Useful Properties of Laplace Transform limit functions

3 Useful Properties of Laplace Transform limit functions 

Back transformation from the transfer function to a differential equation in the time-domain, can be achieved by substitution of 5 -x(5 ) = dx(t)/dt. The variable s is independent of the time and indicates more or less the rate of change. This can be explained by the final-value theorem and the initial value theorem apphed to a variation in the input variable of a system. 

The final steady state value can be found by taking s— 0  

The initial slope of the time response ciuve can be determined easily, as differentiation corresponds to multiplication by 5  

For a step change apphed to the first-order system of Eqn. (5.10), the initial and final values become  

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