# EEE 302 Lecture 13 - University of Nevada, Las Vegas 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

L f t F s f t e st dt 0 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) 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 0 u (t ) 1 Graphical illustration u(t) t 0 t 0 1

0 t Extensions of the Unit Step Function A more general unit step function is u(t-a) 0 u (t a ) 1 t a t a 1

The gate function can be constructed from u(t) t a rectangular pulse that starts at t= and0ends aat t= +T like an on/off switch 1 u(t-) - u(t- -T) 0 +TT t

Delta or Unit Impulse Function, (t) The delta or unit impulse function, (t) Mathematical definition (non-pure version) 0 (t t 0 ) 1 t t 0 t t 0 Graphical illustration (t) 1

0 t0 t Transform Pairs The Laplace transforms pairs in Table 13.1 are important, and the most important are repeated here. (t)

F (s ) (t) 1 u (t) {a c o n s ta n t} 1 s e -a t t te

-a t 1 sa 1 s2 1 s a 2 Laplace Transform Properties T h e o re m P ro p e rty (t)

F (s ) 1 S c a lin g A (t) A F (s ) 2 L in e a r it y 1 (t) 2 (t)

F 1(s ) F 2(s ) 3 T im e S c a l i n g ( a t) 4 T im e S h i f t i n g (t-t0 ) u ( t-t0 ) 6 F r e q u e n c y S h if t in g

e 9 T im e D o m a i n D if f e r e n t ia t io n d f (t) dt s F (s ) - (0 ) 7 F r e q u e n c y D o m a in D if f e r e n t ia t io n

t ( t) 10 T im e D o m a i n I n t e g r a t io n d F (s) ds 1 F (s) s 11 C o n v o lu t io n

0 0 t t - a t (t)

f ( ) d f 1 ( ) f 2 ( t ) d 1 s F a a e - s t 0 a 0 F (s ) t0 0 F (s + a ) F 1(s ) F 2(s )

Block Diagram Reduction Block Diagram Reduction Block Diagram Reduction Block Diagram Reduction Reference K Sum R(s) -

Gplant Forward Path H(s) Closed Loop Y(s) = ___K*G(s) R(s) 1+K*G*H(s) Y is the 'Controlled Output' Y(s) Closed Loop

Y(s) = ___K*G(s) R(s) 1+K*G*H(s) Characteristic Equation: Den(s) = 1+K*GH(s) = 0 Characteristic Equation: Den(s) = 1+K*GH(s) = 0 Closed Loop Poles are the roots of the Characteristic Equation, i.e. 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

Poles and Stability Poles and Stability Poles and Stability Underdamped System (2 Order) nd