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Bài giảng Tín hiệu và hệ thống - Chương 6: Continuous-time system analysis using the laplace transform

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Ch-6: Continuous-Time System Analysis Using the Laplace Transform  
P6.1. Consider the signal f(t)=e-5tu(t-1) and denote its Laplace  
transform by F(s).  
a) Using analysis function, evaluate F(s) and specify its ROC.  
b) Determine the values of the finite numbers A and t0 such that the  
Laplace transform G(s) of g(t)=Ae-5tu(-t-t0) has the same algebraic  
form as F(s). What is the ROC corresponding to G(s)  
P6.2. Consider the signal f(t)=e-5tu(t)+e-βtu(t) and denote its  
Laplace transform by F(s). What are the constraints placed on the  
real and imaginary parts of β if the ROC of F(s) is Re{s}>-3?  
t
e sin2t; t ≤ 0  
P6.3. For the Laplace transform of  
f(t)=  
0;  
t>0  
Indicate the location of its poles and its ROC  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
Ch-6: Continuous-Time System Analysis Using the Laplace Transform  
P6.4. How many signals have a Laplace transform that may be  
expressed as  
s −1  
(s+2)(s+3)(s2 +s+1)  
in its ROC?  
P6.5. Given that  
1
e−atu(t) ↔  
; ROC: Re{s}>Re{-a}  
s+a  
determine the inverse Laplace transform of  
2(s + 2)  
s2 +7s+12  
F(s) =  
; ROC:Re{s}>-3  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
1
Ch-6: Continuous-Time System Analysis Using the Laplace Transform  
P6.6. Let g(t)=f(t)+af(-t) where f(t)=be-tu(t) and Laplace transform  
of g(t) is  
s
G(s) =  
; ROC: −1< Re{s} <1  
s2 -1  
determine the values of the constant a and b.  
P6.7. Determine the Laplace transform and the associated ROC  
a) f(t)=e−2tu(t) + e−3tu(t)  
b) f(t)=e−4tu(t) + e−5tsin(5t)u(t)  
c) f(t)=e2tu( − t) + e3tu( − t) d) f(t)=te−2|t|  
e) f(t)=|t|e−2|t|  
f) f(t)=|t|e2tu(−t)  
h) f(t)=t.rect(t-1/2)+(2-t)rect(t-3/2)  
g) f(t)=rect(t −1/2)  
i) f(t)=δ (t) + u(t)  
j) f(t)=δ (3t) + u(3t)  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
Ch-6: Continuous-Time System Analysis Using the Laplace Transform  
P6.8. Determine the Laplace transform of the following signals:  
e) f(t)=te-tu(t-t0 )  
a) f(t)=u(t)-u(t-1)  
b) f(t)=e-(t-t )u(t-t0 )  
0
f) f(t)=sin[ω0 (t-t0 )]u(t-t0 )  
g) f(t)=sin[ω0 (t-t0 )]u(t)  
c) f(t)=e-(t-t )u(t)  
0
d) f(t)=e-tu(t-t0 )  
h) f(t)=sin(ω0t)u(t-t0 )  
P6.9. Determine the function of time, f(t), for each of the following  
Laplace transforms and their associated ROC:  
1
s
a) F(s)=  
; Re{s} > 0  
b) F(s)=  
; Re{s}< 0  
s2 + 9  
s +1  
(s +1)2 + 9  
(s +1)2  
s2 − s +1  
s2 +9  
s + 2  
s2 + 7s +12  
s2 − s +1  
(s +1)2  
d) F(s)=  
; -4<Re{s}< −3  
c) F(s)=  
; Re{s}< −1  
1
f) F(s)=  
; Re{s} > −1  
e) F(s)=  
; Re{s} >  
2
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
2
Ch-6: Continuous-Time System Analysis Using the Laplace Transform  
P6.10. Determine the function of time, f(t), for each of the  
following one-side Laplace transforms  
1
5
2s+5  
s2 +5s+6  
3s+5  
g)  
h)  
i)  
d)  
e)  
f)  
a)  
b)  
c)  
(s+1)(s+2)4  
s2 (s+2)  
s+1  
s(s+2)2 (s2 +4s+5)  
2s+1  
(s+1)(s2 +2s+2)  
s2 +4s+13  
2
s3  
s+1  
s2 -s-6  
s+2  
s(s+1)2  
(
)
(s+1)2 (s2 +2s+5)  
P6.11. Determine the transfer function and step response of the  
system depicted in FigP6.11  
1Ω  
1H  
vi (t)  
1F  
v0 (t)  
1Ω  
FigP6.11  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
Ch-6: Continuous-Time System Analysis Using the Laplace Transform  
P6.12. Determine transfer function of the system shown in  
FigP6.12(a), and (b)  
R
C
vi (t)  
v (t)  
v0 (t)  
R
v0 (t)  
C
i
R2  
R2  
R1  
R1  
(a)  
(b)  
FigP6.12  
P6.13. The input f(t) and output y(t) of a causal LTI system are  
related through the block diagram representation shown in  
FigP6.13.  
