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