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

2. 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
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EEE 202

3. 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 sec, the angular frequency is 377 rads/sec)
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EEE 202
EEE 202

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

5. 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
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EEE 202

6. More Complex Numbers Polar coordinates: A = z  θ
Rectangular coordinates: A = x + jy
Polar ↔ rectangular conversion relations:
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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
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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)
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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)
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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
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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
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12. Unit Step Function, u(t)
The unit step function, u(t)
Mathematical definition
Graphical illustration 1 t
u(t)
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13. 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
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EEE 202

14. Delta or Unit Impulse Function, (t)
The delta or unit impulse function, (t)
Mathematical definition (non-purist version)
Graphical illustration 1 t
(t) t0
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15. 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)
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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)
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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)
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18. Laplace Transform Pairs
Most of the time we find Laplace transforms from tables like Table 5.1 rather than using the defining integral
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19. Some Laplace Transform Properties
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20. Class Examples
Drill Problem P5-2 (solve using both the defining Laplace integral, and the table of integrals with appropriate properties)
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