**2.1. Broken bars stator current modulation and its contribution to the MCS formation**

Introduced theory and rotor fault analysis and diagnostics come from the basic principle when periodical changes in rotor magnetomotive force (MMF) cause the periodical changes of IM stator current amplitude and phase, and thus the stator current AM and PM.

Broken rotor bars, as an electric fault, cause the rotor asymmetry, the distortion of the rotor current distribution, rotor current pulsation and its amplitude modulation by the slip frequency *fslip*. Rotor bars current amplitude changes are transformed to stator current on slip pole frequency *fsp* and appears here as a stator current AM. This modulation can be interpreted as a primary modulation.

Rotor bars current amplitude changes cause the changes in force on coils moving in a magnetic field. The force can be obtained from the vector cross product of the current vector and the flux density vector *Fx= NI /x,* where *NI* is MMF*,*  is linked magnetic flux and *x* is the force direction. Subsequently, the total force on a current carrying rotor coils (bars), moving in a magnetic field, changes and electromagnetic torque oscillation appears. Torsion vibration also can appear. Oscillating torque causes periodical changes in the rotating phase angle and therefore the stator current PM. This modulation can be interpreted as a subsequent or secondary modulation, so PM cannot originate without AM and JAPM always exists. The JAPM can be interpreted as a stator current modulation by a complex modulating vector which changes both its amplitude and phase. Angular speed oscillation as the derivation of phase oscillation also appears.

The frequency of periodical changes is a slip pole frequency *fsp* (suffix*sp*) which is independent on the number of motor poles

$$f\_{sp} = p f\_{slip} = p s f\_{sync} = p s \mathcal{D} f\_l \text{ / p = } \mathcal{D} s f\_l = \mathcal{D} f\_l - f\_r p \tag{1}$$

Rotor Cage Fault Detection in Induction Motors by Motor Current Demodulation Analysis 525

, / 2 *L H l sp P PL PH l spp f f f a a a II* (3)

Unlike AM, PM is not observable from the stator current time course at so small modulation indexes to 0.15. PM can be visible as stator current time course compression and decompression from modulation indexes greater than 2, which are approximately 10 times

The spectrum of phase modulation (suffix *P*) is dependent on the modulation index size. If the modulation index is greater than *1*, the PM spectrum consists of many side components, if it is lower than *0.4*, the spectrum contains only a few side components. The spectral magnitudes computation of phase-modulated signals requires the use of the Bessel

The real modulation indexes at rotor faults, expressed by *Ispp*, are very low about to max. *0.15* at large rotor faults and in this case the stator current autospectrum contains only two significant sideband components *J1(Ispp)* with the same magnitudes *aPL,H = aP* equal to the half

The result is that the autospectrum of PM current looks the same as the autospectrum of AM current. But the substantial difference is in the initial phases of sidebands magnitudes at *fl -fsp* and *fl+fsp* frequencies, Table 1., 2nd row, bold, and Fig.2. Spectral sideband magnitudes

> Component frequency *fl-2fsp fl-fsp fl fl+fsp fl+2fsp* Initial phase *π-2φ π/2-ϕ 0 π/2+ϕ π+2ϕ* Component amplitude *J2(Ispp) J1(Ispp) J0(Ispp) J1(Ispp) J2(Ispp)*

> > *sp*

*sp*

)(1 *spp IJ* )(1 *spp IJ*

of the *Ispp* multiplied by phase current amplitude *Il.* (Bessel functions are not needed).

*aPL*, *aPH* increase both owing to *Ispp* and also owing to *Il*, unlike at AM, see (2).

**Table 1.** Frequencies and phases of the first 2 sidebands components of PM

*l*

**Figure 2.** Vector representation of PM for broken bars

1 *J*<sup>0</sup>

for the component at *fl+fsp* is –*ϕ* (a highlight and bold in Table 1, middle row)*.* 

Table 1 and Fig.2 show that the initial phase of Bessel function *J1* is *π/2* so the resulting vector of PM is perpendicular to the resulting vector of AM. The initial phase shift for the component at *fl+fsp* is the original phase shift +*ϕ* of the modulating signal, the initial phase

*2.1.2. Phase modulation-PM* 

functions *Ji*, *i=1, 2,..,M* (Randall, 1987).

higher than the usual modulation indexes at broken bars.

