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Transcript
Laplace Transformations
Dr. Holbert
February 25, 2008
Lect10
EEE 202
1
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 = 2p f
– Radians per second
• Amplitude: A
– For example, could be volts or amps
Lect10
EEE 202
2
Example Sinusoid
• What is the amplitude, period, frequency,
and angular (radian) frequency of this
sinusoid?
8
6
4
2
0
-2 0
0.01
0.02
0.03
0.04
0.05
-4
-6
-8
Time (sec)
Lect10
EEE 202
3
Phasors
• A phasor is a complex number that
represents the magnitude and phase
angle of a sinusoidal signal:
X M cost   
X  X M 
Lect10
EEE 202
4
Complex Numbers
imaginary
axis
• x is the real part
• y is the imaginary
part
• z is the
magnitude
y

x
Lect10
real
axis
• θ is the phase
angle
EEE 202
5
More Complex Numbers
• Polar coordinates: A = z  θ
• Rectangular coordinates: A = x + jy
• Polar ↔ rectangular conversion relations:
y  z sin 
x  z cos
z x y
2
Lect10
  tan
2
EEE 202
1
y
x
6
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
7
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
8
Complex Number
Multiplication and Division
• Multiplication is most easily performed in polar
coordinates:
A = AM  
B = BM  f
A  B = (AM  BM)  (  f)
• Division is also most easily performed in polar
coordinates:
A / B = (AM / BM)  (  f)
Lect10
EEE 202
9
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
10
Singularity Functions
• Singularity functions are either not finite or
don't have finite derivatives everywhere
• The two singularity functions of interest
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
Lect10
EEE 202
11
Unit Step Function, u(t)
• The unit step function, u(t)
– Mathematical definition
0
u (t )  
1
t0
t 0
– Graphical illustration
u(t)
1
t
0
Lect10
EEE 202
12
Extensions of Unit Step Function
• A more general unit
step function is u(t-a)
0
u (t  a)  
1
1
ta
ta
0
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
Lect10
EEE 202
1
0

+T
t
13
Delta or Unit Impulse Function, (t)
• The delta or unit impulse function, (t)
– Mathematical definition (non-purist version)
t  t0
0
 (t  t 0 )  
1
t  t0
– Graphical illustration
(t)
1
0
Lect10
EEE 202
t0
t
14
Extensions of the Delta Function
• An important property of the unit impulse
function is its sampling property
– Mathematical definition (non-purist version)
t  t0
 0
f (t )  (t  t 0 )  
 f (t 0 )
t  t0
f(t)
f(t) (t-t0)
0
Lect10
EEE 202
t0
t
15
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:
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 (we’ll wait to the next class time for this)
Lect10
EEE 202
16
Definition of Laplace Transform
• Definition of the unilateral (one-sided) Laplace
transform

L f t   Fs    f t  e
 st
dt
0
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
17
Laplace Transform Pairs
Most of the time
we find Laplace
transforms from
tables like Table
5.1 rather than
using the
defining integral
(t)
F(s)
δ(t)
1
u(t) {a constant}
1
s
e–at
t
t e–at
Lect10
EEE 202
1
sa
1
s2
1
s  a  2
18
Some
Laplace
Transform
Properties
Property
(t)
F(s)
Scaling
A (t)
A F(s)
Linearity
1(t) ± 2(t)
F1(s) ± F2(s)
(a·t)
Time Scaling
a0
(t–t0) u(t–t0)
e–s·t0 F(s) t00
Frequency Shifting
e–a·t (t)
F(s+a)
Time Domain
Differentiation
d f (t )
dt
s F(s) – (0)
Time Shifting (delay)
Frequency Domain
Differentiation
t (t)

Time Domain
Integration
Convolution
Lect10
1 s
F 
a a
t
0

t
0
f ( ) d
f1 ( ) f 2 (t   ) d
EEE 202
 d F (s)
ds
1
F ( s)
s
F1(s) F2(s)
19
Class Examples
• Drill Problem P5-2 (solve using both the
defining Laplace integral, and the table of
integrals with appropriate properties)
Lect10
EEE 202
20