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)

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 1 t 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 1 t a 1 t  +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 1 t (t) t0

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

Laplace Transform Properties

Block Diagram Reduction

Block Diagram Reduction

Block Diagram Reduction

Block Diagram Reduction

Poles and Stability

Poles and Stability

Poles and Stability

Underdamped System (2nd Order)