IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 27, NO. 1, MARCH 2012 71 An Active–Reactive Power Method for the Diagnosis of Rotor Faults in Three-Phase Induction Motors Operating Under Time-Varying Load Conditions S´ ergio M. A. Cruz, Member, IEEE Abstract—This paper presents a new method to diagnose rotor faults in operating three-phase induction motors under the pres- ence of time-varying loads. The proposed diagnostic strategy relies on a combined analysis of the amplitude and phase spectra of the instantaneous active and reactive powers of the motor, and allows to discriminate the effects introduced by a rotor fault from the ones caused by an oscillating load torque, even when these phenomena occur simultaneously. A theoretical analysis carried out using a lin- earized model of the induction motor in a synchronous reference frame, complemented with several simulation and experimental results, confirms the validity of the proposed diagnostic approach. Index Terms—Active–reactive power, condition monitoring, in- duction machines, load torque oscillations, rotor faults. NOMENCLATURE Δ ˜ I sp , Δ ˜ I sn Positive (negative) sequence component of the stator incremental currents. Δx Instantaneous incremental value of quantity x. Δˆ x Amplitude of Δx. ω e Angular supply frequency. ω L Load torque oscillation angular frequency. ω x Arbitrary oscillation angular frequency. ϕ 1 , ϕ 2 Initial phase angles of load torque oscillation and rotor slip variations, respectively. σ Total leakage factor. ¯ τ r = L r ¯ R r Rotor time constant for a faulty motor. τ s = L s R s Stator time constant. θ e Angular position of the dq axes in a syn- chronously rotating reference frame. θ r Rotor electrical position. θ sω Slip angle. f e Fundamental supply frequency. i ds ,i qs , Stator/rotor dq current components in i dr ,i qr a synchronous reference frame. Manuscript received March 31, 2011; revised September 1, 2011; accepted November 13, 2011. Date of publication February 3, 2012; date of current ver- sion February 17, 2012. This work was supported by the Portuguese Foundation for Science and Technology under Project PTDC/EEA-ENE/67350/2006. Paper no. TEC-00163-2011. The author is with the Department of Electrical and Computer Engineering and the Instituto de Telecomunicac ¸˜ oes, University of Coimbra, 3030-290 Coim- bra, Portugal (e-mail: smacruz@ieee.org). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEC.2011.2178027 ¯ i s , ¯ i r Space phasor of the steady-state stator (rotor) currents. J Moment of inertia. L m Mutual inductance between stator and rotor. L r Rotor inductance. L s Stator inductance. M Current ratio. n F Number or broken rotor bars. N b Number of rotor bars. p Total instantaneous active power. p 0 No-load active power. p = d/dt Time derivative operator. p p Number of pole pairs. q Total instantaneous reactive power. q 0 No-load reactive power. R F Additional resistance due to a rotor fault. R r Rotor resistance in healthy conditions. R r Average value of rotor resistance with a rotor fault. R s Stator resistance. s Rotor slip. t em ,t L Electromagnetic / load torque. u ds ,u qs Supply voltage components in dq axes. U Amplitude of the motor supply voltages. ¯ x Steady-state value of quantity x. x ∗ Conjugate of complex x. ˜ X Phasor of x. Z rp ,Z rn Positive (negative) sequence component impedance of the motor seen from the rotor side. I. INTRODUCTION I N the past decades, several methods have been proposed with the aim to diagnose rotor faults in operating three- phase induction motors [1]. Techniques such as the motor cur- rent signature analysis (MCSA) [2]–[8], spectral analysis of the electromagnetic torque [9], active power [10]–[12], reac- tive power [13], axial vibrations [14], analysis of the swing angle [15], or the extended Park’s vector approach [16] are just some examples of the vast research done in this field [17], [18]. In spite of the wide range of diagnostic techniques available to detect a rotor fault, most of them rely on the assumption that the motor is coupled to a constant load. If this is not verified, incon- clusive or erroneous results can be obtained, thus compromising the reliability of the diagnostic process. 0885-8969/$26.00 © 2011 IEEE