This document contains mathematical calculations and derivations involving parameters such as s, K, and z. It determines that certain expressions equal zero, finds limits, and identifies poles and zeros of transfer functions. The document also contains comments on closed-loop stability analysis and indicates that the poles are real when K is less than one.

1. 6ou
J
ch
t
L
A,-
{
Sra
# +ltCt

2. 7
-r-
sJ rS AT COMPK
in

3. "tD;t
r;tclio o
* Lern i
I
dlte?r
NC
: /€
he:l. Qn
ir) rrlr.b O o
Tqh h
in th Tf
ejJ rrern&vs
t suj4;r ;e n t
g^r ;s ti
fa the
the Ch
Crite'rriQO
Ch d.ho cte ?f t
Fts) €- gn
,trfherfe Coe
eca
:n
Alt
o'r-.[-d h
rnatio n
l+ io
h
ha,l
od +h nomf

4. TO
u
I
in
L:
L

5. ? H c 6iterrioo
al I in r-be .l+
/ r ha-l fl -
Lab u latio n
"o
,] oJl-
Lh
Ftn
hr z the
s[St 1(s+-z) +-r = rr
1[st t) (stz) _
S{ *+r-s+ q+ z) f 3 = o
- a
.l-
:-
,4.s +ae-ru a'k
no Sf
3

6. T-
---- cornnne nL olStAbj4il
f 'uite'uict, if +he Jb 'Z-F1oS
L t>;bt-
5
-1)(ro) -[t(7 =-
< 2,
J
'----+- t__
_J
t
3
,L
71.-ls
017
I St r
r ,-o
-=----

7. I
TT
Ca
rl
T c '-l he 1>r
-!-
{7
--<- ouUh tctb ioo
p)aff1€
0 Lh
"J
coe oon- Z
CLfO
ne9
rtb f{it
De e.'T m il! +
5 q-r
2_
-5
=-
)-
2:
t
=
Jr
ir
f
-=L
T(
€s k
7
l-!
O
->
; L>L e)enent rc
.-'Ae Qon-zeTc-
f f", . t otD -ffri:
f o s eci
-u de,:
(i t-{t-
-g-c e e rl
r'lu" e-;v-il9

8. -t
__
o
_:-
-L
9e+
3
t
t
t
.D
ax lf: h{ e )<-
.--e.-<
Qn
F--
t -<
-Thr ,:,rJ
iS
u'th bu

9. on th
-joe
_2
7
'I-
L
L
L
Z __ 25 S 5C'=
-2!
-5
=
DiVi'
-25
L
.-z
L
ul ion the
intJ
U" O
a
t,/Dit_

10. q
.lb
.-r
LL)iLh ch@
-+

11. L-6 = -3t
3'|
I L'12 :Q 61^
s
S
s2
rtil +h< ala-lrrL
?al cat
he Q.u
Pa's
= aC4s zsJ=O
2
5
3
j6-j:/
-g-
7
,'u lsitit{- n (

12. dar =L
L
.t
>-
b
a-ile or eo
tLs o
bz
b=z
- 2J
,tcl Yite rpotS
:
-----
./-
7
--------
? ----
-_-_ +
= -2
T
L
2

13. t
L
--
^
+
o
tr
rroobs
2
t3TT iS
aDeflL o n t
r q srd-b rA
of) i_s;
7
2
6-71 = -7
tl s
iri
I
t
6-S
L
'g
T?aht

14. not
e r. ' -v]^ o 1gb
uth'
d;soo I paharneK?r
c-o -ld;;

15. 0r)
I
q
Dh.L)_5K
b* +IY
ri ?r-io o *o
i'0 D'.
"2)*5
K>o
'17<z,z6 -
a<
Sfo
q
CT itt?r ro n

16. ;+
inrli ate s th
C.to" 1t 0,25s
=C

17. L+ o"qt)(t +o'L53,
o"z5s t- e j5
B
0,65
.65
3
Z
L
=6,5
T
---*J
,,
(
I
--06
o
?'-D o "65-
'is
aq
o
TDuc__4
=O
0,65 n-
I

18. o,€5 -5-o
"' O"6 L= -68
= -F i3-x62=
iVEN J
9tc ,'6-l
o"Las
S
K
S'

19. 2-rK -f rt9; o
ztn=-U
-_ a tK
ztK
T
=O
1
,r
cD= L ra/lct c
t )
KtT
T
p.

20. K
Er__
Lirnitqff
;d on -f- rr
,'r) )e ! L--Da-T',
L
-o.t
Lonna

21. 'dt'Vr"
h
rr) ctbOtrF
7
sra)(Sf2
abiJ-l
S---r An
rs )---
s( sr1J$Iz

22. -
jtD
io tb-< 'ha-hactqJU
5* 331 ZS
- 1 )' -1
3 a.
1- 3_ 1) + 3 '+ -J-2
Z)L
'-)
-r1)_ L e
+ -_37
3
a
, 1--
1
U

23. >oab
e+o
S) = qt
-L =-F
o-Ya
hc|
icl z - 7 -.r"n"
D+
Ti
Zs-Q
= 21= i
55
J:--
-@
2
"L
I
F,rp"rr q1 ?rccd
L4 t,/
Fronn
o
-Tlr
3" 36
a))
16.9

