b) y 3

2­

1\

c) The. ~ ~e.no~ ~fe9u.etlCi.eJ.> {cn­lO'hic.h im.. hnnSfu: bUxu:r\or rrdca.b~r .. QO" (t(J;)ono.l1ca..') .

1he.. t:~ ol~ inoee- f~ue.(')ci~ I \P'fUch ~ b", ~-+hro~ -the. S~.

d) PlOT" - 6~

OoE

-r:. 'i + 'I .. k' ( 1I .. ~ J'U d-l:. +-Iö ü)

e) HQJhe.rnaJ'ica..Q ~".i"'on ph~ rnosain ~R

<D R 180· + 'fo (LOs)w;:

A-rneJti-h....ole.. m o.r8~ n..

A :.__1__ ft 'b 0 ~ (,Oe) I

Prob\e.m -1

e)JG-Ic(ß

o

a) Pr2, h"o-n6f.e.r be..ho..l.rior

Tl. '1 ~ ~ 'i ... y = k· '1.t

et {cfl.~) 3 = 1\

'F es) ~ GGs) . o/j [dL.f:)J =. Go (6} . - 1

~o~ <f' n.o.R. \Jo.L.a.

~ f Ce) - 1.Urt 8· G(~)-l:. -c~ - 8;:> 0

= 0 ~

b) J)()..rT'\fin~ 0t a.. fa...U- cf .eA8QnV~

d. cos ( a.io..n. tlm t'\ i '] ) ~~Ai 3

A" : - Dl;t Q,d A2. :::; - q. 1: ~

cl "-1 ~ c.cs (~-kLn (4-)) ~~ = cos (~ C:.t)) ~ o. 8Slt~

:::. ­~

ol2.2. ~ cos (a.~ c*))0\ "'2, =. cos (o..~ (~ ~Q. )) := o. RS 4­

-@ fit = Q., The.. .l1ear.kc::h~nß i.s c4./etr2. ~ 4,l.

PC"Dble.m ~

c.)

A ~ !:J:. .~ ..".

i-l§2.b=. 11 .'"~T·I I .. a i

t-:>-1- "5 "3 -11 -1 3 5 T

d)

Pr2, - h-ansf-u fu.ncrticn

<;(5) _ -1 ~ Tz. s"2. ... (~~T2.) S'- t-4

G (5) -=- YCs) l{ (s,)

~? 1 ~ -1 .-1 ~1i.~ -t(r.+li)s+s Ttr"2. S4+(l;.T"li)5......, s

AU(.~):: --:5 l.t (~). ~ (t)

~ s ~ + \.j ~ ~ ( LI ... 106 "\A ) 't -=; kz.· U

<;~ (&) Q 1-'\.4' (Ä ... .0 S ) -1 ..... l.-t S

3

e..) Re~CR... l-ra..nsF fu..nch'or1

bW(S.) = ~R~s ..Go f\JRN6 ..... 2-R~s 1 +- Go

J).i&hJ..c ba.nc.e h-0.t\6fe.r tu.ndiOtl

tJR. ~s Gs ~~) .. :

kJ~Ns .... eR.1: s /I + Go

\0<2. • ( TÄe» -i" 4 ) c

~5+'1 T \(~K", (1+iös)

eho.ro..c..~ eft-lc, ~ u.a...4-1'on

1Ä ~ + -1 -t- k", K2 ... K;z "" 10 s "7 0

-1 ... K,.. K, "7 0

/\ K A k'2. "7 - -1 _17t-

Tl>

~ -InQV1SreJt~)UVlCliOV'l of1he j)?cl,nkeJ InJ1e7t S~~+em

Gi cs) = YilSjjUUS) :

Ui'l+J-~"....a.------~

G,{s) Gl(S)

__ r G'2.(S)

T Yl't-HG,(S)Vi (tl (;2(S) GJ(sJ ....... .,.1+ . .....- ) -,'"> G2(SJ - ~

G4(S) jf!(i I

U,Hl 1+ Qis) G~(sJ GJ(sJ Yi{f ....... -. .,. ... ~

G'2.(S) 1 + G'2 (s) G3(S) b4(s-) I

,

I j

. G.2(S!G,3(S) + "" Cl(<;)~' G;(s) f' ~.

bi(sj =-----------­t + GZ(S)G3(5J (;4(5)

