Laplace Transform

• Applications of the Laplace transform– solve differential equations (both ordinary and partial)

– application to RLC circuit analysis

• Laplace transform converts differential equations in the time domain to algebraic equations in the frequency domain, thus 3 important processes:(1) transformation from the time to frequency domain

(2) manipulate the algebraic equations to form a solution

(3) inverse transformation from the frequency to time domain

Definition of Laplace Transform

• Definition of the unilateral (one-sided) Laplace transform

where s=+j is the complex frequency, and f(t)=0 for t<0

• The inverse Laplace transform requires a course in complex variables analysis (e.g., MAT 461)

0

dtetfstf stFL

Singularity Functions

• Singularity functions are either not finite or don't have finite derivatives everywhere

• The two singularity functions of interest here are

(1) unit step function, u(t)(2) delta or unit impulse function, (t)

Unit Step Function, u(t)

• The unit step function, u(t)– Mathematical definition

– Graphical illustration

01

00)(

t

ttu

1

t0

u(t)

Extensions of the Unit Step Function• A more general unit step function is u(t-a)

• The gate function can be constructed from u(t)– a rectangular pulse that starts at t= and ends at t= +T

– like an on/off switch

at

atatu

1

0)(

1

t0 a

1

t0 +T

u(t-) - u(t- -T)

Delta or Unit Impulse Function, (t)

• The delta or unit impulse function, (t)– Mathematical definition (non-pure version)

– Graphical illustration

0

00 1

0)(

tt

tttt

1

t0

(t)

t0

Transform Pairs

The Laplace transforms pairs in Table 13.1 are important, and the most important are repeated here.

(t) F (s )

δ (t) 1

u (t) {a co ns ta n t}

s

1

e -a t

as 1

t2

1

s

t e -a t

2

1

as

Laplace Transform PropertiesT h e o r e m P r o p e r t y ( t ) F ( s )

1 S c a l i n g A ( t ) A F ( s )

2 L i n e a r i t y 1 ( t ) ±

2 ( t ) F 1 ( s ) ± F 2 ( s )

3 T i m e S c a l i n g ( a · t ) 01

aa

s

aF

4 T i m e S h i f t i n g ( t - t 0 ) u ( t - t 0 ) e - s · t 0 F ( s ) t 0 0

6 F r e q u e n c y S h i f t i n g e - a · t ( t ) F ( s + a )

9 T i m e D o m a i nD i f f e r e n t i a t i o n dt

tfd )( s F ( s ) - ( 0 )

7 F r e q u e n c y D o m a i nD i f f e r e n t i a t i o n

t ( t )ds

sd )(F

1 0 T i m e D o m a i nI n t e g r a t i o n

tdf

0)( )(

1s

sF

1 1 C o n v o l u t i o n t

dtff0 21 )()( F 1 ( s ) F 2 ( s )

Block Diagram Reduction

Block Diagram Reduction

Block Diagram Reduction

Block Diagram Reduction

Reference

Y(s) = ___K*G(s)R(s) 1+K*G*H(s)Closed Loop

KSum

H(s)

Y(s)R(s)-

Gplant

Y is the 'ControlledOutput'

Forward Path

Y(s) = ___K*G(s)R(s) 1+K*G*H(s)Closed Loop

Characteristic Equation:Den(s) = 1+K*GH(s) = 0

Stability: The response y(t) reverts to

zero if input r = 0.All roots (poles of Y/R) must have Re(pi) < 0

Characteristic Equation:Den(s) = 1+K*GH(s) = 0

Closed Loop Poles are theroots of the Characteristic

Equation, i.e. 1+K*GH(s) = 0

Poles and Stability

Poles and Stability

Poles and Stability

Underdamped System (2nd Order)