Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev

Signal and System

Electrical Engineering (EE) : Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev

The document Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev is a part of the Electrical Engineering (EE) Course Signal and System.
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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.

  • Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev 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 Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev 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
    Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
  • 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 > σc and if lim e-σt|x(t)|=∞ for σ > σc then σc is called abscissa of convergence, (where σc is a point on real axis in s-plane).
  • The value of s for which the integral Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev 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{pk}}
  • If x(t) is a left sided signal x(t) = 0, t1 > t0, the ROC is of the form R{s} > min {R{pk}}
  • 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.
    Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
  • Where, When one or more poles have multiplicity r.
    In this case, X(s) has the term (s-p)r.
    Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
    The coefficient λk can be found as
    Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
  • Transfer Function of a Network having output Y(t) & Input X(t) can be computed as Laplace Transform
    Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
    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
    Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
  • Inversion of Laplace Using Standard Laplace Transform Table.

(Magic) A: Derivatives in t → Multiplication by s.
Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
(Magic) B: Multiplication by t → Derivatives in s.
Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev

Laplace Transform of Some Standard Signals
Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev

Some Standard Laplace Transform Pairs
Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev

Properties of Laplace Transform
Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
Study Notes for Laplace Transform & Their Properties Electrical Engineering (EE) Notes | EduRev
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.
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