The Effect of Pole and Zero Locations on the Transient Response of Linear Dynamic Systems

The effect of the locations of the poles and zeros of the transfer function of a linear dynamic system on the locations and the magnitudes of the maxima and minima of the transient response resulting from the application of a step-function input to the system is studied. Consideration is given to the necessary conditions for the production of a monotonic time response, expressed in terms of the pole and zero locations. In general, the results of the investigation are limited to stable low-pass systems, having only first-order poles and no poles on the jω axis. A method of computing the locations and magnitudes of the maxima and minima in the time response is given which allows the calculations to be made in a straightforward and efficient manner. The evaluation of the transient performance of many practical lowpass systems can be simplified considerably by the use of this method. It is shown that, under certain conditions of pole and zero locations, the normalized time response may be well approximated by a single dominant time term. Methods of ascertaining from the pole and zero pattern whether these conditions exist are given. On the basis of the dominant-term approximation, a method is outlined for the design of pole and zero patterns to yield prescribed time-response characteristics of a certain class to step-function inputs. Constant overshoot-factor curves and charts are provided for this purpose and for rapid solution of analysis problems when applicable.