Detection of Mechanical Abnormalities in Induction Motors by Electric Measurements

The paper gives an overview of the issues and means of detection of mechanical abnormalities in induction motors by electric measurements. If undetected and untreated, the worn or damaged bearings, rotor imbalance and eccentricity, broken bars of the rotor cage, and torsional and lateral vibration lead to roughly a half of all failures of induction motor drives. The detection of abnormalities is based on the fact that they cause periodic disturbance of motor variables, such as the speed, torque, current, and magnetic flux. Thus, spectral analysis of those or related quantities may yield a warning about an incipient failure of the drive system. Although the traditional non-invasive diagnostics has mostly been based on the signature analysis of the stator current, other media can also be employed. In particular, the partial instantaneous input power is shown, theoretically and experimentally, to offer distinct advantages under noisy operating conditions. Use of torque and flux estimates is also discussed.


INTRODUCTION
Induction motors, the primary workhorse of industry, consume over 60% of the total electric power produced in the USA.They are robust and inexpensive, insensitive to overloads, and maintain an approximately constant speed within a wide range of the load torque.The only significant weakness of induction motors consists in their sensitivity to magnitude of the supply voltage, which has a quadratic impact on the developed torque.Although the number of frequency- controlled adjustable-speed ac drives with induc- tion motors has recently been increasing at a fast pace, over 90% of induction motors in industry still operate in the uncontrolled mode, with a constant supply frequency.
The commonness of induction motors makes them to appear in various critical applications, in which a motor failure may cause huge material losses due to process interruption and spoilage, and even the safety endangerment.Direct and indirect   Tel." (702) 784-6927.Fax: (707)  784-6627.costs of such failure may largely exceed the expenses of repair or replacement of the motor.It is the cause of the increasing need for health monitoring of induction machines, facilitated by the progress in signal processing methods and hardware.In this paper, issues of detection of mechanical abnormalities in induction motor drives by means of non-invasive electric measure- ments are discussed.Most of the abnormalities affect the normally smooth operation of the motor, so that the resultant vibration and speed oscillations can serve as warning signals.However, for the majority of industrial medium-and low- power drives, direct mechanical sensors of vibra- tion and speed are impractical.In contrast, measurements of the supply (stator) voltage and current are easy and inexpensive.Actually, in most practical drive systems those two quantities are already measured and displayed as indicators of operating conditions.The problem is how to process the voltage and current signals, especially in a noisy environment, to obtain meaningful and reliable diagnostic information.An adequately accurate mathematical model of the induction motor is the necessary starting point.

FIGURE
Per-phase steady-state equivalent circuit of the induction motor.
The equivalent load resistance, RL, and, conse- quently, the currents and flux linkages, depend on slip, s, of the motor, which is defined as the relative difference between the synchronous angular veloc- ity, COsyn, and rotor velocity, toM, that is,

STEADY-STATE AND DYNAMIC MODELS OF THE INDUCTION MOTOR
Several mathematical models of the induction motor are known.Here, only two common repre- sentations are briefly described, namely the steady-state T equivalent circuit and its dynamic counterpart in the stator reference frame.The per-phase steady-state T equivalent circuit, shown in Fig. 1, is based on the analogy between the induction motor and the transformer, as explained, for instance, by Sharma (1994).where ccs denotes the slip velocity.
The synchronous angular velocity, cosyn, is the speed of the revolving magnetic field of the stator, proportional to the supply radian frequency, co, and inversely proportional to the number, p, of magnetic poles of the stator.Specifically, cosyn 2co/p, where co 2rCjup and j-up denotes the supply frequency in hertz.The slip of the motor increases with the load torque.In the steady-state or quasi-steady-state, the average load torque is matched by the average torque, Tare, developed by the motor.The latter quantity can be calculated from the equivalent circuit in Fig. as the ratio of the total output power, Pot,, dissipated in the equivalent load resistances of all the three phases.The per-phase equivalent load resistance, RL, is given by RI-Rr(-1). (2) As Pout 3RII 2, where Ir denotes the rms value of rotor current, then Tave-3 RL/r2 3 12Rr-. ( In the ideal steady-state of a drive system, the instantaneous torque, T, is constant and equals Tave.However, to obtain a general expression for the instantaneous torque, valid also under transient operating conditions of the motor, a dynamic model must be used in which electric time constants are taken into account.The dynamic equivalent circuit of the induction motor in the stator reference flame is shown in Fig. 2, taken from author's book (1994).The circuit is based on the idea of space vectors, denoted here by bold letters.Three phase windings of the stator of an induction machine produce a magnetic field that revolves in space with the synchronous speed.Thus, the magnetomotive force and magnetic flux are true space vectors rotating in the hypothetical reference flame affixed on the stator.Since the magnetomotive force is generated by stator currents, a vecfor, Is, of stator current can easily be defined.The definition can also be extended on space vectors of the stator voltage, Vs, rotor current, I, and stator and rotor flux linkages, Xs and 1, commonly, although imprecisely, called stator and rotor fluxes.Any space vector, , in the induction motor can be expressed as a geometric sum of its horizontal (direct-axis) and vertical (quadrature-axis) compo- nents, rd and !)q, as II ]O d + jl/tq, (4) where 1//d and l/jrq are related to the actual phase quantities, rA, 1//B and Oc, as The matrix differential equation of the electrical part of the motor can now be written as di (Av + Bi) (6) dt L where L2 LsL,.L2m, Symbols Ls and L denote the stator inductance and rotor inductance, respectively, where Ls-Lls+Lm and L,-=LI,-+Lm, while co0=PWM/2 represents the angular velocity of an equivalent two-pole motor.Components dl" and Vq,-of the rotor voltage vector are zero in the most common, squirrel-cage induction machines.The flux vectors, ks and r, are given by Ls Lrn] Lm Lr Ir]" (12)   Finally, the torque, T, developed in the motor can be expressed as T_P P Lm (iqs/dr ids iqr) (iqsAds ids/qs).(! 3)

