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
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 <: