Skip to Main Content
The paper describes the z-transform theory and its application. The background theory is built up in four stages from the continuous convolution integral, through the real-time impulse theory, to the z-transform. It is stressed that, unlike the Laplace transform, the z-transform is only an approximation. Engineering units are used throughout and emphasis is given to the correct representation of the sampler. The finite time of sampling is taken into account and it is shown how to make practical use of it. Illustrations of practical systems show good agreement with the calculations and attention is called to some common errors in the literature. The stability criteria are stated and methods of stability investigations are described. It is shown that for systems with zero-order hold circuits, the continuous Laplace transform is a more useful tool than the z-transform. The paper concludes with a practical approach to the design of discrete controllers.