Application of the zeros and poles matched method for modeling of electrical systems

: pp. 72 - 87
Lviv Polytechnic National University
Lviv Polytechnic National University

The widespread use of mathematical applications, which include differential equations solvers, and the increase in the speed of computing devices have led to a decrease in interest in operator methods, in particular, the z-transform. Nevertheless, the use of the z-transform capabilities allows the implementation of efficient high-speed computing schemes with high numerical stability. The need for this may arise in the case of real-time simulation or the synthesis of digital control systems. Based on the analysis of literary sources, the relevance and advantages of using the z-transform for modeling the dynamics of electrical engineering systems are shown.

The method of computer modeling is considered, the basis of which is the use of the method of matching zeros and poles of an equivalent continuous transfer function to build a computer model. The process of implementing the modeling recurrent formulas obtained by this method is shown for three elementary dynamic blocks, which are obtained as a result of the expansion of the transfer function according to the Heaviside residue theorem: integral (zero pole), first-order inertial (real pole) and second-order blocks with a real zero and by a pair of complex-conjugated poles. In this way, the parallel decomposition of the researched system is implemented, which makes it possible to reduce the negative impact of the limited bit precision of the system and facilitate the execution of parallel calculations. A discrete transfer function and a simulation recurrent equation were obtained for each such block.

The practical use and advantages of this method are shown on two examples: a simple elastic joint mechanical system, which is described by a second-order differential equation, and a nonlinear model of an asynchronous machine based on a single-phase T-shaped equivalent circuit. Both problems are illustrated by examples of solutions in the environment of the Mathcad mathematical application. The effectiveness of the zeros and poles matched method of compared to classical numerical methods for solving ordinary differential equations is shown.

The use of this method of mathematical modeling makes it possible to provide a stable numerical solution with a specified accuracy for a wide range of solution steps.

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