EEE 302 Electrical Networks II Dr. Keith E. Holbert Summer 2001 Lecture 14 Solving Differential Equations • Laplace transform approach automatically includes initial conditions in the.
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Transcript EEE 302 Electrical Networks II Dr. Keith E. Holbert Summer 2001 Lecture 14 Solving Differential Equations • Laplace transform approach automatically includes initial conditions in the.
EEE 302
Electrical Networks II
Dr. Keith E. Holbert
Summer 2001
Lecture 14
1
Solving Differential Equations
• Laplace transform approach automatically includes
initial conditions in the solution
L d x(t ) s X(s) x(0)
dt
d 2 y (t )
L 2 s 2 Y(s) s y(0) y' (0)
dt
• Exercise: For zero initial conditions, solve
d 2 y(t )
d y (t )
11
30 y (t ) 4 u (t )
2
dt
dt
Lecture 14
2
Inverse Laplace Transform
• Consider that F(s) is a ratio of polynomial
expressions
N ( s)
F ( s)
D ( s)
• The roots of the denominator, D(s) are called the
poles
– Poles really determine the response and stability of the
system
• The roots of the numerator, N(s), are called the zeros
Lecture 14
3
Inverse Laplace Transform
• We will use partial fractions expansion with the
method of residues to determine the inverse Laplace
transform
• Three possible cases (need proper rational, i.e., n>m)
(1) simple poles (real and unequal)
(2) simple complex roots (conjugate pair)
(3) repeated roots of same value
Lecture 14
4
Simple Poles
• Simple poles are placed in a partial fractions expansion
K 0 s z1 s z m
Kn
K1
K2
F ( s)
s p1 s p2 s pn s p1 s p2
s pn
• The constants, Ki, can be found from (use method of residues)
K i ( s pi ) F ( s) s p
i
• Finally, tabulated Laplace transform pairs are used to invert
expression, but this is a nice form since the solution is
f (t ) K1 e p1 t K 2 e p2 t K n e pn t
Lecture 14
5
Class Examples
• Extension Exercise E13.9
• Extension Exercise E13.10
Lecture 14
6
Complex Conjugate Poles
• Complex poles result in a Laplace transform of the form
K1
K1
K1
K1*
F ( s)
s ( j ) s ( j )
s ( j ) s ( j )
• The K1 can be found using the same method as for simple
poles
K 1 ( s j ) F ( s ) s j
WARNING: the "positive" pole of the form -+j MUST be
the one that is used
• The corresponding time domain function is
f (t ) 2 K1 e t cos t
Lecture 14
7
Class Example
• Extension Exercise E13.11
Lecture 14
8
Repeated Poles
• When F(s) has a pole of multiplicity r, then F(s) is
written as
F ( s)
P1 (s)
Q 1 (s) s p1
r
K11
K12
K1r
2
r
s p1 s p1
s p1
• Where the time domain function is then
f (t ) K11 e p1t K12 t e p1t
t r 1 p1t
K1r
e
r 1!
• That is we get the usual exponential multiplied by t's
Lecture 14
9
Repeated Poles (cont’d.)
• The K1j terms are evaluated from
K1 j
1
d r j
r
s
p
F ( s)
1
r j
r j ! ds
s p1
• This actually simplifies nicely until you reach s³
terms, that is for a double root (s+p1)²
K12 s p1 F (s)
2
s p1
d
s p1 2 F(s)
K11
ds
s p1
• Thus K12 is found just like for simple roots
• Note this reverse order of solving for the K values
Lecture 14
10
Class Examples
• Extension Exercise E13.12
• Extension Exercise E13.13
Lecture 14
11