For better understanding of a real IM state, which comes in IM current at dynamic rotor faults and for the explanation of *aAPL* and *aAPH* origination and formation from AM and PM, the properties of AM and PM have to be firstly well known. So the basic properties of AM and PM will be firstly presented before the explanation of the real IM state - JAMP.

## *2.1.1. Amplitude modulation-AM*

AM is clearly visible from the time course of an IM current amplitude which is not stable, but changes according to the modulating current amplitude. It can be clearly seen as the IM current time course envelope already from about 2% deep of modulation *Ispa/Il* representing approximately 2 broken bars.

Spectrum of AM (suffix *A*) is derived from the Euler formula *cos(ωt)=½(exp(jωt)+exp(-jωt))*  expressing the decomposition of a harmonic signal into the pair of rotating vectors (phasors) -Fig.1. Phasors' amplitudes are the half of the original harmonic signal amplitude; one rotates positive direction, the other rotates negative direction.

**Figure 1.** Vector representation of AM for broken bars

Spectra of modulated signals are always connected with two sidebands peaks around the carrier signal spectral peak. AM appears in the autospectrum of the modulated signal as three peaks: one at the carrier frequency with the magnitude equal to the amplitude of carrier signal and two sidebands spaced by modulating frequency from the carrier frequency each of them with the half amplitude of the modulating signal.

For broken bars there are two sideband components at low and high frequency *fL,H* and their low and high sidebands magnitudes *aAL, aAH* have the same size *aA* which equals the half of the amplitude of the modulating current *Ispa*

$$f\_{L,H} = f\_l \pm f\_{sp} \qquad \qquad a\_A = a\_{AL} = a\_{AH} = I\_{spa} / \,\,\mathcal{D} \tag{2}$$

### *2.1.2. Phase modulation-PM*

524 Induction Motors – Modelling and Control

as the derivation of phase oscillation also appears.

independent on the number of motor poles

*2.1.1. Amplitude modulation-AM* 

approximately 2 broken bars.

subsequent or secondary modulation, so PM cannot originate without AM and JAPM always exists. The JAPM can be interpreted as a stator current modulation by a complex modulating vector which changes both its amplitude and phase. Angular speed oscillation

The frequency of periodical changes is a slip pole frequency *fsp* (suffix*sp*) which is

For better understanding of a real IM state, which comes in IM current at dynamic rotor faults and for the explanation of *aAPL* and *aAPH* origination and formation from AM and PM, the properties of AM and PM have to be firstly well known. So the basic properties of AM

AM is clearly visible from the time course of an IM current amplitude which is not stable, but changes according to the modulating current amplitude. It can be clearly seen as the IM current time course envelope already from about 2% deep of modulation *Ispa/Il* representing

Spectrum of AM (suffix *A*) is derived from the Euler formula *cos(ωt)=½(exp(jωt)+exp(-jωt))*  expressing the decomposition of a harmonic signal into the pair of rotating vectors (phasors) -Fig.1. Phasors' amplitudes are the half of the original harmonic signal amplitude; one

Spectra of modulated signals are always connected with two sidebands peaks around the carrier signal spectral peak. AM appears in the autospectrum of the modulated signal as three peaks: one at the carrier frequency with the magnitude equal to the amplitude of carrier signal and two sidebands spaced by modulating frequency from the carrier

For broken bars there are two sideband components at low and high frequency *fL,H* and their low and high sidebands magnitudes *aAL, aAH* have the same size *aA* which equals the half of

 *sp*

2/ *spa I*

2/ *spa I*

*sp*

, / 2 *L H l sp A AL AH spa f ff a a a I* (2)

rotates positive direction, the other rotates negative direction.

**Figure 1.** Vector representation of AM for broken bars

the amplitude of the modulating current *Ispa*

*l*

*l I*

frequency each of them with the half amplitude of the modulating signal.

and PM will be firstly presented before the explanation of the real IM state - JAMP.

2/ 2 2 *sp slip sync l l lr f pf psf ps f p sf f f p* (1)

Unlike AM, PM is not observable from the stator current time course at so small modulation indexes to 0.15. PM can be visible as stator current time course compression and decompression from modulation indexes greater than 2, which are approximately 10 times higher than the usual modulation indexes at broken bars.

The spectrum of phase modulation (suffix *P*) is dependent on the modulation index size. If the modulation index is greater than *1*, the PM spectrum consists of many side components, if it is lower than *0.4*, the spectrum contains only a few side components. The spectral magnitudes computation of phase-modulated signals requires the use of the Bessel functions *Ji*, *i=1, 2,..,M* (Randall, 1987).

