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EEE 302 Electrical Networks II

EEE 302 Electrical Networks II. Dr. Keith E. Holbert Summer 2001. Laplace Transform. Applications of the Laplace transform solve differential equations (both ordinary and partial) application to RLC circuit analysis

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EEE 302 Electrical Networks II

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  1. EEE 302Electrical Networks II Dr. Keith E. Holbert Summer 2001 Lecture 13

  2. 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 13

  3. 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 13

  4. 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 13

  5. u(t) 1 t 0 Unit Step Function, u(t) • The unit step function, u(t) • Mathematical definition • Graphical illustration Lecture 13

  6. 1 t 0 a 1 t 0  +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 u(t-) - u(t- -T) Lecture 13

  7. (t) 1 t0 t 0 Delta or Unit Impulse Function, (t) • The delta or unit impulse function, (t) • Mathematical definition (non-pure version) • Graphical illustration Lecture 13

  8. f(t) f(t) (t-t0) t 0 t0 Extensions of the Delta Function • An important property of the unit impulse function is its sampling property • Mathematical definition (non-pure version) Lecture 13

  9. Transform Pairs The Laplace transforms pairs in Table 13.1 are important, and the most important are repeated here. Lecture 13

  10. Class Examples • Extension Exercise E13.1 • Extension Exercise E13.2 Lecture 13

  11. Laplace Transform Properties Lecture 13

  12. Class Examples • Extension Exercise E13.3 • Extension Exercise E13.4 • Extension Exercise E13.5 • Extension Exercise E13.6 • Extension Exercise E13.8 Lecture 13

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