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8/10/2019 (408177384) FinalExam Solution (1)
1/8
NAME MASTER
(PRINT)
EE105 FINAL
FALL 10
1
2
3
4
5
6
7
TOTAL
TOTAL
1.4
GRADE
Put your answers in the space provie
!appy !o"iays
PROBLEM 1
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Given that Y(s) = (1s) = (s1)
!(s) (s3)(s4) s27s12
"#$%&ete the 'ieentia& e*+ati#n ,e- that e&ates (t) t# /(t)
0 (''t)4 (''t)3 1 (''t)2 7 (''t) 12 (t) =
0 (''t)4 (''t)3 (''t)2 1 (''t) 1 /(t)
Given /(t) = e/%(3t) in' (t)
t 8 (t) = t 8
Given /(t) = e/%(3t) in' (t)
t (t) =
t
Given that /(t) = e/%(3t) in' (t) # a&& t.
Y(s) = (s1) = 2 3 3
(s3)2(s4) (s3)2 (s3) (s4)
(t) = (32t)e/%(3t) 9 3e/%(4t)
:in' the 'ete$inant # the ;a%h in :i;.1.
< = 1666 = 1
:in' the tans$issi#n Y! in the ;a%h # :i;.1.
Y! = 012(16) 121 = 721 Y! = 3.7PROBLEM 2
56
1
2
1
! 56 1
4
3 Y
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:in' the tanse +n>ti#n ?(s) # the sste$ -ith the @#'e %t # :i;.1.
?(s) = 1
s2as1
?(1) = 1 ',
?(s) = 1
s21s1
See the lab experiments
?(s) =
:in' the #'e # the >haa>teisti>%#&n#$ia& # the 'ieentia& e*+ati#n
that ;#vens the sste$ in :i;.2.
O'e = 0 1 "#$%&ete the - ;a%h # :i;.3 that
'es>i,es the 'na$i>s # the sste$ in :i;.2.
The $#t# >#nstant is B$.
Cs ( 1$ ) $ ( B$ ) T1( 1 ) T2 ( 12) T3
( 1$ ) ( @ ) 10@2sF
e; ( B$ ) 1( 12) 2 ( 1s ) H2PROBLEM 3
F
@2
H2 T3$
Cs @1 E1D$ 1 E2
e; T1 T2
FIG 2
R(s) ?(s) G(s) "(s)
5
FIG 1
FIG 1
8/10/2019 (408177384) FinalExam Solution (1)
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n the >#nt#& sste$ # :i;.1 G(s) = s an' ? has the #$ ? = (As @)s.
Ihat t%e # >#nt#&&e is ?(s). "i>&e #+ >h#i>e
A. J#%#ti#na& (J)
@. J#%#ti#na&nte;a& (J) !". J#%#ti#na&Deivative (JD)
D. J#%#ti#na&nte;a&Deivative (JD)
"h##se A an' @ s# that the >haa>teisti> %#&n#$ia& # the sste$ # :i;.1 is
J(s) = (s4)2
J(s) = s2As @ = s2Ks16
A = K = .KK @ = 16 = 1.7K
The %#siti#n >#nt#& sste$ # :i;.2 >#ntains ateee',a>.
The >#nt#&&e has the tanse +n>ti#n ?=(s1)s.:in' the #'e # the sste$ >haa>teisti>
%#&n#$ia&.
O'e = 3
:in' the sste$ >haa>teisti> %#&n#$ia& in te$s
# A.
= 1 2A(s1) 2(s1)
s2 s3
J(s) = s3 2As22As 2s 2 = s3 2As2 2(As1) 2
J(s) =
:in' the va&+e # A # -hi>h the sste$ #s>i&&ates.
4(A2A)= 2 A2 A .5 =
A = .5 .75.5= .366
PROBLEM 4
HR(s) ?(s) 23s 13s "(s)
5 A
FIG 2
N(s) '(s)
R(s) ?(s) G(s) "(s)
FIG 1
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The +nitee',a> sste$ # :i;.1 has the %&ant G(s) = 2
s(s1)an' the >#nt#& &a- ?(s) = 5(ss1)
s
:in' the stea' state e# N(t8) # s1= 5 i (t) = 32t an' '(t) =
N(t8) =
:in' the stea' state #+t%+t >(t8) # s1= 5 i (t) = 32t an' '(t) =
>(t8) = 32t
:in' the stea' state e# N(t8) # s1= 5 i (t) = an' '(t) = 23t
sN(s) = (23s) 20s(s1) sN = (23s) 2s 11(s5)0s2(s1) s2(s1)1(s5)
sN(s) = 65 = .12s
N(t8) =
:in' the $a/i$+$ va&+e # s1# -hi>h the sste$ is sta,&e.
J(s) = s3 1s2 1s 1s1
1s1MA!= 1 s1MA!= 1
s1MA!=
PROBLEM 5
A >etain sste$ has the #%en#% tanse +n>ti#n
G? = B(s2) s(s6)2
As B vaies the ##ts # the sste$ have >i. :in' = 3
As B ten's t# ininit M >i ten' t# ininit. :in' M M = 2
L#>i e/ist #n the ea& a/is ,et-een s1an' s2. :in' s1an' s2.
8/10/2019 (408177384) FinalExam Solution (1)
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s1 =
s2= 2L#>i &eave the #i;in at an;&es H1an' H2. :in' H1an' H2.
H1 =
H = P1KO
(31) = PO
. H2 =
The >i that ;# t# ininit ae as$%t#ti> t# stai;ht &ines.
These &ines intese>t the ea& a/is at s3. :in' s3
(122)(31) = 5
s3=
"ae+&& set>h the ##t >i an' thei as$%t#ti> &ines #n the ;a%h ,e-.
135
1K
225
27
PROBLEM 6
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The %t a,#ve is the @#'e %t # the #%en#% tanse +n>ti#n # a >etain >#nt#&
sste$.
Ma #n the %t the sste$ %hase $a;in. The sste$ has t-# ;ain $a;ins. the ;ain
is 'e>ease' the sste$ -i&& #s>i&&ate. Ma #n the ;a%h this ;ain $a;in.
Esti$ate the ;ain in>ease e*+ie' # #s>i&&ati#ns an' the #s>i&&ati#n e*+en> .
:e*+en> # #s>i&&ati#n = 2a'se>
the ;ain is in>ease' the sste$ -i&& #s>i&&ate. Ma #n the ;a%h this ;ain $a;in.
Esti$ate the ;ain 'e>ease e*+ie' # #s>i&&ati#ns an' the #s>i&&ati#n e*+en>.
:e*+en> # #s>i&&ati#n = 11 a'se>
PROBLEM
A >etain sste$ has the #%en#% tanse +n>ti#n
G? = M(1sQ)
s2(1s%)
is t# have a %hase $a;in # 6Oat a +nit ;ain e*+en> # 3a'se>.
:in' M an' Q an' %.
An;(G?) = 1K6 = 1K 2atn(a)
8/10/2019 (408177384) FinalExam Solution (1)
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2atn(a) = 15O a = 3.73
Q = 33.73 = .K
% = 11.2
3.73M = 1 M = 2.41
M = 2.41
Q = .K
% = 11.2
A >etain sste$ has the #%en#% tanse +n>ti#n
G? = M(1sQ)
s2(1s1)
is t# have a %hase $a;in # 45Oat a +nit ;ain e*+en> # 3a'se>.
:in' M an' Q.
An;(G?) = 1K atn(31) atn(1Q) = 1K 45
atn(3Q) = 61.7 Q = 1.62
M = 1.3 = .4 = 4.4K 11.K5 2.1
M = 4.4K
Q = 1.62