The is a powerful tool for converting functions from the s-domain back to the . It's crucial for analyzing system behavior after performing operations in the s-domain, like working with transfer functions or .
and are key techniques for finding inverse transforms. These methods allow us to break down complex s-domain functions into simpler terms, making it easier to convert back to the time domain and understand system responses.
Inverse Laplace Transform
Inverse Laplace transform definition
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Converts a function from the s-domain () back to the time domain (t-domain)
Denoted as L−1{F(s)}=f(t), where F(s) is the Laplace transform of the time-domain function f(t)
Recovers the original time-domain function from its Laplace transform representation
Essential for analyzing the behavior of systems in the time domain after performing operations in the s-domain (transfer functions, convolution)
Partial fraction expansion for inversion
Decomposes rational Laplace transforms into a sum of simpler terms
is a ratio of two polynomials in the s-domain, F(s)=Q(s)P(s)
Factors the denominator polynomial Q(s) and expresses the rational function as a sum of partial fractions
Each partial fraction has a denominator that is a linear or quadratic factor of Q(s) (, )
Resulting partial fractions are easier to inverse Laplace transform individually using Laplace transform pairs or tables
Time-domain function f(t) is obtained by summing the inverse Laplace transforms of each partial fraction term
Laplace transform pairs and tables
Set of correspondences between time-domain functions and their respective Laplace transforms
Derived from the properties of the Laplace transform and well-established for common functions (exponential, sinusoidal)
Compilation of these pairs used as a reference for finding the inverse Laplace transform of a given function
Tables list the time-domain function f(t) and its corresponding Laplace transform F(s)
Locate the entry in the table that matches the given Laplace transform and read off the corresponding time-domain function
Common Laplace transform pairs:
L{1}=s1
L{tn}=sn+1n!, where n is a non-negative integer
L{eat}=s−a1
L{sin(at)}=s2+a2a
L{cos(at)}=s2+a2s
System Response in Time Domain
Time-domain response from Laplace transforms
Determined by finding the inverse Laplace transform of the system's
Transfer function H(s) is the ratio of the Laplace transform of the output Y(s) to the Laplace transform of the input X(s), H(s)=X(s)Y(s)
Given H(s) and X(s), the Laplace transform of the output is calculated as Y(s)=H(s)⋅X(s)
To find the time-domain output y(t):
Take the inverse Laplace transform of Y(s) using partial fraction expansion (if necessary)
Use Laplace transform pairs or tables
y(t)=L−1{Y(s)}=L−1{H(s)⋅X(s)}
Resulting time-domain function y(t) represents the system's response to the given input in the time domain (step response, impulse response)