International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 7 (2018) pp. 5029-5033 © Research India Publications. http://www.ripublication.com 5029 Mathematical Model of a Synchronous Machine under Complicated Fault Conditions Prof. Hani Obeid PhD EE, P.Eng.,SMIEEE, Applied Sciences University, P.O.Box 950674, Amman 11195- Jordan. Abstract This paper presents a mathematical model of a synchronous machine when a complicated fault occurs in the external network. The equations of stator and rotor circuits are written in space vector using Clarke’s transformation. The equations are subjected to mathematical manipulations based on phasor concepts, and series expansion of algebraic and differential equations, neglecting high order harmonics, in order to obtain a final model of the machine. This model of the machine is an appropriate one to obtain the negative sequence impedance of the synchronous machine. Keywords: synchronous machine; Clarke’s transformation; stator and rotor equations. INTRODUCTION At present time short circuit calculations are performed on the basis of IEC 60909, where initial symmetrical short circuit current is calculated for fundamental frequency, and consequently, all other quantities, such as, asymmetrical, peak, making and breaking currents are obtained for the evaluation of thermal and mechanical effects of short circuit [1]. Symmetrical components transformation has been used for decades in calculation of short circuit currents in power systems. The application of time dependent symmetrical components makes sense, as the network parameters are available in this form [2]. Usually, these calculations are limited to short period of time as the protection will operate, and in a fast manner, clear the short circuit. In these situations, the assumption of constant emf of synchronous machines is accepted and doesn’t lead to appreciable errors. At the same time, in evaluation of electromechanical transient processes, positive sequence emf is implemented in calculations, which in turn is considered constant during the time of fault. There are cases when the protection fails to clear the fault, and the fault duration is extended, therefore, the assumption of constant emf will lead to errors. In such cases, it required to take into consideration the electromechanical transient processes in combination with short circuit calculations [3]. Mathematical model of a synchronous machine is considered to be of prime factor among all models of network elements, especially, when complicated fault conditions occurred in the external network. Under these conditions, pulsating magnetic field in rotor circuit will be created, which in turns will produce full spectrum of higher harmonics. When a complicated short circuit occurs in external network, asymmetrical current will flow in stator and rotor circuits of the machine, and emf of odd harmonics will be induced in stator circuits, while even harmonics in terms of rotor position will be created. When asymmetrical current of frequency ω flows through stator winding, in the rotor circuit rotating with , emf and current will appear, rotational speed of which can be determined by the following formula: Where harmonic order. At the same time, in stator circuit emf and current will be created with rotational speed calculated by the following formula: Mathematical model of a synchronous machine The following model of a synchronous machine is used to perform calculations of any complicated fault conditions in power system [4]. This model is based on transforming equations from time domain phase quantities [5] to space vector by using Clarke transformation: (1) (2) (3) (4) (5) (6)