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By:http://admission.edhole.com
MCA Admission in India
Dr. HolbertFebruary 27, 2008
Lect11EEE 2022
Inverse Laplace Transformations
Inverse Laplace Transform
Lect11EEE 2023
Consider that F(s) is a ratio of polynomial expressions
The n roots of the denominator, D(s) are called the polesPoles really determine the response and
stability of the systemThe m roots of the numerator, N(s), are
called the zeros
)(
)()(
s
ss
DN
F
Inverse Laplace Transform
Lect11EEE 2024
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
1. Simple Poles
Lect11EEE 2025
Simple poles are placed in a partial fractions expansion
The constants, Ki, can be found from (use method of residues)
Finally, tabulated Laplace transform pairs are used to invert expression, but this is a nice form since the solution is
n
n
n
m
ps
K
ps
K
ps
K
pspsps
zszsKs
2
2
1
1
21
10)(F
ipsii spsK
)()( F
tpn
tptp neKeKeKtf 2121)(
2. Complex Conjugate Poles
Lect11EEE 2026
Complex poles result in a Laplace transform of the form
The K1 can be found using the same method as for simple poles
WARNING: the "positive" pole of the form –+j MUST be the one that is used
The corresponding time domain function is
)()()()(
)( 11*11
js
K
js
K
js
K
js
Ks F
jssjsK
)()(1 F
teKtf t cos2)( 1
3. Repeated Poles
Lect11EEE 2027
When F(s) has a pole of multiplicity r, then F(s) is written as
Where the time domain function is then
That is, we obtain the usual exponential but multiplied by t's
rr
r ps
K
ps
K
ps
K
pss
ss
1
12
1
12
1
11
11
1
)(
)()(
Q
PF
tpr
rtptp e
r
tKetKeKtf 111
!1)(
1
11211
3. Repeated Poles (cont’d.)
Lect11EEE 2028
The K1j terms are evaluated from
This actually simplifies nicely until you reach s³ terms, that is for a double root (s+p1)²
Thus K12 is found just like for simple rootsNote this reverse order of solving for the K values
1
)(!
111
ps
r
jr
jr
j spsds
d
jrK
F
1
1
)()( 2111
2112
psps
spsds
dKspsK
FF
The “Finger” Method
Let’s suppose we want to find the inverse Laplace transform of
We’ll use the “finger” method which is an easy way of visualizing the method of residues for the case of simple roots (non-repeated)
We note immediately that the poles ares1 = 0 ; s2 = –2 ; s3 = –3
)3)(2(
)1(5)(
sss
ssF
Lect11 EEE 202 9
The Finger Method (cont’d)For each pole (root), we will write down the
function F(s) and put our finger over the term that caused that particular root, and then substitute that pole (root) value into every other occurrence of ‘s’ in F(s); let’s start with s1=0
This result gives us the constant coefficient for the inverse transform of that pole; here: e–0·t
Lect11 EEE 202 10
6
5
)3)(2(
)1(5
)30)(20)((
)10(5
)3)(2(
)1(5)(
ssss
ssF
The Finger Method (cont’d)
Let’s ‘finger’ the 2nd and 3rd poles (s2 & s3)
They have inverses of e–2·t and e–3·t
The final answer is then
tt eetf 32
3
10
2
5
6
5)(
Lect11 EEE 202 11
3
10
)1)(3(
)2(5
)3)(23)(3(
)13(5
)3)(2(
)1(5)(
2
5
)1)(2(
)1(5
)32)(2)(2(
)12(5
)3)(2(
)1(5)(
ssss
ss
ssss
ss
F
F
Initial Value Theorem
Lect11EEE 20212
The initial value theorem states
Oftentimes we must use L'Hopital's Rule:If g(x)/h(x) has the indeterminate form 0/0 or
/ at x=c, then
)(lim)(lim0
s s tfst
F
)('
)('lim
)(
)(lim
xh
xg
xh
xg
cxcx
Final Value Theorem
Lect11EEE 20213
The final value theorem states
The initial and final value theorems are useful for determining initial and steady-state conditions, respectively, for transient circuit solutions when we don’t need the entire time domain answer and we don’t want to perform the inverse Laplace transform
)(lim)(lim s s tf 0stF
Initial and Final Value Theorems
The initial and final value theorems also provide quick ways to somewhat check our answers
Example: the ‘finger’ method solution gave
Substituting t=0 and t=∞ yields
tt eetf 32
3
10
2
5
6
5)(
6
5
2
1510
6
5)(
06
20155
3
10
2
5
6
5)0( 00
eetf
eetf
Lect11 EEE 202 14
Initial and Final Value TheoremsWhat would initial and final value theorems
find?First, try the initial value theorem (L'Hopital's
too)
Next, employ final value theorem
This gives us confidence with our earlier answer
05
52
5lim
65
)1(5lim)0(
)3)(2(
)1(5lim)(lim)0(
2
sss
sf
ss
ss s f
sdsd
dsd
s
ssF
6
5
)3)(2(
)1(5
)3)(2(
)1(5lim)(lim)(
00
ss
ss s f
ssF
Lect11 EEE 202 15
Solving Differential Equations
Lect11EEE 20216
Laplace transform approach automatically includes initial conditions in the solution
Example: For zero initial conditions, solve)0(')0()()(
)0()()(
22
2
yysssdt
tyd
xssdt
txd
Y
X
L
L
)(4)(30)(
11)(
2
2
tutydt
tyd
dt
tyd
Class Examples
Find inverse Laplace transforms of
Drill Problems P5-3, P5-5 (if time permits)
84)(
)1()(
2
2
ss
ss
s
ss
Z
Y
Lect11 EEE 202 17
