Laplace Transformations Dr. Holbert Lecture 10 Laplace Transformations Dr. Holbert February 25, 2008 Lect10 EEE 202 EEE 202

Sinusoids Period: T = 2p/ω = 1/f Frequency: f = 1/T Time necessary to go through one cycle Frequency: f = 1/T Cycles per second (Hertz, Hz) Angular frequency: ω = 2p f Radians per second Amplitude: A For example, could be volts or amps Lect10 EEE 202

Dr. Holbert Lecture 10 Example Sinusoid What is the amplitude, period, frequency, and angular (radian) frequency of this sinusoid? The amplitude is 7, the cyclic frequency is 60 Hz, the period is 0.016667 sec, the angular frequency is 377 rads/sec) Lect10 EEE 202 EEE 202

Phasors A phasor is a complex number that represents the magnitude and phase angle of a sinusoidal signal: Lect10 EEE 202

Complex Numbers θ is the phase angle x is the real part imaginary axis x is the real part y is the imaginary part z is the magnitude θ is the phase angle y z q real axis x Lect10 EEE 202

More Complex Numbers Polar coordinates: A = z  θ Rectangular coordinates: A = x + jy Polar ↔ rectangular conversion relations: Lect10 EEE 202

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 Lect10 EEE 202

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) Lect10 EEE 202

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) Lect10 EEE 202

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 Lect10 EEE 202

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 Lect10 EEE 202

Unit Step Function, u(t) The unit step function, u(t) Mathematical definition Graphical illustration 1 t u(t) Lect10 EEE 202

Extensions of 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 : u(t-) – u(t--T) like an on/off switch 1 t a 1 t  +T Lect10 EEE 202

Delta or Unit Impulse Function, (t) The delta or unit impulse function, (t) Mathematical definition (non-purist version) Graphical illustration 1 t (t) t0 Lect10 EEE 202

Extensions of the Delta Function An important property of the unit impulse function is its sampling property Mathematical definition (non-purist version) f(t) t t0 f(t) (t-t0) Lect10 EEE 202

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) Lect10 EEE 202

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) Lect10 EEE 202

Laplace Transform Pairs Most of the time we find Laplace transforms from tables like Table 5.1 rather than using the defining integral Lect10 EEE 202

Some Laplace Transform Properties Lect10 EEE 202

Class Examples Drill Problem P5-2 (solve using both the defining Laplace integral, and the table of integrals with appropriate properties) Lect10 EEE 202