Study Notes for Laplace Transform & Their Properties | Signals and Systems - Electrical Engineering (EE) PDF Download

Laplace Transform
The Laplace Transform is very important tool to analyze any electrical containing by which we can convert the Integro-Differential Equation in Algebraic by converting the given situation in Time Domain to Frequency Domain.

•  is also called bilateral or two-sided Laplace transform.
• If x(t) is defined for t≥0, [i.e., if x(t) is causal], then  is also called unilateral or one-sided Laplace transform.

Below we are listed the Following advantage of accepting Laplace transform:

• Analysis of general R-L-C circuits become more easy.
• Natural and Forced response can be easily analyzed.
• Circuit can be analyzed with impedances.
• Analysis of stability can be done easiest way.

Statement of Laplace Transform

• The direct Laplace transform or the Laplace integral of a function f(t) defined for 0 ≤ t < ∞ is the ordinary calculus integration problem for a given function f(t) .
• Its Laplace transform is the function, denoted F(s) = L{f}(s), defined by
  
• A causal signal x(t) is said to be of exponential order if a real, positive constant σ (where σ is real part of s) exists such that the function, e- σt|X(t)| approaches zero as t approaches infinity.
• For a causal signal, if lim e^-σt|x(t)|=0,  for σ > σ_c and if lim e^-σt|x(t)|=∞ for σ > σ_c then σ_c is called abscissa of convergence, (where σc is a point on real axis in s-plane).
• The value of s for which the integral  converges is called Region of Convergence (ROC).
• For a causal signal, the ROC includes all points on the s-plane to the right of abscissa of convergence.
• For an anti-causal signal, the ROC includes all points on the s-plane to the left of abscissa of convergence.
• For a two-sided signal, the ROC includes all points on the s-plane in the region in between two abscissa of convergence.

Properties of the ROC
The region of convergence has the following properties

• ROC consists of strips parallel to the jω-axis in the s-plane.
• ROC does not contain any poles.
• If x(t) is a finite duration signal, x(t) ≠ 0, t₁ < t < t₂ and is absolutely integrable, the ROC is the entire s-plane.
• If x(t) is a right sided signal, x(t) = 0, t₁ < t₀, the ROC is of the form R{s} > max {R{p_k}}
• If x(t) is a left sided signal x(t) = 0, t₁ > t₀, the ROC is of the form R{s} > min {R{p_k}}
• If x(t) is a double sided signal, the ROC is of the form p₁ < R{s} < p₂
• If the ROC includes the jω-axis. Fourier transform exists and the system is stable.

Partial Fraction Expansion in Laplace Transform

• All poles have multiplicity of 1.
  
• Where, When one or more poles have multiplicity r.
  In this case, X(s) has the term (s-p)^r.
  
  The coefficient λ_k can be found as
  
• Transfer Function of a Network having output Y(t) & Input X(t) can be computed as Laplace Transform
  
  where N(s) is the Laplace transform of Output Y(t).
  D(s) Laplace Transform of Input X(t).

Inverse Laplace Transform

• It is the process of finding x(t) given X(s)
  X(t) = L^-1{X(s)}
  There are two methods to obtain the inverse Laplace transform.
• Inversion using Complex Line Integral
  
• Inversion of Laplace Using Standard Laplace Transform Table.

(Magic) A: Derivatives in t → Multiplication by s.




(Magic) B: Multiplication by t → Derivatives in s.


Laplace Transform of Some Standard Signals





Some Standard Laplace Transform Pairs





Properties of Laplace Transform



Key Points

• The convolution theorem of Laplace transform says that, Laplace transform of convolution of two time domain signals is given by the product of the Laplace transform of the individual signals.
• The zeros and poles are two critical complex frequencies at which a rational function of a takes two extreme value zero and infinity respectively.