Characteristic Equation and Function:

Laplace Transformation

We shall first present a definition of the Laplace transformation and a brief discussion of the condition for the existence of the Laplace transform and then provide examples for illustrating the derivation of Laplace transforms of several common functions.

Let us define

f ( t ) = a function of time t such that f ( t ) = 0 for t < 0

s = a complex variable

L = an operational symbol indicating that the quantity that it prefixes is to be transformed by the Laplace integral

image

F(s) = Laplace transform of f( t )

Then the Laplace transform off ( t ) is given by

image

The reverse process of finding the time function f ( t ) from the Laplace transform F(s) is called the inverse Laplace transformation. The notation for the inverse Laplace transformation is L-1, and the inverse Laplace transform can be found from F(s) by the following inversion integral:

image

where c, the abscissa of convergence, is a real constant and is chosen larger than the real parts of all singular points of F(s). Thus, the path of integration is parallel to the jw axis and is displaced by the amount c from it.This path of integration is to the right of all singular points.

Evaluating the inversion integral appears complicated. In practice, we seldom use this integral for finding f (t ). There are simpler methods for finding f ( t ) .

It is noted that in this book the time function f ( t ) is always assumed to be zero for negative time; that is,

imageimageimageimageimageimageimageimageimage

First Order Systems:

imageimage

Fig: Block Diagram of first order systems

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Fig: Exponential Response Curve:

imageimageimage

Figure 5-3

Unit-ramp response of the system shown in above figure.imageimage

Figure 5-41

Unit-impulse response of the system shown in above figure.image

SECOND-ORDER SYSTEMS:imageimage

Figure 5-5

(a) Servo system;

(b) block diagram;

(c) simplified block diagram.imageimage

Figure 5-6 Second-order system.imageimageimageimageimageimageimage

Figure 5-7

Unit-step response curves of the system shown in above figure.

imageimageimageimage

Figure 5-8

Unit-step response curve showing td, tr, tp, Mp and ts.

The time-domain specifications just given are quite important since most control systems are time-domain systems; that is, they must exhibit acceptable time responses. (This means that, the control system must be modified until the transient response is satisfactory.)

Note that not all these specifications necessarily apply to any given case. For example, for an over damped system, the terms peak time and maximum overshoot do not apply.

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