'1, (SJCi(sh Q3(S) ( G'2(S}-t GI(SJ) _ :: +lU'(sj1 ..... G2lS) G3ls)bQ(S)

--------~ G&{ S) ~---•

. J

.c) -me 'S~te $~e moJef of 1Tie s~s~ '-·0 I o~ 0

-, ' ~(S)

\lot})

.J B =- 0A= 0 o I " ' o -6 -5 ~'-'

:1 d .::::. 0c=[~'.t 6 2.] o

\ 0o o l Xl. -+ 0

:- 0

-6 -s )(~ 12o

6 2-] [~:'l + [0] uy=[-\

-.. d "2 ~(+J -LfU.(t}-12 )u.H)dt ~-la~H) -\2~(-O+2SU.ltlcH'+12'UtJ

~

2 ~'n-) -1.\ ü. (+ J- \2 u.(.+i oll = _\t.')~ H\- \ 1. ~ (t ) -t·2 lUt}f l2 U(-\--)

=J> ... " u .

. ~(t\+5~(+)+6~H) ::2Ü(-\) +60,lH +U(+}

~ [~·(.o 5 ~ l-ti +6~l tl] :: rL2 Lic+)+6~(+J+U.{+]-I­

L0~ + 552 -I' 6:sJ Ytsj ~[ '2. 52 +6S+1] llLs)

G(s)=. '1is) -= 2:2-t6~+l l u(s) s 3..... 5S -\-.6S i

2e) G(s j = 2. S -\- 6' s-\- I S~-i 1?.s2. -t~ 6$

Chqnect~ \Is-l~ e9\.t~t\ön t

$'3-+_ 55 2+6s == ~ es (S ~ 155>+6) ::00

$\ ~O J S 2. ~ - 2 ;J 'S3 :; -,3

~ 1l)e S~5{eM is ~ovvJ4n~ -skbie,

[.oJ

I I

-to .1---_-L---+---4---L... ~._,...j-.~I-

0..)

with GRJ'\

(;01(5); Gr, C~) GCH (»

_ 51<107 (~TS") (S+/fo) ~_

- ( St -1 ) (~tS'o) (5t-""IOo) (s t $:(0) (~. "t.rf'N)O)

( +;-1) (;0 + -1 )

(5;-1) (~f 1)('1~t> +1) ( t--s +"1) (2-+ 1 ) ~v . ~oo 1~QQ

......i---1--~~-"""O-"'-.-- ...-·-·....-·. "'---7> "'1 . 5 10 S'o -1f't:' (.(.J

'I J r'Wftfl l7rR:l

Gof S ') ;:

(S +-1) (St ~o) ( St Icv) (5+500) ('>t10c:0)

ßoo ( g1-1) (*+-1) (!~T7"t /f)-- (*,'2 )

P~obk1r1 S

Ct) w\.J1 er Q2

\Gldß _L",1 n.Yl [oJ r__ ~~..,

[,,1 \.- ~'-. 1_' """J

'~4iJ

, [-].;J

p~t o<?

-9l o

-180

I I

:: s~<> g"" 4"'''.:-....,. ~:---r---t-+-I--~ ...-----.:J~.[,' , .--.-_.

h) Phase ~JfI (GrO'7 )

tpJn of G~1(S,): kiJr"1= '1 t{nJrn Ej. (..j(./fJ )

[Goi f O )/-= 0 dB =7L6<>1 (0) -== 0°

~jr101= 0 0

_ (-1go"');;: 19,/"

Arnrpbt«de f1l./ArEi n [ Gu;) ) Q

Y'LtI7J.. eteSIl/.t GtC;,-055 - -1 ßO f( u): 0 -=1 ~

C) Vles ,

- -tJifL ofen Lop15 Clrc g;tp.b1e. / beawse tk8 hAve sam-e.

0 0 000~f9.b&. PO&-S 5'1;;; -/] S'~= -5'0 ~3= -1 5'f=-~oO s;:s-= -1 •

.- M f0k.s Oe. t tluz. o'0;'n.

- ne3c<tf've f~clbc< cl::.

d) T~ cLs<? - [, 0 f w/tA Gp (S) an) t?r. (s) /s stJ;Le./

becaf!-Se t~ ;lta..s€ tna'(1/I1. ~~ G01 >0 '"'.

