INTRODUCTIONLaplace transforms provide a method for representing and analyzing linear systems using algebraic methods. In systems that begin undeflected and at rest the Laplace ’s’ can directly replace the d/dt operator in differential equations. It is a superset of the phasor representation in that it has both a complex part, for the steady state response, but also a real part, representing the transient part. As with the other representations the Laplace s is related to the rate of change in the system. ![]() The basic definition of the Laplace transform is shown in Figure 17.2 The Laplace transform. The normal convention is to show the function of time with a lower case letter, while the same function in the s-domain is shown in upper case. Another useful observation is that the transform starts at t=0s. Examples of the application of the transform are shown in Figure 17.3 Proof of the step function transform for a step function and in Figure 17.4 Proof of the first order derivative transform for a first order derivative. ![]() Figure 17.2 The Laplace transform ![]() Figure 17.3 Proof of the step function transform ![]() Figure 17.4 Proof of the first order derivative transform The previous proofs were presented to establish the theoretical basis for this method, however tables of values will be presented in a later section for the most popular transforms. |