Post on 17-Jan-2016
Continuous-Time System Analysis Using The Laplace Transform
Dr. Mohamed Bingabr
University of Central OklahomaSlides For Lathi’s Textbook Provided by Dr. Peter Cheung
Outline
• Introduction• Properties of Laplace Transform• Solution of Differential Equations• Analysis of Electrical Networks• Block Diagrams and System Realization• Frequency Response of an LTIC System• Filter Design by Placement of Poles and Zeros of H(s)
The materials in these slides are covered in the Lathi Textbook all Ch 4 except sections 4.4, 4.7, 4.9, 4.11
x(t) = 3e-5t
22)5(
3)(
5
3)(
sX
ssX ts
ii
iesXtx
)()(
|X(s)|
LT
Sigma
Omega
u = e-st dv = dx
HW5_Ch4: 4.1-1 (a, b, c, d), 4.1-3 (a, b, c, d, f), 4.2-1 (a, b, e, g), 4.2-3 (a,c), 4.2-6
Where is H(s)?
ExampleIn the circuit, the switch is in the closed position for a long time before t=0, when it is opened instantaneously. Find the inductor current y(t) for t 0.
10 V
2
t=0
5
1 H
0.2 F
y(t)
x(t)
t
tudyC
tRydt
dyL )(10)(
1)(
ss
dy
s
sYsYyssY
10)(5
)(5)(2)0()(
0
Ay 25
10)0( 2)0()(
0
CVqdy c
sss
sYsYyssY
1010)(5)(2)0()(
)()6.262cos(5)( tutety ot
ExampleFind the response y(t) of an LTIC system described by the equation
if the input x(t) = 3e-5tu(t) and all the initial conditions are zero; that is the system is in the zero state (relaxed).
Answer :
)()32()( 325 tueeety ttt
)()(
)(6)(
5)(
2
2
txdt
tdxty
dt
tdy
dt
tyd
Internal Stability• Internal Stability (Asymptotic)
– If and only if all the poles are in the LHP– Unstable if, and only if, one or both of the
following conditions exist:• At least one pole is in the RHP• There are repeated poles on the imaginary axis
– Marginally stable if, and only if, there are no poles in the RHP, and there are some unrepeated poles on the imaginary axis.
External Stability BIBO
The transfer function H(s) can only indicate the external stability of the system BIBO.
NNN
MMM
asas
bsbsbsH
...
...)(
11
110
Example
Is the system below BIBO and asymptotically (internally) stable?
1
1
S 1
1
S
Sx(t) y(t)
BIBO stable if M N and all poles are in the LHP
Block Diagrams
System Realization
NNN
MMM
asas
bsbsbsH
...
...)(
11
110
• Realization is a synthesis problem, so there is no unique way of realizing a system.
• A common realization of H(s) is using• Integrator• Scalar multiplier• Adders
Direct Form I Realization
322
13
322
13
0)(asasas
bsbsbsbsH
Divide every term by s with the highest order s3
33
221
33
221
0
1)(
sa
sa
sa
sb
sb
sb
bsH
33
221
33
221
0
1
1)(
s
a
sa
sas
b
s
b
s
bbsH
H1(s) H2(s)X(s) W(s) Y(s)
Direct Form I RealizationH1(s) H2(s)
X(s) W(s) Y(s)
33
221
33
221
0
1
1)(
s
a
sa
sas
b
s
b
s
bbsH
Direct Form II Realization
33
221
0
33
2211
1)(
s
b
s
b
s
bb
s
a
sa
sa
sH
H2(s) H1(s)X(s) W(s) Y(s)
Example
Find the canonic direct form realization of the following transfer functions:
56
284d)
7
5c)
7b)
7
5a)
2
ss
ss
ss
ss
Cascade and Parallel Realizations
56
284)(
2
ss
ssH
Cascade Realization
5
1
1
284
)5)(1(
284)(
ss
s
ss
ssH
Parallel Realization
5
2
1
6
)5)(1(
284)(
ssss
ssH
The complex poles in H(s) should be realized as a second-order system.
Using Operational Amplifier for System Realization
Example
Use Op-Amp circuits to realize the canonic direct form of the transfer function
104
52)(
2
ss
ssH
HW6_Ch4: 4.3-1 (b,c), 4.3-2 (b,c), 4.3-4, 4.3-7, 4.3-10, 4.4-1, 4.5-2, 4.6-1