REFLECTION OF TORQUE AND SPEED OSCILLATIONS IN THE SPECTRA OF STATOR CURRENT AND INSTANTANEOUS INPUT POWER
Diagnostics of mechanical abnormalities in an induction motor by electric measurements is based on the premise that the abnormalities cause oscil- lation of torque and speed of the motor.The frequency and amplitude of that oscillation are indicative of the type and extent of the abnormality.
Certain afflictions, such as rotor imbalance and/ or eccentricity demonstrate themselves by oscilla- tions synchronous with the rotational frequency, fM, of the motor.Broken bars in the rotor cage produce torque oscillation at even multiples of the frequency, fr, of rotor currents, equal Sfsup, as shown by Tavner and Penman (1987).Worn or damaged ball bearings cause high-frequency, low- amplitude disturbance of the motor speed, and the fundamental frequency of the resulting oscilla- tions is specific for the type and parameters of bearings, as explained by Schoen et al. (1995).
The lateral or torsional vibration, or mechanical problems in the drive system outside the motor will also have an impact on the torque and speed.
Any changes in speed of the motor affect the waveform of stator current because of the close interrelations of motor variables, as expressed by Eqs.
(3), ( 6), ( 12) and (13).In practice, voltages and currents are measured in the three-phase line supplying the motor.In the subsequent consid- erations, the line-to-line supply voltage, VAB(/), between terminals A and B of the stator is assumed to be FAB(/) x/VLL COS(CO/), ( 14) where VLL denotes the rms value of this voltage.
Then, in the case of a perfectly healthy motor, current ia,0(t) in line A is given by iA,0(t) /IL COS cot 99 (15 where IL is the rms value of this current and 9 denotes the load angle of the motor.However, if a periodic speed oscillation with the radian frequency col develops due to an abnormality in the motor, the current is modulated with the same frequency.The resultant waveform, iA(t), can be expressed as where m denotes the modulation index, dependent on severity of the abnormality.It can be seen from ( 16) that two sideband components appear in the spectrum of current, at frequencies f Aup hi--fl and f=fsup-fl, where f=co/(27r).Spectral analysis of the supply current, commonly called current signature analysis, is an established method of non-invasive health monitoring of induction motors.See, for instance, Kliman and  Stein (1992)  or Schoen et al. (1994).
If the value off1 is low in comparison with fsup, as with a broken bar in the rotor or with low- frequency vibration, the sideband components may be difficult to discern, especially in a noisy spectrum.Noisy spectra are typical for such drive systems as those of grinders, impeller pumps, or conveyor belts, in which the load torque is char- acterized by high-frequency random variations.
The noise can also be a result of electromagnetic interference, for instance that generated by power electronic converters.Therefore, other diagnostic media are worth considering.In particular, as demonstrated by Thomas and Hamilton (1994)  or Legowski et al. (1996), spectral analysis of the instantaneous input power seems to offer a promising alternative to the current signature analysis.
The instantaneous power in a three-phase line is measured using two voltage sensors connected between lines A and B and between lines C and B, and two current sensors, in lines A and C. The total power, PABe(t), drawn by the motor is a sum of partial powers, PAB(t) and peB(t) wtiich, in turn, are products of the respective voltages and currents, i.e., PAB(t):VAB(t)iA(t) and pe(t): vc(t)ie(t).Under ideal conditions, waveforms of the partial powers consist of a dc component and a sinusoidal ac component whose frequency equals 2v, while the total power has only the dc component constituting the average power (real power) consumed by the motor.With the modulation of current as in ( 16), the partial and total instantaneous powers drawn by the motor are given by pA(t) pA,0(t) + mVLIL where PAB,0(t), Pcu,0(t) and PAC,0 denote ideal waveforms of the respective powers.
It can be seen from ( 17)-( 19) that three oscilla- tion-induced components appear in the spectra of partial instantaneous powers: at f fsup n t-fl, f= Aup--fl and at f= fl.The latter spectral com- ponent, subsequently called a characteristic com- ponent, carries precious direct information about the oscillation.The spectrum of total power contains only the dc component, PAge,0, and the characteristic component.The amplitudes, AAB, AcB and AABC, of the characteristic components ofpAB, PCB and PABC are given by AAB m VLLIL COS g) -+- ACB m VLLIL COS (2) AABC v/mVLLIL COS(g)). ( Since the load angle in uncontrolled induction motors is always positive (current lags voltage), it can be inferred from (20)-( 22) that the character- istic component of partial power PcB is stronger than that of PAB.When g)> 60 , i.e., when motor runs light, AcB is even greater than AABC.Thus, proper selection of the measured partial power is important.
The question which instantaneous electric power, partial or total, is better as a diagnostic medium is more difficult to answer.On one hand, since the total power is ideally of a dc quality, then any ac component would indicate non-ideal conditions of the machine, unless the load intro- duces a periodic disturbance.On the other hand, spectrum of the partial power displays as many as three components related to the speed and torque oscillation.Consequently, the characteristic component can be employed as the oscillation indicator, with the sideband components used for verification.The verification of the characteristic component is crucial in noisy spectra, which is demonstrated in the subsequent description of laboratory experiments on a low-power induction motor.