The real modulation indexes at rotor faults, expressed by *Ispp*, are very low about to max. *0.15* at large rotor faults and in this case the stator current autospectrum contains only two significant sideband components *J1(Ispp)* with the same magnitudes *aPL,H = aP* equal to the half of the *Ispp* multiplied by phase current amplitude *Il.* (Bessel functions are not needed).

$$f\_{L,H} = f\_l \pm f\_{sp} \qquad \qquad a\_P = a\_{PL} = a\_{PH} = I\_l I\_{spp} / 2 \tag{3}$$

The result is that the autospectrum of PM current looks the same as the autospectrum of AM current. But the substantial difference is in the initial phases of sidebands magnitudes at *fl -fsp* and *fl+fsp* frequencies, Table 1., 2nd row, bold, and Fig.2. Spectral sideband magnitudes *aPL*, *aPH* increase both owing to *Ispp* and also owing to *Il*, unlike at AM, see (2).


**Table 1.** Frequencies and phases of the first 2 sidebands components of PM

**Figure 2.** Vector representation of PM for broken bars

Table 1 and Fig.2 show that the initial phase of Bessel function *J1* is *π/2* so the resulting vector of PM is perpendicular to the resulting vector of AM. The initial phase shift for the component at *fl+fsp* is the original phase shift +*ϕ* of the modulating signal, the initial phase for the component at *fl+fsp* is –*ϕ* (a highlight and bold in Table 1, middle row)*.* 

### *2.1.3. JAPM and MCS–the real motor state at broken bars*

This type of modulation expresses exactly the real motor state at broken bars. Both AM and PM have the same frequencies and their amplitudes and phase shifts between them are in certain relations depending on IM load and inertia. Providing sinusoidal currents, the IM current of healthy motor *ia=Il cos(ωlt)* changes at rotor broken bars to

$$\dot{q}\_a = \left(I\_l + I\_{spa}\cos\alpha\_{sp}t\right)\cos\{\alpha\_l t - I\_{spp}\cos\{\alpha\_{sp}t + \varphi\}\}\tag{4}$$

Rotor Cage Fault Detection in Induction Motors by Motor Current Demodulation Analysis 527

*fl+fsp aA*

+ϕ *aAPH aP*

(6)

Generally the inertia influences dynamic behavior of systems. Great inertia causes the increase of the mechanical time constant of the motor rotating system and therefore the delay *–ϕ* (*ϕ* <0) of PM behind AM on *fsp*. IM current spectrum symmetry and *aAPL, aAPH* equality disappears. The initial phase shift for the positive component at *fl + fsp* frequency is +*ϕ*, see bold in Table I, and therefore *aAPH* decreases, the initial phases shift for the negative component at *fl - fsp* is *–ϕ* and therefore the *aAPL* increases -Fig.4. The angle between AM and PM modulation vectors is not π/2, (5) is not valid and the resulting *aAPL,* 

Spectral magnitudes *aAPL,H* of MCS can be computed according to the modified cosine low.

=0 this general equation for MCS computation, equation (6) changes to equation (5).

2 2 , 2 sin( ) *APL H A P A P a a a aa*

**Figure 4.** The formation of MCS *aAPL, aAPH* at broken bars from JAPM at great inertia, ϕ= -/6

right sideband component is almost zero*.* The result is only theoretical.

*aA fl-fsp* 

*aAPL aP*


breaking and their formation to *aAPL* and *aAPH* can be explained.

**2.2. Dynamic eccentricity IM current modulation** 

In the case of the extremely large inertia the shift between AM and PM is almost *ϕ= -π/2.*  The result is that only the low spectral sideband component appears, see (6), with and the

*fl* 

The same comes in the case of overloaded and insufficiently fed IM by the voltage substantially lower than nominal AC voltage, which can be named as "abnormal working condition", when PM cannot follow AM and *ϕ<*0. In all other cases of normal IM working condition *aAPL, aAPH* should be the same size, in the range of possible Digital signal processing

Based on the JAMP the phenomena which appear in dynamic motor modes like start-up or

The rotor cage faults can also lead to the mechanical stresses, shaft torques, bearing failures and therefore air gap dynamic eccentricity (Joksimovic, 2005; Drif & Cardoso, 2008). So the dynamic eccentricity, as a mechanical fault, closely relates to the rotor cage faults. Dynamic eccentricity is the condition of the unequal air gap between the stator and rotor caused namely by loose or bent rotor, worn bearing etc. IM current analysis is the base method, but the vibration as the cause of unbalance and the torsional vibration as the cause of the torque oscillation can be also performed as supporting method, but IM current related detection

*aAPH*, are the vector sum of AM and PM rotating vectors.