a) Determine a differential equation relating y(t) and f(t).  
b) Is this system stable?  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
3
Ch-6: Continuous-Time System Analysis Using the Laplace Transform  
P6.14. Draw a direct-form representation for the causal LTI system  
with the following system functions:  
s2 −5s + 6  
s2 + 7s +10  
s +1  
s2 + 5s + 6  
5
a) H1(s)=  
b) H2 (s)=  
c) H3 (s)=  
(s + 2)2  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
Ch-6: Continuous-Time System Analysis Using the Laplace Transform  
P6.15. Realize following transfer functions:  
s(s+2)  
3s(s+2)  
(s+1)(s2 +2s+2)  
2s+3  
5s(s+2)2 (s+3)  
s3  
a) H(s)=  
c) H(s)=  
e) H(s)=  
b) H(s)=  
d) H(s)=  
f) H(s)=  
(s+1)(s+3)(s+4)  
2s-4  
(s+2)(s2 +4)  
s(s+1)(s+2)  
(s+5)(s+6)(s+8)  
(s+1)2 (s+2)(s+3)  
by canonical, series, and parallel forms.  
P6.16. In this problem we show how a pair of complex conjugate  
poles may be using a cascade of two first-order transfer functions.  
Show that the of transfer function of the block diagrams in Fig  
P6.16a, b, and c are  
1
s+a  
(s+a)2 +b2  
As+B  
(s+a)2 +b2  
a) H(s)=  
b) H(s)=  
c) H(s)=  
(s+a)2 +b2  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
4
Ch-6: Continuous-Time System Analysis Using the Laplace Transform  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
Ch-6: Continuous-Time System Analysis Using the Laplace Transform  
P6.17. Show op-amp realization of the following transfer functions:  
-10  
10  
s+2  
s+5  
a) H(s)=  
b) H(s)=  
c) H(s)=  
s+5  
s+5  
P6.18. Show two different op-amp realization of the transfer  
function:  
s+2  
s+5  
3
H(s)=  
=1-  
s+5  
P6.19. Show op-amp canonical realization of the following transfer  
functions:  
s2 +5s+2  
3s+7  
s2 +4s+10  
b) H(s)=  
a) H(s)=  
s2 +4s+13  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
5
Ch-6: Continuous-Time System Analysis Using the Laplace Transform  
P6.20. Determine the rise time tr, the settling time ts, the PO and the  
steady-state errors es, er and ep for each of the following systems,  
whose transfer functions are:  
9
4
95  
s2 +10s+100  
a) H(s)=  
b) H(s)=  
c) H(s)=  
s2 +3s+9  
s2 +3s+4  
P6.21. For a position control system depicted in Fig P6.21, the unit  
step response shows the peak time tp=π/4, the PO=9%, and the  
steady-state value of the output for the unit step input is yss=2.  
Determine K1, K2 and a  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
Ch-6: Continuous-Time System Analysis Using the Laplace Transform  
P6.22. For a position control system depicted in Fig P6.22, the  
following specifications are impose: tr≤0.3, ts≤0.1, PO≤30%, and  
es=0. Which of these specifications cannot be met by the system for  
any value of K? Which specifications can be met by simple  
adjustment of K?  
P6.23. Open loop transfer functions of four closed-loop system are  
given below. In each case, give a rough sketch of the root locus.  
K(s+1)  
K(s+5)  
a) H(s)=  
c) H(s)=  
b) H(s)=  
d) H(s)=  
s(s+3)(s+5)  
K(s+1)  
s(s+3)  
K(s+1)  
s(s+4)(s2 +2s+2)  
s(s+3)(s+5)(s+7)  
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11  
6
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