EXPERIMENTAL EXAMPLES
A well-worn 3-hp, 220-V, 60-Hz, four-pole induc- tion motor was experimented with.Voltage and current measurements were stored and processed in a personal computer.Spectrum of partial power of the motor on no load is shown in Fig. 3 in the logarithmic amplitude scale.Since the rotational frequency of the motor was 29.8 Hz (1788 rpm), the dominant spectral components are seen to appear at 29.8 (characteristic component), 90.2 and 149.8 (sideband components) and 120Hz (supply- frequency component).Synchronousness of the frequency of the characteristic component with the rotational frequency of the motor implies minor imbalance and eccentricity of the rotor, conditions typical for essentially healthy but old machines.To confirm the above conclusion, the rotor imbal- ance was purposefully worsened by attaching a small weight to the shaft surface.The resultant spectrum of partial power is shown in Fig. 4. The three indicative spectral components are seen to have increased by ten and more decibels.
In another experiment, the imbalancing weight was removed, but one bar of the rotor cage was separated from the end ring.The motor ran with the rotational frequency of 29.73 Hz (1784 rpm), i.e., with the slip, s, of 0.009, and the frequency, .f, of rotor currents of 0.54 Hz.Thus, the resultant dis- turbances of the developed torque were expected at multiples of 1.08Hz.Indeed, as seen in Fig. 5, the most prominent sideband components in the spectrum of partial instantaneous power appear at about 118.9 and 121.1 Hz, with smaller components at 116.7, 117.8, 122.2 and 123.3Hz.All these components are so close to the central supplyfrequency component of 120 Hz that they would be difficult to discern if the frequency range was wider, e.g., 0-200 Hz as in Figs. 3 and 4.However, as shown in Fig. 6, the characteristic component at 1.1 Hz directly points to the frequency of speed oscillation and, indirectly, to the type of abnorm- ality responsible for that oscillation.
For the last experiment presented, the motor was coupled to a dc generator with an electrically controlled load such that the load torque displayed Frequency, Hz FIGURE 6 Spectrum of partial power drawn by the motor with a separated rotor bar (frequency range: 0-10 Hz).
high-frequency random variations.The purpose of that arrangement was to "drown" the prominent frequency components seen in Fig. 3 in the load- generated noise.In this respect, the experiment was quite successful, as seen in the spectra of the total power and partial power shown in Figs.7 and 8, respectively.
To show the advantage of three indicative spectral components of the partial instantaneous power over the single characteristic component of the total power, an attempt was made to determine the fundamental frequency of speed oscillation from the spectrum of partial power.The effort of recovering the 29.8-HZ characteristic component seen in Fig. from the noisy spectrum in Fig. 8 was based on the fact that the characteristic component is accompanied by two sideband components, and that amplitudes of all the three components, as follows from (17), do not differ much.Indeed, if 30 , a value typical for induction motors (unless on no load), all three amplitudes are equal.
The spectrum filtering procedure comprised the following four steps: (1) Determination of coordinates of all peaks of the spectrum.
(2) Elimination ofpeaks that fall below an assumed average noise level.Here, it was taken as the average value (in dB) of all the peaks, including the dc and fundamental components.
(3) Elimination of all peaks not accompanied by the corresponding sideband components.
(4) Elimination of all the remaining peaks that differ significantly from their sideband counter- parts.
With the 6-dB allowable amplitude difference in step 4, five frequency components, at 16, 19, 30, 84   and 90 Hz (the resolution of Hz was employed) survived the elimination.The first two components were generated by the load torque, while the 90-Hz frequency was that of the sideband of the expected 30-Hz characteristic component.With the allow- able difference reduced to 5 dB, the 16-Hz compo- nent was eliminated.The difference of 4 dB had left only the 19 and 30-Hz components, while with the difference reduced to 3 dB only the expected 30-Hz   component had remained.The described pro- cedure was quite unsophisticated, and better results could be expected from more advanced methods of signal processing.

