+2)di+dt # 5(2) 155i / di+dt # 5 di+dt / 155i # 5 %olving b the method of integr!ting f!ctor * # 155 -*t *t i(t) # e e .Udt / ke-*t i(t) # e-*t e*t.Udt / ke-*t i(t) # e-155t e155t.(5)dt / ke-155t i(t) # 5e-155t e155tdt / ke-155t i(t) # 5e-155t(e155t+155) / ke-155t i(t) # += / ke -155t :ling initi!l condition i(5/) # += / ke-5(5/) i(5/) # += / ke5 i(5/) # += / k(1) 1+2 # += / k 1+2 - += # k 5.= 6 5. # k - 5.1 # k 3herefore i(t) # += / ke -155t i(t) # += - 5.1e -155t 3ime const!nt # C # 1+5 # 5.52= secs. i(t) # += - 5.1e -155t # += - 5.1e-t+C
U # 5
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 1+C # 155 C # 1+155 # 5.51 secs.
5.
%te!d st!te lus tr!nsien t
5.=
- 5.1 %te!d st!te lus tr!nsien t
efore switching&
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
:fter switching&
n c!se of D.C. inductor beh!ves like ! short circuit
->. %olution& niti!l condition v c(5-) # vc(5/) # v2(5/) # 5 for t 5 (v2 6 v1)+ 1 / Cdv2+dt / v2+ 2 # 5 v2+ 1 6 v1+ 1 / Cdv2+dt / v2+ 2 # 5 v2+ 1 / Cdv2+dt / v2+ 2 # v1+ 1 v2+ 1 / v2+ 2 / Cdv2+dt # v1+ 1 v2(1+ 1 / 1+ 2) / Cdv2+dt # v1+ 1 Dividing both the sides b BC7 v2(1+ 1 / 1+ 2)+C / Cdv2+Cdt # v1+C 1 v2(1+ 1 / 1+ 2)+C / dv2+dt # v1+C 1 C # (1+25) F
1 # 15-ohm
2 # 25-ohm
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi v2(1+15 / 1+25)+(1+25) / dv 2+dt # e-t+H(1+25)(15)I v2(5.1 / 5.5=)+(5.5=) / dv 2+dt # e-t+H5.=I v2(5.1=)+(5.5=) / dv 2+dt # e-t+H5.=I v2 / dv2+dt # 2e-t
v2(t) # e-t - e-t %ketch v2(t)
U # 2e-t
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi 2.5
2
2
1.5
e g a t l
Series1 Series2
o v
1
1
0.5
0.319
0.133
0
0 1
2
3
Series1
0
1
2
Series2
0
0.319
0.133
time
ou should imlement ! rogr!m using W:$: for the solution of the e?u!tion v 2(t)
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
import java.io.*; public class Addition { public static void main (String args []) throws I!"c#ption { $u%%#r#dad#r stdin ' n#w $u%%#r#dad#r (n#w InputStr#amad#r(Sst#m.in)); doubl# # ' .+,; doubl# a- b; String string- string+; int num+- num; Sst#m.out.println(#nt#r th# valu# o% /0); string ' stdin.r#ad1in#(); num ' Int#g#r.pars#Int (string); %or(int c ' 2; c 3' num; c44){ Sst#m.out.println(#nt#r th# valu# o% t0); string+ ' stdin.r#ad1in#(); num+ ' Int#g#r.pars#Int (string+); a '(doubl#)(+56ath.pow(#- num+)); b '(doubl#)(+56ath.pow(#- 7*num+)); Sst#m.out.println(8h# solution is0 4 (a 9 b)); :55%or loop :55m#thod main :55class Addition -A. %olution& Network !tt!ins ! ste!d st!te 3herefore i2(5-) # $5+ 1 / 2 i2(5-) # +15 / = # +1= # 1+= :m. v!(5/) # i2(5/)( 2) v!(5/) # (1+=)(=) # 1 $olt
for t ≥ 6 :ccording to kirchhoffs current l!w&
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi (v! 6 $5)+ 1 / v!+ 2 / (1+9) v!dt # 5 utting 1 # 15, 2 # =, $5 # 9 # J (v! 6 )+15 / v!+= / (1+(1+2)) v!dt # 5 (v! 6 )+15 / v!+= / 2 v!dt # 5 v!+15 6 +15 / v!+= / 2 v!dt # 5 v!+15 / 2 v!dt # +15 Differenti!ting with resect to Bt7 d+dtHv!+15 / 2 v!dtI # d+dtH+15I d+dtHv!+15I / d+dtH2 v!dtI # d+dtH+15I (+15)d+dtHv !I / 2v! # 5 (+15)d+dtHv !I # - 2v! d+dtHv !I # - 2v!+(+15) d+dtHv !I # - 25v!+ dv!+v! # - 25dt+ ntegr!ting both the sides dv!+v! # - 25dt+ lnv! # - 25t+ / C v! # e-25t+ / C v! # e-25t+ eC v! # ke-25t+ :ling initi!l condition v!(5/) # ke-25(5/)+ v!(5/) # ke5 v!(5/) # k(1) 1 # k 3herefore v! # ke-25t+ v! # (1)e-25t+ v! # e-25t+
efore switching
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
/ -
:fter switching
/ -
n c!se of D.C. inductor beh!ves like ! short circuit
/ -
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
-15. %olution& G is oened !t t # 5 ut v2(5-) # v2(5/) # (1+)5 for t 5, GC9 v2+1 / (1+2)dv2+dt # 5 v2+1 / (1+2)dv2+dt # 5 @ultiling both the sides b B27 2v2+1 / 2(1+2)dv 2+dt # 25 2v2 / dv2+dt # 25 dv2+dt / 2v2 # 25 %olving b integr!ting f!ctor method *# 2 -*t *t v2(t) # e e .Udt / ke-*t v2(t) # e-2t e2t.(25)dt / ke-2t v2(t) # 25e-2t e2tdt / ke -2t v2(t) # 25e-2t(e2t)+2 / ke-2t v2(t) # 5e5 / ke-2t v2(t) # 5(1) / ke-2t v2(t) # 5 / ke-2t :ling initi!l condition v2(t) # 5 / ke-2t v2(5/) # 5 / ke-2(5/) v2(5/) # 5 / ke5 v2(5/) # 5 / k(1) (1+) 5 # 5 / k (1+) 5 - 5 # k -(2+)5 # k v2(t) # 5 / ke-2t v2(t) # 5 / (-(2+) 5)e-2t v2(t) # 5(1 - (2+)e -2t) efore switching
U # 25
Muhammad Irfan Yousuf Dedicated to: Prof. Dr. Sohail Aftab Qureshi
:fter switching
n c!se of D.C. c!!citor beh!ves like !n oen circuit -12. %olution& %witch closed !t t # 5 niti!l condition&i9(5-) # $+( 1 / 2) i9(5-) # i9(5/) # $+( 1 / 2) for t 5, G$9 1i / 9di+dt # $ Dividing both the sides b B97 1i+9 / di+dt # $+9 %olving b integr!ting f!ctor method * # 1+9 -*t *t i(t) # e e .Udt / ke-*t i(t) # e-(1+9)t e(1+9)t($+9)dt / ke -(1+9)t
U # $+9