2.5.  USEFUL LAPLACE TRANSFORMS

The Laplace transforms for various time functions will now be considered. These are readily obtainable through a direct application of Eq. (2.54).

A.  Laplace Transform of a Unit Step

For the unit step function defined by

the Laplace transform is

Therefore,

From here on we assume that f(t) = 0 for t < 0.

B.  Laplace Transform of an Exponential Decay

For the function

we have the Laplace transform

Therefore,

C.  Laplace Transform of a Unit Ramp

For the function

the Laplace transform is

Integrating by parts,

with u = t, dv = e^−st dt, the following is obtained:

Therefore,

D.  Laplace Transform of a Sinusoidal Function

For the function

The Laplace transform is

The solution to Eq. (2.63) is simplified by using the exponential form of sin ωt,

Therefore,

Therefore,

Once the Laplace transform for any function f(t) is obtained and tabulated, it need not be derived again. The foregoing results and other important transform pairs useful to the control engineer appear in Table 2.1. An extended table is shown in Appendix A. In addition, the location of the poles of the transformed function in the s-plane is listed in Table 2.1.