**3. Demodulation methods for rotor faults**

528 Induction Motors – Modelling and Control

rotor dynamic eccentricity to

which contains both AM and PM.

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

Hanning window approximately 4 discrete step

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

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

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

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

*i I I t tI t a l ra r l rp r* cos cos( cos( )) 

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

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

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

*f=1/T= 0.25* Hz*,* so a minimal difference between external *fload* and rotor faults frequencies *fsp,*

gearbox transmissions, the gear-ratio has to be counted for *fload* determination).

stator current envelope or in the time course of amplitude demodulated current.

*LH l r* , *f ff* (7)

 

(8)

*f*. Usually used acquisition time is *T= 4s,* 

stator and the rotor radius. The values of dynamic eccentricity are 0-1 or 0-100%.

rotation frequency *fr* (suffix *r*). IM spectrum contains two sidebands around *fl*.

higher harmonics, unlike modulating current for broken bars.

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

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

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 rotor fault diagnostics.

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

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 sideband current harmonics. Because of the interaction of time harmonics with a space harmonics, a saturation related permeance harmonics together with phase shifts of AM and PM means that some sidebands harmonics are suppressed and only certain ones can appear, so the above introduced formula is not generally valid.

Demodulation in the region of higher *k*-harmonics of the supply frequency requires the shift of the supply carrier frequency *kfl* to zero before the demodulation. It means the spectral frequency resolution *f* increasing which is often called Band Selectable Fourier Analysis (BSFA) or Zoom. Dynamic signal analyzers are equipped with this function (zoom mode) and a maximal *f* = 1mHz. However, this analysis has a little practical sense, because of above mentioned problems with higher order harmonics. In addition the modulating currents there usually have smaller amplitudes.
