Lecture 131 EEE 302 Electrical Networks II Dr. Keith E. Holbert Summer 2001

Lecture 132 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

Lecture 133 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)

Lecture 134 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) and its construct: the gate function (2) delta or unit impulse function,  (t) and its construct: the sampling function

Lecture 135 Unit Step Function, u(t) The unit step function, u(t) –Mathematical definition –Graphical illustration 1 t 0 u(t)u(t)

Lecture 136 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 0a 1 t 0  +T u(t-  ) - u(t-  -T)

Lecture 137 Delta or Unit Impulse Function,  (t) The delta or unit impulse function,  (t) –Mathematical definition (non-pure version) –Graphical illustration 1 t 0 (t)(t) t0t0

Lecture 138 Extensions of the Delta Function An important property of the unit impulse function is its sampling property –Mathematical definition (non-pure version) f(t)f(t) t 0t0t0 f(t)  (t-t 0 )

Lecture 139 Transform Pairs The Laplace transforms pairs in Table 13.1 are important, and the most important are repeated here.

Lecture 1310 Class Examples Extension Exercise E13.1 Extension Exercise E13.2

Lecture 1311 Laplace Transform Properties

Lecture 1312 Class Examples Extension Exercise E13.3 Extension Exercise E13.4 Extension Exercise E13.5 Extension Exercise E13.6 Extension Exercise E13.8