24. Locu s ;
c.-I) sSi b&
cne o
-L-he -!,".
De
(,
-4
t,'l P" tr
tni
_f
Ih
ide
h e 70 o_t5
h
art(K
Roc
oLTF =. ts) Ht
i>
c(s)
cLT F =
(s)
rD
rt (5)
K
J
Q=
S
s(sr

25. ,1
,/
gtr Lf
ct
cL>
9 :n
_(, = 2rJ -K =-2
--vLl-6
DC"te
J.c uJes ,1 o,.t,
-*2l-j+-Y,
': -2 t
O tc c^r
- L<- Rcal I
-J I jl 6
-2 - Jtu.2--
'rla{
Z
Si = C.' 26+
lf@
'
f----
:
2 T-l icP

26. rh
oJes
-r, cJeq t't,, rlhen K--1-1
Josed
ho t he P oJ es a-e TeaJ
| ^l
tejh e O
dh( o S K<ce, th
ni
o-les- a>re TtaJ
I
J es a-he Ca
'rl
:
t
:! LrcLt S te JJs Lts h u
llas rr)c,,/( i h
1* C slH(s) '=
US{

27. ft(.:-.li-l (s)
G,Is
n oir- c,r-r ri ifr c'n
: ri cr
t l-L
ruS tC
( _,(
.i-
A
-Lf
<-/
-
* l'tr
/c / 't/,
Lf'
I- zti tt 2t

28. OLT F
v cah;
o
o
'=5
{= 1-e
SI SI
he
j;, o th TO 0t
g i'o *L
=7
K :7
-i-5 -L5 t
1
+
in *t + i
TL
Y V<n
steJGr

29. /)ro
't./ |
Iil-r
-R
urle r : lbg -:{c,r' t l*_'c uS
-1--gto
R tr-X-
+_a:
i+:---L:
iE ZZ
Ncte ]riil lcci te rrn;na?
?r ?enJ c
cn
tdcl nrt
1R L Rt-
--T-
Lt-[ j

30. ints'
*l
A
ToDb }CIcU QNC] =&
Fq -
Zevo
^d =C -I
€ nd
T DO 0k
(4
| .t-
Real
c c LTi=
CT AL CA,
-[oc i s
-[oc
nt-l mbe
-z bra
-r hr
.l
-rb
Le-t
r-n ort< to q2
c tcs ,"ve S
b-r
I
Z
-7- 7
SS"
tE "o)
7t(
h
6
vc= ->P=Tz
0l es
Z<T CL

31. T
I
'6{r)ir-le b{QaFc+4
clr
i[3
6ry1:4ur bve
'F, nd -the
--sdj?='-
c hatr q c+ *'r ;st-ic E g n
" -'--
1 r q(sjHrs)=
1,)rsf 3) f K(Sr6):O
);.: 6(st 6
ni i- Lr l p(1J--. ,--!-'A /J
=_t3)

32. LrS f3
T2
IL|j
$= -2.L
'{S be ttr-,
=
:1a, bz
bet-r-o ee n Z(YC,
7'-LE:
-: JeS
lo
ffi
(n1< )<sve 4
?rDot *cci
cn
v
c
I
CLI Qn CLN
I9c'
,-S +h e nu o f rrJt
Lhe bt e cuL) oi"Q
cio

33. r{
- __ -Lc--r he ?iqht - h, fhp q-
cl}
i'n
is ,.,)t
'l
t rt(
Ag-hcl t scun th
Cc a>t'oqCt tht
in < th
r-k E rrtoile tocrh
C{ar-) cL ) i'Ot !
;c nS , Ar,
hei
'L9 ?
A-o'ra
;n tt D Sr'Cf;
Je-
s )t's
-' t.r > l-hgl '-Ln qU t
ht I ar)

34. 74' = (,3clri bLrh'Co S
1r-,o?,ff
D'b -5ol RooL, k!
--t
;nd
ID he
Lhe otbe t b'tartChr S !.nci
: nd, 'd r_f'
c ; cil +il nb
zeus cn S r'troo
6,7
Ir
€
n,
h
!
t
:C)
lbt
ju-/
?'D S
--1
I
*'-{
I
Ia
-i t
I
i
e
red
(i.
TT t^V
brJr't
an

35. J'.c tancht
S(S+3 (sf 6
t
AE S=Ar-3,-6
-Ln o
1!o -[oc i = =3
<z
RL
*>
;
:_*{
K=o
-b
K: o
oJe
0n the_ recF
<>
--- AS ghoorQ iQ he b cve
.
-t-- l
A a_ / r a
?-r.(Tl = -b LU

36. )
19D : 6C)o
e= 1_)+I) tW
g'
7( D+t
ir-L .'ji/ n.-'
f ..^,j
'S t htze
1 9oo
ri
It Is
K
{s.r3)(s+e
J
t€) c

37. sf 7 g
L +:gs'
,)
- + Jgs r j g I
i
+
+-
r>
<>
<t
b
t-c
t fS th
noU
ev i5 tic
=O
s(s tg(ste
K .- LJ
$- -I"27 .pbt-ucrs-co
+ G,
Sf (2 1g
8s
i xloLL) >T rn A Rc,Ufh

38. ind TOGJ O.
rq[l)-K = o
-., d; t(
-K: O
.-- |
F'qT
LCL= C
=-16L
nsL ?r2Q)
t)
'l-
I-
7
he'
=aGZ
R> L6L
fuc,r'