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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.

- 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, t
_{1}< t < t_{2}and is absolutely integrable, the ROC is the entire s-plane. - If x(t) is a right sided signal, x(t) = 0, t
_{1}< t_{0}, the ROC is of the form R{s} > max {R{p_{k}}} - If x(t) is a left sided signal x(t) = 0, t
_{1}> t_{0}, 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 p
_{1}< R{s} < p_{2} - 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.

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