For 

(DSP) errors.

which contains both AM and PM. Amplitude modulating current *iAM =Ispa cos spt*, phase modulating current *iPM = - Ispp cos (spt* +*)*. The term *Ispa/Il* represents the deep of modulation of AM and *Ispp* represents the modulation index of PM.

At low inertia and normal and stable working conditions the AM and PM currents have exactly opposite phases *ϕ0=, ϕ=0.* This results from the IM torque-speed characteristic, where torque changes induced by MMF changes induce opposite changes in speed.

The necessary condition for both MCS low *aAPL* and high *aAPH* autospectral magnitudes equality is the mutual perpendicularity of vectors forming AM and PM, Fig.1, 2 and Table 1. It occurs only in the case of exactly coincident or exactly opposite phases of AM and PM.

Since the necessary condition is fulfilled, the resulting low *aAPL* and high *aAPH* autospectral sideband magnitudes of JAPM have the same size given by (5).

$$a\_{APL} = a\_{APH} = \sqrt{a\_A^2 + a\_P^2} \tag{5}$$

Unfortunately from sideband magnitudes *aAPL, aAPH*, which are the results of the widely used MCS, the contribution of AM and PM cannot be found out. Only demodulation techniques can find them.

Increasing IM load causes the increase of IM current *Il*. Previous investigations and experiments (Jaksch & Zalud, 2010) proved also an increasing oscillation of rotor magnetic field at *fsp* with increasing load which means the increase of *Ispp.* It means consequently the increase of *aAPL, aAPH* according to (3) at the same rotor fault size (dash line in Fig.3). The result is that PM is the main reason of *aAPL, aAPH* dependence on IM load.

**Figure 3.** The formation of MCS *aAPL, aAPH* at broken bars from JAPM, *ϕ*=0, increasing motor load means the increase of PM –dash line.

Generally the inertia influences dynamic behavior of systems. Great inertia causes the increase of the mechanical time constant of the motor rotating system and therefore the delay *–ϕ* (*ϕ* <0) of PM behind AM on *fsp*. IM current spectrum symmetry and *aAPL, aAPH* equality disappears. The initial phase shift for the positive component at *fl + fsp* frequency is +*ϕ*, see bold in Table I, and therefore *aAPH* decreases, the initial phases shift for the negative component at *fl - fsp* is *–ϕ* and therefore the *aAPL* increases -Fig.4. The angle between AM and PM modulation vectors is not π/2, (5) is not valid and the resulting *aAPL, aAPH*, are the vector sum of AM and PM rotating vectors.

526 Induction Motors – Modelling and Control

modulating current *iPM = - Ispp cos (*

exactly opposite phases *ϕ0=*

can find them.

the increase of PM –dash line.

*2.1.3. JAPM and MCS–the real motor state at broken bars* 

current of healthy motor *ia=Il cos(ωlt)* changes at rotor broken bars to

*spt* +

of AM and *Ispp* represents the modulation index of PM.

sideband magnitudes of JAPM have the same size given by (5).

result is that PM is the main reason of *aAPL, aAPH* dependence on IM load.

*aA*

*aP aAPL*

which contains both AM and PM. Amplitude modulating current *iAM =Ispa cos* 

where torque changes induced by MMF changes induce opposite changes in speed.

This type of modulation expresses exactly the real motor state at broken bars. Both AM and PM have the same frequencies and their amplitudes and phase shifts between them are in certain relations depending on IM load and inertia. Providing sinusoidal currents, the IM

cos cos( cos( ))

At low inertia and normal and stable working conditions the AM and PM currents have

The necessary condition for both MCS low *aAPL* and high *aAPH* autospectral magnitudes equality is the mutual perpendicularity of vectors forming AM and PM, Fig.1, 2 and Table 1. It occurs only in the case of exactly coincident or exactly opposite phases of AM and PM.

