Detection of Mechanical Load Faults in Induction Motors at Variable Speed Using Stator Current Time-Frequency Analysis Martin Blodt l , Marie Chabert2 , Jeremi Regnier l , Jean Faucher l and Bruno Dagues l 1Laboratoire
d 'Electrotechnique et d 'Electronique Industrielle (LEEI) Unite mixte de recherche INPT-ENSEEIHT / CNRS No. 5828 2ENSEEIHT / IRIT / TeSA 2, Rue C. Camichel BP 7122 31071 Toulouse Cedex 7, France Email:
[email protected]@enseeiht.fr.
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Abstract- This paper examines the detection of mechanical load faults in induction motors during speed transients by stator current analysis. Mechanical load faults generally lead to load torque oscillations at specific frequencies. these frequencies are related to the mechanical rotor speed. The torque oscillations produce a characteristic sinusoidal phase modulation of the stator current. Speed transients result in time-varYing supply frequencies that prevent the classical, Fourier transform based spectral estimation. This problem can be overcome using timefrequency signal analysis. The methods applied in this paper are instantaneous frequency estimation and the Wigner Distribution. Furthermore, an adaptive demodulation method is proposed. The theoretical considerations are validated on signals obtained from an experimental setup. I. INTRODUCTION
Induction motors are nowadays used in a wide variety of industrial applications. In order to increase the productivity, reliability and safety of an installation containing induction motors, permanent and automatic motor condition monitoring is desired. Stator current based condition monitoring is often desirable due to easy and economical implementation. The monitoring is in most cases done in steady operation state using classical spectral analysis tools. However, a lot of drives are adjustable speed drives where mechanical speed transients may be present during a long time period. The speed transients are often linear and lead therefore to a linear evolution of the supply frequency. The time-varying supply frequency prevents the use of classical spectral analysis. The application of other signal processing methods like time-frequency analysis overcomes this problem and makes condition monitoring possible during speed transients. This paper investigates the detection of torque oscillations caused by mechanical faults in induction machines using stator current time-frequency analysis. In a general way, a fault in the load part of the drive will be seen from the induction machine by a periodic variation of the load torque that is no longer
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constant. Examples for such faults causing torque oscillations include: • general fault in the load part of the drive system e.g. load imbalance, shaft misalignment • gearbox faults e.g. broken tooth • bearing faults Torque oscillations already exist in a healthy motor due to space and time harmonics of the airgap field, but the considered fault related torque oscillations are present at particular frequencies, often related to the mechanical motor speed. Thomson mentioned in [1] that mechanical problems in the load may cause speed oscillations that modulate the motor input current and lead to additional frequencies in the current spectrum. Schoen et. al. have shown in [2] that load torque oscillations appearing at multiples of the rotational speed lead to frequencies in the stator current given by :
Iload == Is ± nIr
(1)
where Is is the stator supply frequency, Ir the rotational frequency and n == 1,2,3, ... The proposed fault detection schemes (e.g. [3]) may only be applied in steady motor operation i.e. at constant supply frequency. In this paper, the stator current is analyzed during transients i.e. variable supply frequencies using time-frequency signal analysis. In section II, the considered signal model for the stator current under fault is shortly presented. Different signal processing methods for detection such as instantaneous frequency estimation and the Wigner Distribution are discussed in section III. In section IV, these methods are applied to experimental signals in order to show their effectiveness. II. STATOR CURRENT SIGNAL MODEL UNDER MECHANICAL FAULT
The method used to study the influence of the load torque oscillation on the stator current is based on the magnetomotive force (MMF) and permeance wave approach [4] [5]. This approach is traditionally used for the calculation of the magnetic
airgap field with respect to rotor and stator slotting or static and dynamic eccentricity [6] [7]. The detailed theoretical development for the stator current in case of load torque oscillation has been given in [8] to identify the consequence of bearing faults and in [9] for the general case. The results will be shortly resumed in the following. As this paper considers variable speed drives, the supply frequency f s and the fault frequency f c can be considered variable. Note than fc can be for example the time-varying rotational frequency fr. The theoretical analysis of the stator current under fault, however, is identical to the steady state if relatively slow frequency variations are considered. Under a mechanical fault, the load torque as a function of time is assumed to be described by a constant component r canst and an additional component varying at the fault characteristic frequency fc. The first term of the variable component Fourier series is a cosine with frequency fc. For the sake of clarity, higher order terms are neglected in the following and only the fundamental term is considered. The load torque can therefore be described by:
(2) where r c is the amplitude of the load torque oscillation and W c == 21r fc. Considering the mechanical equation of the machine, the oscillating load torque leads to periodic oscillations at fc of the mechanical rotor speed. The consequence is an oscillation at the same frequency on the mechanical rotor position. If the fundamental rotor MMF is calculated in the stator reference frame by using the transformation between the two reference frames, the oscillating mechanical rotor position produces an oscillating rotor MMF F r ( (), t) that can be expressed as follows:
leads to the following stator current expression (for an arbitrary machine phase): i (t)
== i st (t) + irt (t) == 1st sin [ws(t)t + CPs]
i st (t) and irt (t) denote the stator current components resulting from the stator and rotor MMF. The amplitudes 1st and lrt are supposed quasi-constant. The healthy case is obtained considering f3 == O. The supply and fault frequencies are modelled as affine time functions as a consequence of typical linear speed profiles:
ws(t) == 211" (as + f3 st) wc(t) == 211" (a c + f3ct)
where p is the pole pair number, J the total inertia and s == 21r Is. The fault effect on the rotor MMF can be seen as a sinusoidal phase modulation at the characteristic fault frequency. The stator MMF F s ( (), t) does not change and takes the same expression as in the healthy case:
W
The previous section has shown that the load torque oscillations cause a phase modulation on one stator current component (6). When variable speed is considered, the supply frequency and the fault frequency vary with respect to time. Traditional methods of spectral analysis [10] using the Fourier transform on a long observation window, e.g. the periodogram, can therefore not be applied. In the following, for a proper instantaneous frequency definition, all signals will be considered in their complex form, the so-called analytical signal [11] [12]. The analytical signal z (t) is related to the real signal x (t) via the Hilbert Transform
+ jH {x(t)}
(9)
The analytical signal contains the same information as the real signal but its Fourier transform is zero at negative frequencies. The analytical form of the current signal is therefore:
i( t) == 1st exp j [21r Is (t)t + 'Ps] + I rt exp j [21r Is (t)t + f3 cos (211" I c ( t)t) ]
(10)
A. Instantaneous Frequency For a complex monocomponent signal z(t) == a(t)ej
