1. Continuous-Time System Analysis Using The Laplace Transform
Dr. Mohamed Bingabr
University of Central Oklahoma
Slides For Lathi’s Textbook Provided by Dr. Peter Cheung

2. 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

5. LT
x(t) = 3e-5t
|X(s)|
Omega
Sigma

20. u = e-st
dv = dx

26. HW5_Ch4: 4. 1-1 (a, b, c, d), 4. 1-3 (a, b, c, d, f), 4
HW5_Ch4: (a, b, c, d), (a, b, c, d, f), (a, b, e, g), (a,c), 4.2-6

32. Where is H(s)?

33. Example
In 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)

35. Example
Find 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 :

37. 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.

38. External Stability BIBO
The transfer function H(s) can only indicate the external stability of the system BIBO.
BIBO stable if M  N and all poles are in the LHP
Example
Is the system below BIBO and asymptotically (internally) stable?
y(t)
x(t)

39. Block Diagrams

40. System Realization
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

41. Direct Form I Realization
Divide every term by s with the highest order s3
H1(s)
H2(s)
X(s)
W(s)
Y(s)

42. Direct Form I Realization
H1(s)
H2(s)
X(s)
W(s)
Y(s)

43. Direct Form II Realization
H2(s)
H1(s)
X(s)
W(s)
Y(s)

44. Example
Find the canonic direct form realization of the following transfer functions:

45. Cascade and Parallel Realizations
Cascade Realization
Parallel Realization
The complex poles in H(s) should be realized as a second-order system.

46. Using Operational Amplifier for System Realization

47. Example
Use Op-Amp circuits to realize the canonic direct form of the transfer function
HW6_Ch4: (b,c), (b,c), , , , 4.4-1, ,