Since the necessary condition is fulfilled, the resulting low *aAPL* and high *aAPH* autospectral

Unfortunately from sideband magnitudes *aAPL, aAPH*, which are the results of the widely used MCS, the contribution of AM and PM cannot be found out. Only demodulation techniques

Increasing IM load causes the increase of IM current *Il*. Previous investigations and experiments (Jaksch & Zalud, 2010) proved also an increasing oscillation of rotor magnetic field at *fsp* with increasing load which means the increase of *Ispp.* It means consequently the increase of *aAPL, aAPH* according to (3) at the same rotor fault size (dash line in Fig.3). The

**Figure 3.** The formation of MCS *aAPL, aAPH* at broken bars from JAPM, *ϕ*=0, increasing motor load means

*fl fl-fsp aA*

*aAPH*

*fl+fsp* 

*aP*

2 2

*a l spa sp l spp sp*

 

*, ϕ=0.* This results from the IM torque-speed characteristic,

*APL APH A P a a aa* (5)

*)*. The term *Ispa/Il* represents the deep of modulation

*spt*, phase

*i I I t tI t* (4)

Spectral magnitudes *aAPL,H* of MCS can be computed according to the modified cosine low.

$$a\_{APL,H} = \sqrt{a\_A^2 + a\_P^2 - 2a\_A a\_P \sin(\mp \phi)}\tag{6}$$

For =0 this general equation for MCS computation, equation (6) changes to equation (5).

**Figure 4.** The formation of MCS *aAPL, aAPH* at broken bars from JAPM at great inertia, ϕ= -/6

In the case of the extremely large inertia the shift between AM and PM is almost *ϕ= -π/2.*  The result is that only the low spectral sideband component appears, see (6), with and the right sideband component is almost zero*.* The result is only theoretical.

The same comes in the case of overloaded and insufficiently fed IM by the voltage substantially lower than nominal AC voltage, which can be named as "abnormal working condition", when PM cannot follow AM and *ϕ<*0. In all other cases of normal IM working condition *aAPL, aAPH* should be the same size, in the range of possible Digital signal processing (DSP) errors.

Based on the JAMP the phenomena which appear in dynamic motor modes like start-up or breaking and their formation to *aAPL* and *aAPH* can be explained.

### **2.2. Dynamic eccentricity IM current modulation**

The rotor cage faults can also lead to the mechanical stresses, shaft torques, bearing failures and therefore air gap dynamic eccentricity (Joksimovic, 2005; Drif & Cardoso, 2008). So the dynamic eccentricity, as a mechanical fault, closely relates to the rotor cage faults. Dynamic eccentricity is the condition of the unequal air gap between the stator and rotor caused namely by loose or bent rotor, worn bearing etc. IM current analysis is the base method, but the vibration as the cause of unbalance and the torsional vibration as the cause of the torque oscillation can be also performed as supporting method, but IM current related detection methods in most cases give very good results. Relative dynamic eccentricity is defined as the ratio of the difference between the rotor and the stator center to the difference between the stator and the rotor radius. The values of dynamic eccentricity are 0-1 or 0-100%.

Dynamic or combined eccentricity and subsequently the air gap alternation changes the rotor electromagnetic field ones per IM revolution, so the modulating frequency is the rotation frequency *fr* (suffix *r*). IM spectrum contains two sidebands around *fl*.

$$f\_{L,H} = f\_l \pm f\_r \tag{7}$$

Rotor Cage Fault Detection in Induction Motors by Motor Current Demodulation Analysis 529

*fr* is roughly 1Hz for the rotor fault diagnostic not influenced by the time varying load. The *fsp* computation from *fr* (1), enables the differentiation and subsequently the omission of an

Small external time varying load can be also caused by the fault of IM driven devices as a gearbox. In MCDA spectrum another peak appears and by the allocation of this peak frequency to the gearbox relevant mechanism frequency, the device fault can be identified.

Generally, the demodulation methods extract the original AM and PM signals using special computation methods. Now the demodulated signals are original modulating signals.

