Electric Power Systems Research 53 (2000) 149 – 156 Application of the reference frame theory to the dynamic analysis of a three-phase induction motor fed from a single-phase supply Yusuf A. Al-Turki a, *, Hussain Al-Umari b a Department of Electrical and Computer Engineering, Faculty of Engineering, K.A.U., PO Box 9027, Jeddah 21413, Saudi Arabia b Saudi Aramco, Jeddah, Saudi Arabia Received 19 August 1998; accepted 15 February 1999 Abstract Dynamic analysis of a three-phase induction motor fed from a single-phase source is presented. This analysis is based on the reference frame theory where real quantities are used instead of the instantaneous symmetrical components theory that deals with instantaneous complex quantities. Various criteria for capacitor selection are reviewed and dynamic as well as steady state analysis is carried out for the major criteria (zero V 2 , minimum V 2 and minimum U V ). The paper also shows that minimum U I criteria is essentially the same as minimum U V . © 2000 Elsevier Science S.A. All rights reserved. Keywords: Reference frame theory; Dynamic analysis; Three-phase induction www.elsevier.com/locate/epsr 1. Introduction Numerous studies have been conducted to investigate the operational analysis of a balanced three-phase in- duction motor. Such studies dealt mostly with motor behavior under steady-state conditions. A case that may occur in real life is the operation of a three-phase induction motor connected to a single-phase supply. This may happen if one of the phases is open-circuited either accidentally or intentionally. In rural areas, where the distribution system might be single-phase, the three-phase motor is operated using an auxiliary means to start the motor. Many authors have studied the steady-state opertion of the three-phase induction motor when fed from a single-phase source. To start the motor, a phase conver- tor (usually a capacitor) is used. Brown and Jha [1] studied motor starting and applied certain criteria to select capacitor size. Jha and Jha [2] selected three capacitor values: one at the starting, the second in the middle region of the torque-speed curve and the third at the normal running region. Oliveira [3] discussed the starting of such motor and suggested the use of a capacitor which gives zero negative sequence voltage at a certain point. In his study, Oliveira recommended switching-off the capacitor when the motor reaches a certain speed. He introduced current unbalance factor U I in addition to voltage unbalance factor U v . Mo- hamadein et al. [4] suggested other criteria for the selection of capacitor size. These criteria are: maximum torque, maximum power factor and maximum effi- ciency. However, they found that the new criteria lead to higher phase voltage which may damage stator wind- ing. Pillary and Brzezinski [5] considered different schemes of phase converters. They discussed motor derating in case of unbalanced voltages or unbalanced currents. Regarding the transient analysis of the motor when fed from a single-phase supply, Murthy et al. [6] pre- sented a good paper to address the problem. They used the instantaneous symmetrical components method to investigate motor performance and select capacitor size which gives minimum unbalance factor. Al-Ohaly et al. [7] have also used the instataneous symmetrical compo- nents method in the analysis. However, the positive and negative instantaneous symmetrical components are complex conjugates which does not match with the fact that instantaneous three-phase quantities are real. Also, * Corresponding author. 0378-7796/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII:S0378-7796(99)00027-9