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Applications of the Laplace Transform

ECE 2221/MCT 2210 Signals and Systems

(Analysis) Sem. I (03/04)

- Br. Shahrul Naim Sidek
- Department of Mechatronics Engineering
- International Islamic University Malaysia

Lecture Chapter 6

http//eng.iiu.edu.my/snaim

Topic covered

- Analysis of Electrical Networks
- Concept of Transfer function
- Stability via Routh Test
- Bode plot to display Frequency Response

Analysis of Electrical Networks Passive Circuit

Elements

- Laplace transforms with zero-valued initial

conditions - Capacitor

- Inductor
- Resistor

Transfer Function

Ex First-Order RC Lowpass Filter

R

x(t)

y(t)

C

i(t)

Time domain

R

X(s)

Y(s)

I(s)

Laplace domain

Passive Circuit Elements

- Laplace transforms with non-zero initial

conditions - Capacitor

- Inductor
- Resistor ??

Initial and Final Values

- Values of f(t) as t ? 0 and t ? ? may be computed

from its Laplace transform F(s) - Initial value theorem
- If f(t) and its derivative df/dt have Laplace

transforms, then provided

that the limit on the right- - hand side of the equation exists.
- Final value theorem
- If both f(t) and df/dt have Laplace transforms,

then provided that s

F(s) has no poles in the - RHP or on the imaginary axis.

Final and Initial Values Example

- Transfer function
- Poles at s 0, s -1 ? j2
- Zero at s -3/2

Initial Value

Final Value

Transfer Function

- Provides a complete behaviors of a specified LTI

system. - T.F. is independent of the particular input and

is a property of the circuit only. - T.F. is obtained for the case of zero initial

conditions. - T.F. for LTI circuit is a rational function of s
- Frequency response is H(s) and let s jw

Component of response

The Laplace transform of differential equation

with initial condition is given

Component of response

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