Demodulation methods can be used without precarious presumption whether the signal is modulated or not and in the case of no modulation, the demodulation results are zeros (PM)

For the determination of the modulated signal instantaneous amplitude and phase, a complex analytical signal has to be defined and created. An analytical complex signal created by mathematical formula is the base for the demodulation analysis. The most used methods are Hilbert transform, Hilbert-Huang transform or quadrate mixing. For 3-phase

The rotor fault amplitude demodulation extracts the original AM current. The AM current appears in stator current as an envelope of this current -Fig.5. Therefore the amplitude demodulation is known as an envelope analysis (Jaksch, 2003). It is the base for dynamic

At broken bars, phase demodulation extracts PM current *Ispp* [A] which, as an argument of harmonic function (4), really represents the phase angle ripple or phase swinging [rad]. The phase demodulation gives the time course of the instantaneous swinging angle or instantaneous angular speed and represents a huge tool for the research of the rotor magnetic field oscillation, sensor-less angular speed, speed variation or other irregularities.

The demodulation analysis should be used for band pass filtered signals with the center in a carrier frequency and span corresponding to the maximal modulating frequency. Spectrum of demodulated current outside this bandwidth is shifted by a carrier frequency towards to the low frequencies. For IM rotor faults the carrier frequency is usually a supply frequency *fl* and the maximal modulating frequency is *fr*, so the basic bandwidth *0-2fl* is suitable. In the case when analog bandwidth *0-2fl* cannot be kept it is possible to use higher bandwidths, but Shannon sampling theorem has to be strictly kept and demodulated spectrum must be

Higher order harmonics of supply current and also modulating broken bar current should appear as sideband components at frequencies *fk,l = kfl ± 2lsfl, k=3,5,7, l= 1,2,3* where *k* represents the index for stator current harmonics and *l* represents the index for broken bar

evaluated only in the range of *0-fl* because for higher frequencies is not valid.

motors rotor faults a new method based on the space transform was developed.

additional disturbing spectral peak on *fload*.

**3. Demodulation methods for rotor faults** 

or constants (AM) and by removing DC also zeros.

rotor fault diagnostics.

Spectrum of demodulated current contains peaks on direct frequencies *fr.* The air gap changes do not contain stepping changes and the change usually pass subsequently during one revolution, so modulating current is often almost harmonic and contains only small higher harmonics, unlike modulating current for broken bars.

Providing sinusoidal currents, the motor current of healthy motor *ia=Il cos(ωlt)* changes at rotor dynamic eccentricity to

$$\dot{a}\_a = \left(I\_l + I\_{ra}\cos\alpha\_r t\right)\cos(\alpha\_l t - I\_{rp}\cos(\alpha\_r t))\tag{8}$$

which contains both AM and PM.

### **2.3. The influence of the time varying load**

Until now the modulations caused by the internal IM rotor faults were solved providing IM different, but constant load when MCDA spectrum contains two significant peaks on *fsp* and *f*r , see Fig.6. In the case of the external periodical harmonic time varying load, which varies with the frequency *fload* < *fl*, both additional AM and PM of IM current arise on the *fload* frequency. It can come e.g. in the case when IM drives machines with various machine cycles e.g. textile machines, machine tools etc. (In the case of the speed reducing devices as gearbox transmissions, the gear-ratio has to be counted for *fload* determination).

In the IM full current spectrum - MCS two additional spectral sidebands peaks appear on frequencies *fl ± fload*. In MCDA spectrum the time varying load appears as a one spectral peak on *fload* with the magnitude proportional to a load torque difference which can be expressed as an *Iload*. So together 3 significant spectral peaks on *fsp*, *f*r and *fload* appear in MCDA spectrum of IM with rotor faults. The AM can be also observed in the time course of IM current as the stator current envelope or in the time course of amplitude demodulated current.

If *fload* equals exactly *fsp* or *fr* the rotor faults diagnostics is not correct because the resulting spectral magnitudes are the vector sum of the corresponding modulating amplitudes. The situation when *fload* =*fr* can come when IM directly drives a machine with uneven load during one revolution e.g. a cam mounted on the main shaft. But practically it is a minimal probability that *fload* equals *fsp* or *fr*. The minimal difference between *fload* and *fsp* or *fr* is at Hanning window approximately 4 discrete step *f*. Usually used acquisition time is *T= 4s, f=1/T= 0.25* Hz*,* so a minimal difference between external *fload* and rotor faults frequencies *fsp,*

*fr* is roughly 1Hz for the rotor fault diagnostic not influenced by the time varying load. The *fsp* computation from *fr* (1), enables the differentiation and subsequently the omission of an additional disturbing spectral peak on *fload*.

Small external time varying load can be also caused by the fault of IM driven devices as a gearbox. In MCDA spectrum another peak appears and by the allocation of this peak frequency to the gearbox relevant mechanism frequency, the device fault can be identified.
