Continuous-Time System Analysis Using The Laplace Transform

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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

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

LT

X

(

s

)

s

3

5

X

(

s

)

(

3

5 )

2

2

x

(

t

)

i



X

(

s i

)

e s i t

|X(s)|

Omega

Sigma

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)?

Example

In the circuit, the switch is in the closed position for a long time

sY

before t=0, when it is opened instantaneously. Find the inductor current y(t) for t

0.

1 H

2

y

(

t

)

t

(

L s

)

dy dt

y

Ry

( 0

(

)

t

)

1

C

2

Y

(

s

)

y

(

)

5

Y d

s

(

s

)

10

u

5

(

0

 

t

)

y

(

s

)

d

10

s x

(

t

)

10 V

5

 t=0

0.2 F

y

( 0

)

10

5

2

A

0

 

y

(

)

d

q c

( 0

)

CV

2

sY

(

s

)

y

( 0

)

2

Y

(

s

)

5

Y

(

s

)

s

10

s

10

s y

(

t

)

5

e

t

cos( 2

t

26 .

6

o

)

u

(

t

)

Example

Find the response y(t) of an LTIC system described by the equation

d

2

y

(

t

)

dt

2

5

dy

(

t dt

)

6

y

(

t

)

dx dt

(

t

)

x

(

t

) if the input x(t) = 3e

-5t u(t) and all the initial conditions are zero; that is the system is in the zero state (relaxed).

Answer :

y

(

t

)

(

2

e

5

t

e

2

t

3

e

3

t

)

u

(

t

)

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.

H

(

s

)

b

0

s s

M

N

a

1

s

M b

1

s

N

1

1

...

...

b a

N

M

BIBO stable if M

N and all poles are in the LHP

Example

Is the system below BIBO and asymptotically (internally) stable?

x(t)

S

1

1

S

S

1

1 y(t)

Block Diagrams

System Realization

H

(

s

)

b

0

s s

M

N

a

1

s

M b

1

s

N

1

1

...

...

b a

N

M

• 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

H

(

s

)

b

0

s

3

s

3

a

1

b

1

s s

2

2

a b

2

s

2

s

b a

3

3

Divide every term by

s

the highest order

s

3 with

H

(

s

)

b

0

b

1

s

b

2

s

2

b

3

s

3

H

(

s

)

b

0

1

b a

1

s

1

s

a

2

s b s

2

2

2

s b s a

3

3

3

3

1

a

1

s

1

a

2

s

2

a

3

s

3

H

1

(s) H

2

(s)

X(s) W(s) Y(s)

Direct Form I Realization

X(s)

H

1

(s)

W(s)

H

2

(s)

H

(

Y(s)

s

)

b

0

b

1

s

b

2

s

2

b

3

s

3

1

a

1

s

1

a

2

s

2

a

3

s

3

Direct Form II Realization

H

(

s

)

1

a

1

s

1

a

2

s

2

a

3

s

3

X(s)

b

0

b

1

s

b

2

s

2

b

3

s

3

H

2

(s)

W(s)

H

1

(s)

Y(s)

Example

Find the canonic direct form realization of the following transfer functions: a) b) c) d)

s

5

7

s s s

7

5

4

s

7

s

2

s

6

s

28

5

Cascade and Parallel Realizations

H

(

s

)

s

2

4

s

28

6

s

5

Cascade Realization

H

(

s

)

(

s

4

s

1 )(

s

28

5 )

4

s s

28

1

 

s

1

5

Parallel Realization

H

(

s

)

(

s

4

s

1 )(

s

28

5 )

s

6

1

s

2

5

The complex poles in H(s) should be realized as a secondorder system.

Using Operational Amplifier for

System Realization

Example

Use Op-Amp circuits to realize the canonic direct form of the transfer function

H

(

s

)

s

2

2

s

5

4

s

10

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

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