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Laplace Transformations

Laplace Transformations. Dr. Holbert February 25, 2008. Sinusoids. Period: T = 2 p / ω = 1/ f Time necessary to go through one cycle Frequency: f = 1/ T Cycles per second (Hertz, Hz) Angular frequency: ω = 2 p f Radians per second Amplitude: A For example, could be volts or amps.

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Laplace Transformations

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  1. Laplace Transformations Dr. Holbert February 25, 2008 EEE 202

  2. Sinusoids • Period: T = 2p/ω = 1/f • Time necessary to go through one cycle • Frequency: f = 1/T • Cycles per second (Hertz, Hz) • Angular frequency: ω = 2pf • Radians per second • Amplitude: A • For example, could be volts or amps EEE 202

  3. Example Sinusoid • What is the amplitude, period, frequency, and angular (radian) frequency of this sinusoid? EEE 202

  4. Phasors • A phasor is a complex number that represents the magnitude and phase angle of a sinusoidal signal: EEE 202

  5. Complex Numbers • x is the real part • y is the imaginary part • z is the magnitude • θ is the phase angle imaginary axis y z q real axis x EEE 202

  6. More Complex Numbers • Polar coordinates: A = z θ • Rectangular coordinates: A = x + jy • Polar ↔ rectangular conversion relations: EEE 202

  7. Arithmetic with Complex Numbers • To compute phasor voltages and currents, we need to be able to perform basic computations with complex numbers • Addition • Subtraction • Multiplication • Division • Appendix A has a review of complex numbers EEE 202

  8. Complex Number Addition and Subtraction • Addition is most easily performed in rectangular coordinates: A = x + jy B = z + jw A + B = (x + z) + j(y + w) • Subtraction is also most easily performed in rectangular coordinates: A - B = (x - z) + j(y - w) EEE 202

  9. Complex Number Multiplication and Division • Multiplication is most easily performed in polar coordinates: A = AMq B = BMf A B = (AM  BM)  (q + f) • Division is also most easily performed in polar coordinates: A / B = (AM / BM)  (q - f) EEE 202

  10. Are You a Technology “Have”? • There is a good chance that your calculator will convert from rectangular to polar, and from polar to rectangular • Convert to polar: 3 + j4 and –3 – j4 • Convert to rectangular: 2 45 & –2 45 • Add: 3 30 + 5 20 • Determine the complex conjugate of the phasor: A = z θ EEE 202

  11. Singularity Functions • Singularity functions are either not finite or don't have finite derivatives everywhere • The two singularity functions of interest are • unit step function, u(t) and its construct: the gate function • delta or unit impulse function, (t) and its construct: the sampling function EEE 202

  12. u(t) 1 t 0 Unit Step Function, u(t) • The unit step function, u(t) • Mathematical definition • Graphical illustration EEE 202

  13. 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 : u(t-) – u(t--T) like an on/off switch 1 t 0 a 1 t 0  +T Extensions of Unit Step Function EEE 202

  14. (t) 1 t0 t 0 Delta or Unit Impulse Function, (t) • The delta or unit impulse function, (t) • Mathematical definition (non-purist version) • Graphical illustration EEE 202

  15. 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-purist version) EEE 202

  16. 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 three important processes: • transformation from the time to frequency domain • manipulate the algebraic equations to form a solution • inverse transformation from the frequency to time domain (we’ll wait to the next class time for this) EEE 202

  17. 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 variable analysis (e.g., MAT 461) EEE 202

  18. Laplace Transform Pairs Most of the time we find Laplace transforms from tables like Table 5.1 rather than using the defining integral EEE 202

  19. SomeLaplace Transform Properties EEE 202

  20. Class Examples • Drill Problem P5-2 (solve using both the defining Laplace integral, and the table of integrals with appropriate properties) EEE 202

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