Tk close - toPp (A);V~ Gr{S) Go (S) /s stai&-./OoYlc!

be07vUscz- v~ OßL-fh' Uuie... f1tt;{'!J/n AR; G-D;) > ~

e) p(j~r plo-t G01 (s)

Re

~e/u5 : ?cles ~

<A) S01 -:. -~ 51 -:. D

'SO'l.. -:. 1 ~"Z.. =. - DlS"

53 =- -z.~o~ -:: <­S't -;. - ~

j:;/b =. -1 !. P __ A -\- .\ - -l_

-" J)l.~ "" c..v =05" S

01. '

]~~ -1

b)

c ) 'y (~) -= kp. V\ (~1

t i~- i iUt- [lk uj) lÄlq ::\)-11

-=::> y') 0 ,{oleS-

V) 0 }e... o~

-1L~ ~ (5) . S +- '( (05) = L<J) . UCS)'j

z k] .~

1

os :: 1 5~ = Z ~ ( ...

te,D.

l5-1) ( 5 -"Z..) ,

J) C. (5) - (; (5) . G (s) ~ p. ~1 =-:} S-t ::: t>0,

5 - -0,­~~1..:.1 l.- •

L<i> t S t ~, (S .. 1) c.:> - Z ) S, ::. - 'l.- - "5 0 '3= "t.-::>'i ::. - .~5( :; t 0\5") ( .s ~"2) (~ t ~:j ( s( +2 5 -\ L) .

5s-/6::' - 1 ~ 1

Gm.(5) -- C~(S) . G] (.~)

-OtS"501( :. - 3 S1 '= "Z..L<J) (5'\"3)

-- (~ :. ­

( ~ ~ 0, S" ) (~ fZ.) (s "" '-t' Ls'l t1.s t 1 ) 5') =- , ,~

5c,.):l - , t 1 ~

e) 5'1) k", 11: . Nv\M bfl·" _.Gi t ?.lc$ vt":: b .fi!fAw: b?,- "t- _t:~'os . W\:."3

-7 t S~ f""- k b,c>-"I c1.eS

J\ "".l. I I I· r: .\ H -M -::. J Qf"-"\c.~e5 J.o. to .tVlt'V\ ~ t-'\{

s~ s\oe ""l 1.. \ }J<A.rr-hr,- . °t ~JCj v\:. S­

j'VoAlI'r\1.:J9" . 0t 1-e'~~' ~ =~

- ~ S S~ f'" t ... k brt)-'1A"l"es " . \ (.. ,V\ - ~ = <t tI ( ..... >"\ c..l-"e:.s ~ -0 \"'\ (":ho l ~ I0

'1 "'" 2: j. _ ~ :50 :

I:" ' ,=1.

:. (0- 0, ~ -L -\- - '1 K -'1~ -l- 3 +-11 -\- 't) )

-}.

. i-- D 'I . tY\-~-'\\ • I" )

o

-=-:/ ~~ ~ 60 ~< ~ {~Oo

~) = ~oo 0

~5~...... <. : t"l." (-0,S" - J - lt -1 rC- 1..v;) - ( -) )

4lz. -=- '13S-°

~t3 ::. <-'-li·

/

= -1

t 14::- " o

llt I ~ 1- / /0

7Dll~ 1 \ -= 1-1 ~ -1 ~ - ~ 1 S" 0 >l<

\ \\")\ -:. \HJ 0." =1~O-l'1~ t l,{ 1 70

1) ::

~ ': C D:>

{os tD"C 1-", l ~));(DJ fd 0(

b -- ­

\1..(

J 1 \.'-: ~dS} ~ Jlo'\l'l

~ ~~ o Je.

l< - I ~ ~~) ~4)1 ~ 6<:-ct) t 1~' I = ((-:)1~'1(,-1<t t~iJr

= 1-3o ( =- 30 -

LJ I :t,i'1{ .....11J·".0l.. l t.l \~ P'l.~ 11' 1'"&I) ,

---e-~+----~\?J) : ~""( &J1..,..o<\.,\ 1r!~ P-r

1 L

M) r: (s) If-.s s (1 +-S'1)T1s (1 t ST,\1 "1c; (~) '::. k [-1 \- s T J) - ~(1+~~~)-l

o I

_b0 '" Jo..,\(

5t ~1 <:­

.1 /( 1) - t - -- ­5 5t-S

- S~\-1S