laplace transform applications of the laplace transform –solve differential equations (both...
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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 3 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
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 variables analysis (e.g., MAT 461)
0
dtetfstf stFL
Singularity Functions
• Singularity functions are either not finite or don't have finite derivatives everywhere
• The two singularity functions of interest here are
(1) unit step function, u(t)(2) delta or unit impulse function, (t)
Unit Step Function, u(t)
• The unit step function, u(t)– Mathematical definition
– Graphical illustration
01
00)(
t
ttu
1
t0
u(t)
Extensions of the 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
– like an on/off switch
at
atatu
1
0)(
1
t0 a
1
t0 +T
u(t-) - u(t- -T)
Delta or Unit Impulse Function, (t)
• The delta or unit impulse function, (t)– Mathematical definition (non-pure version)
– Graphical illustration
0
00 1
0)(
tt
tttt
1
t0
(t)
t0
Transform Pairs
The Laplace transforms pairs in Table 13.1 are important, and the most important are repeated here.
(t) F (s )
δ (t) 1
u (t) {a co ns ta n t}
s
1
e -a t
as 1
t2
1
s
t e -a t
2
1
as
Laplace Transform PropertiesT h e o r e m P r o p e r t y ( t ) F ( s )
1 S c a l i n g A ( t ) A F ( s )
2 L i n e a r i t y 1 ( t ) ±
2 ( t ) F 1 ( s ) ± F 2 ( s )
3 T i m e S c a l i n g ( a · t ) 01
aa
s
aF
4 T i m e S h i f t i n g ( t - t 0 ) u ( t - t 0 ) e - s · t 0 F ( s ) t 0 0
6 F r e q u e n c y S h i f t i n g e - a · t ( t ) F ( s + a )
9 T i m e D o m a i nD i f f e r e n t i a t i o n dt
tfd )( s F ( s ) - ( 0 )
7 F r e q u e n c y D o m a i nD i f f e r e n t i a t i o n
t ( t )ds
sd )(F
1 0 T i m e D o m a i nI n t e g r a t i o n
tdf
0)( )(
1s
sF
1 1 C o n v o l u t i o n t
dtff0 21 )()( F 1 ( s ) F 2 ( s )
Block Diagram Reduction
Block Diagram Reduction
Block Diagram Reduction
Block Diagram Reduction
Reference
Y(s) = ___K*G(s)R(s) 1+K*G*H(s)Closed Loop
KSum
H(s)
Y(s)R(s)-
Gplant
Y is the 'ControlledOutput'
Forward Path
Y(s) = ___K*G(s)R(s) 1+K*G*H(s)Closed Loop
Characteristic Equation:Den(s) = 1+K*GH(s) = 0
Stability: The response y(t) reverts to
zero if input r = 0.All roots (poles of Y/R) must have Re(pi) < 0
Characteristic Equation:Den(s) = 1+K*GH(s) = 0
Closed Loop Poles are theroots of the Characteristic
Equation, i.e. 1+K*GH(s) = 0
Poles and Stability
Poles and Stability
Poles and Stability
Underdamped System (2nd Order)
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