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Laplace Transform β Conditions for Existence, Region of Convergence, Merits & Demerits
Laplace Transform
The Laplace transform is a mathematical tool which is used to convert the differential equations in time domain into the algebraic equations in the frequency domain or s-domain.
Mathematically, if $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ is a time domain function, then its Laplace transform is defined as −
$$\mathrm{\mathit{L\left [ x\left ( \mathrm{t} \right ) \right ]}\mathrm{=} \mathit{X\left ( s \right )}\mathrm{=}\int_{-\infty }^{\infty}\mathit{x\left ( \mathrm{t} \right )e^{-st}\; dt}\; \; ...\left ( 1 \right )}$$
Where, π is a complex variable and it is given by,
$$\mathrm{s = \sigma + j\omega }$$
And the operator L is called the Laplace transform operator which transforms the time domain function into the s-domain function.
Since a linear time invariant (LTI) system is described by differential equations and the response of the system for a given input is obtained by solving the differential equations relating its input and output. But the solution of higher order differential equations is very tedious and time consuming, so the Laplace transform is used to solve these differential equations. The Laplace transform converts the time domain differential equations into the algebraic equations in s-domain, get the solution in s-domain and then the solution in time domain can be obtained by the taking inverse Laplace transform of the solution.
Conditions for Existence of Laplace Transform
The Laplace transform of a function $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$, i.e., function $\mathrm{X \mathit{\left ( \mathit{s} \right )}}$ exists only if
$$\mathrm{\mathit{\int_{-\infty }^{\infty }\left|x\left ( t \right )e^{-\sigma t} \right|dt< \infty }}$$
Or, only if,
$$\mathrm{\mathit{\displaystyle \lim_{t \to \infty }x\left ( t \right )e^{-st}\mathrm{=}\mathrm{0}}}$$
Therefore, the necessary and sufficient conditions for the existence of the Laplace transform are −
The function $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ should be piece-wise continuous in the given closed interval and must be of exponential order.
The function $\mathrm{\mathit{x\left ( t \right )e^{-st}}}$ should be absolutely integrable
Region of Convergence of Laplace Transform
The region of convergence (ROC) is defined as the set of points in the s-plane for which the Laplace transform of function $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ (i.e., the function $\mathrm{X \mathit{\left ( \mathit{s} \right )}}$) converges.
Explanation – For a given function $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$, the Laplace transform as given by the equation (1) may not converge for all values of the complex variable s. Since every value of the variable (π ) corresponds to a particular point in the s-plane. If there is no value corresponding to the variable (π ), i.e., no point on the splane for which the Laplace integral converges. Then, the function $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ does not have a region of convergence (ROC) and hence it is not Laplace transformable.
Properties of Region of Convergence of Laplace Transform
Following are the properties of the ROC of Laplace transform −
The ROC of Laplace transform does not contain any poles.
The ROC of the Laplace transform of $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$, i.e., function $\mathrm{X \mathit{\left ( \mathit{s} \right )}}$ is bounded by poles or extends up to infinity.
The ROC of the sum of two or more signals is equal to the intersection of the ROCs of those signals.
The ROC of Laplace transform must be a connected region.
If the function $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ is a right-sided function, then the ROC of $\mathrm{X \mathit{\left ( \mathit{s} \right )}}$ extends to the right of the right most pole and no-pole is located inside the ROC.
If the signal $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ is a left-sided signal, then the ROC of Laplace transform $\mathrm{X \mathit{\left ( \mathit{s} \right )}}$ extends to the left of the left most pole and no pole is located inside the ROC.
If the signal $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ is a two-sided signal, then the ROC of the Laplace transform $\mathrm{X \mathit{\left ( \mathit{s} \right )}}$ is a strip in the s-plane bounded by the poles and nopole is located inside the ROC.
The impulse signal is the only signal for which the ROC is the entire splane.
The imaginary axis of the s-plane is contained by the ROC of a stable LTI system.
Merits and Demerits of Laplace Transform
Following are some of the merits of using the Laplace transform technique −
The Laplace transform has a convergence factor and hence it is more general than Fourier transform. That means, the signals which are not convergent in Fourier transform are convergent in the Laplace transform.
Using Laplace transform, the differential equations describing a system can be converted into simple algebraic equations. Hence, the analysis of the LTI systems using Laplace transform becomes easier.
The Laplace transform can be used to analyse the unstable systems.
The convolution operation in time domain can be converted into multiplication in s-domain.
The demerits of using Laplace transform are as follows −
$\mathrm{\mathit{s\mathrm{=}j\omega }}$ is used only for steady-state sinusoidal analysis.
Using Laplace transform technique, the frequency response of the system cannot be drawn or estimated. Only the pole-zero plot can be drawn.
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