**Laplace Transform**

The Laplace Transform is a very important** tool** to analyse any electrical containing by which we can convert the Integral-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 have listed the Following advantage of accepting Laplace transform:

- Analysis of general R-L-C circuits become easier.
- Natural and Forced response can be easily analyzed.
- The 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 the 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**, for σ > σ*e*0^{-σt}|x(t)|=_{c}and if**lim**for σ > σ*e*^{-σt}|x(t)|=∞_{c}then σ_{c }is called the abscissa of convergence, (where σ_{c }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 the abscissa of convergence.
- For a two-sided signal, the ROC includes all points on the s-plane in the region in between two abscissae of convergence.

**Properties of the ROC**

*The region of convergence has the following properties*

- ROC consists of strips parallel to the
*j*ω-axis in the s-plane. - ROC does not contain any poles.
- If
*x*(*t*) is a finite duration signal,*x(t) ≠ 0, t*and is absolutely integrable, the ROC is the entire s-plane._{1}< t < t_{2} - If
*x*(*t*) is a right sided signal,*x(t) = 0, t*the ROC is of the form R{s} > max {_{1}< t_{0},*R*{*p*}}_{k} - If
*x*(*t*) is a left sided signal*x(t) = 0, t*the ROC is of the form_{1}> t_{0},*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.

**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.

**Note A:** Derivatives in t → Multiplication by s.

** 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.

## 100 Days Detailed Study Plan of RVUNL JE & AE for Electrical Engg.

You can avail of Gradeup Super for all AE & JE Exams:

## Gradeup Super for AE & JE Exams (12+ Structured LIVE Courses and 160+ Mock Tests)

You can avail of Gradeup Green Card specially designed for all AE & JE Exams:

## Gradeup Green Card AE & JE (160+ Mock Tests)

Thanks,

Sahi Prep Hai Toh Life Set Hai!

## Comments

write a comment