OTHER DIAGNOSTIC MEDIA
Motor variables other than the current or instanta- neous electric power can be estimated and used as media for diagnostics of mechanical abnormalities.The estimations are based on current and voltage measurements, and on the mathematical models of the induction motor presented in Section 2. In particular, the stator flux vector, ks, can be determined from the equation of the equivalent circuit in Fig. 2 dt Vs Rsls. (23) Use of this variable for condition monitoring of large induction motors has been reported by Deng  and Ritchie (1993).More importantly, the knowl- edge of flux allows estimation of the developed torque, T, which, based on (13), can be computed in a simple scheme shown in Fig. 9.As argued by Hsu (1995), the torque constitutes the best medium for diagnostics of induction machines since any abnormality affects this variable in a direct way.
As seen in Fig. 9, only the stator resistance, Rs, must be known for accurate torque estimation from the stator voltage and current measurements.This parameter is easy to measure, but it changes with the temperature.Also, the torque estimator is based on the assumption of linearity of the magnetic circuit of the motor.As a result, ideal tuning of the estimator is not possible, and the estimated torque waveform, T*(t), differs from T(t), that of the actual torque, mainly by an ac component (ripple) having the frequency off, up.This is illustrated in Fig. 10, in which the solid line and the broken line represent T*(t) and T(t), respectively, in a hypothe- tical 10-hp, 230-V, 60-Hz, 6-pole induction motor whose load torque oscillates with the frequency of 10Hz.The torque estimator is detuned by 10% from the actual value of stator resistance.The 60-Hz error component of T*(t) can, however, easily be eliminated using a band-elimination filter blocking the supply frequency.Alternately, the amount of the 60-Hz ripple, could be employed as an accuracy indicator, to be minimized in an.auto- mated tuning process.Use of spectral analysis of the estimated torque as a diagnostic tool is still a novel proposition and further research is required for full evaluation of its usefulness.FIGURE 10 Waveforms of the actual torque (broken line) and estimated torque (solid line) with the estimator detuned by 10% from the actual value of stator resistance.

CONCLUSION
Non-invasive diagnostics of mechanical abnormal- ities in the induction motor by measurements of the stator voltage and current is based on the assumption that in an unhealthy motor the torque and speed oscillate with a frequency that is indicative of the type of abnormality.Simply speaking, the motor acts as a transducer modulating the stator current.However, the current, whose spectrum has traditionally been employed as the motor signature, is not necessarily the best diagnostic medium available.Use of the instantaneous power, prefer- ably the partial power, offers certain advantages over the stator current, especially under noisy conditions.The progress in signal processing equipment and techniques, such as advanced digi- Space vectors of the rotor and stator flux linkages, respectively Direct-axis and quadrature-axis components of the space vector of stator flux linkage, respectively Load angle of an induction motor General symbol of a space vector and its components General symbols of phase variables Supply radian frequency, motor angular velocity, and angular veloc- ity of an equivalent two-pole motor, respectively Slip angular velocity, synchronous angular velocity, and radian fre- quency of speed and torque oscil- lation, respectively

FIGURE 2
FIGURE 2 Dynamic equivalent circuit of the induction lnotor in the stator reference frame.

FIGURE 3 FIGURE 4 FIGURE 5
FIGURE 3 Spectrum of partial power drawn by the healthy motor.

FIGURE 7 FIGURE 8
FIGURE 7 Spectrum of total power drawn by the motor with a randomly varying load.

FIGURE 9
FIGURE 9 Block diagram of a torque estimator.

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