**3.1. The demodulation using Hilbert transformation**

The Hilbert transform (Bendat, 1989) is a well-known tool which enables to create an artificial complex signal *H(t*)= *x(t)+jy(t)*, called analytical signal, from a real input signal *x(t).*  The real part *x(t)* of the analytical signal *H(t)* is the original signal – stator current, the imaginary part *jy(t)* represents the Hilbert transform of a real part *x(t).* The absolute value *magH(t*) representing amplitude demodulation and the phase *(t*) representing phase demodulation can be computed according to (9), (10).

$$
gamma H(t) = \sqrt{\mathbf{x}^2(t) + \mathbf{y}^2(t)}\tag{9}
$$

Rotor Cage Fault Detection in Induction Motors by Motor Current Demodulation Analysis 531

(11)

( ) (2 / 3) ( ) (1 / 3) ( ) (1 / 3) ( )

Space transform was firstly also used for the demodulation (Jaksch, 2003). From the viewpoint of the means necessary for the demodulation process, space vector *P(t)=id(t)+jiq(t)*  represents a complex analytical signal computed from three three currents *ia, ib, ic* similarly like the Hilbert transform creates the artificial complex signal *H(t)* from one phase current. The absolute value *magP(t)=sqrt(id2(t)+iq2(t))* forms the amplitude demodulation,

The space transform requires 3 currents measurement, but only simple computation (11) and no other transformation for the complex analytical signal determination. On the contrary Hilbert transform needs only one current measurement, but *jy(t)* computation.

The space transform creates the analytical signal from 3 currents. In order to obtain the same

Power feeding voltage unbalance or great stator fault, which can cause greater IM

The maximum error should not be greater than in the range of several percent. The experiment showed - see Table II., Table III, *IspaH*, *IspaP* that the differences are up to 5 %.

Amplitude demodulation can be implemented also by the other techniques resulting also from the three phase IM feeding system as an apparent power magnitude or a squared stator current space vector magnitude. However these methods are more complicated than the space transformation method and in addition the results of these methods are in units

Various simulations have been performed. The main aim of the simulation was the verification of (5), (6) for the IM current MCS - *aAPL, aAPH* computation, namely the influence of angle *ϕ* on the sizes of *aAPL, aAPH*. The verification of the equality of MCDA fault indicators

As it was previously derived in the section 2, the IM current of healthy motor *ia=Il cos(ωlt)*

cos cos cos( cos( ) cos( ))

*a l spa sp ra r l spp sp rp r i I I tI t tI t I t* (12)

   

 

and dimensions which are not comparable with the Hilbert transform results.

(lower window in Fig. 5) with input data values *Ispa, Ira* also has been performed.

changes at dynamic rotor faults - rotor broken bars and dynamic eccentricity to

 

*/3* between IM phase currents (space

Small differences between Hilbert and space transforms can occur in the following cases:

sizes of fault indicators as from Hilbert transform, (11), using *Ks*=2/3 must be kept.

*it it it it*

( ) ( 3 / 3) ( ) ( 3 / 3) ( ) *d ab c*

*qbc*

*(t) = arctan (iq(t)/ id (t))* forms the phase demodulation.

The violation of the exact phase shift *a=ej2*

transform supposes exact shift 2π/3).

currents unbalance.

**4. Simulation results** 

**3.3. The comparison of both demodulation methods** 

*it it it* 

$$\beta(t) = \arctan\left(y(t) / \lfloor x(t) \rfloor\right) \text{ } \newline \ln < -\pi \text{ } \newline \text{} \newline \tag{10}$$

The amplitude demodulation computation is quite easy and can be used for continual monitoring and diagnostics in real time.

Phase determination *(t)* [rad] from a complex number position holds only in the range *<-, >* and if the phase overlaps these limits, it is necessary to unwrap it and moreover to remove the phase trend component which increases by*2π* every revolution with the increase of the common phase carrier signal (Randall, 1987). The computation of phase demodulation is a little difficult comparing to amplitude demodulation, but generally there is usually no problem with the phase demodulation computation.

### **3.2. The demodulation using space transformation**

The space (Park) transform, based on the physical motor model, is used primarily for the motor vector control. A three phase *ia,ib,ic*, *ia(t)=Il cos(lt)* system is expressed in one space current vector *i= Ks (ia +a ib +a2 ic), a= e j2π/3* projected to the complex *d-q* plane (11).

From the 3 possible choices of *Ks: Ks =1* - amplitude invariance, *Ks =sqrt(2/3)*- power invariance, *Ks*=2/3 is used. In this case the additional recomputation coefficient between phase currents and transformed currents does not have to be used.

Rotor Cage Fault Detection in Induction Motors by Motor Current Demodulation Analysis 531

$$\begin{aligned} \dot{i}\_d(t) &= (2 \ / \ \Im) \dot{i}\_a(t) - (1 \ / \Im) \dot{i}\_b(t) - (1 \ / \Im) \dot{i}\_c(t) \\ \dot{i}\_q(t) &= (\sqrt{3} \ / \Im) \dot{i}\_b(t) - (\sqrt{3} \ / \Im) \dot{i}\_c(t) \end{aligned} \tag{11}$$

Space transform was firstly also used for the demodulation (Jaksch, 2003). From the viewpoint of the means necessary for the demodulation process, space vector *P(t)=id(t)+jiq(t)*  represents a complex analytical signal computed from three three currents *ia, ib, ic* similarly like the Hilbert transform creates the artificial complex signal *H(t)* from one phase current.

The absolute value *magP(t)=sqrt(id2(t)+iq2(t))* forms the amplitude demodulation, *(t) = arctan (iq(t)/ id (t))* forms the phase demodulation.

### **3.3. The comparison of both demodulation methods**

The space transform requires 3 currents measurement, but only simple computation (11) and no other transformation for the complex analytical signal determination. On the contrary Hilbert transform needs only one current measurement, but *jy(t)* computation.

The space transform creates the analytical signal from 3 currents. In order to obtain the same sizes of fault indicators as from Hilbert transform, (11), using *Ks*=2/3 must be kept.

Small differences between Hilbert and space transforms can occur in the following cases:


The maximum error should not be greater than in the range of several percent. The experiment showed - see Table II., Table III, *IspaH*, *IspaP* that the differences are up to 5 %.

Amplitude demodulation can be implemented also by the other techniques resulting also from the three phase IM feeding system as an apparent power magnitude or a squared stator current space vector magnitude. However these methods are more complicated than the space transformation method and in addition the results of these methods are in units and dimensions which are not comparable with the Hilbert transform results.

### **4. Simulation results**

530 Induction Motors – Modelling and Control

frequency resolution

and a maximal

so the above introduced formula is not generally valid.

currents there usually have smaller amplitudes.

**3.1. The demodulation using Hilbert transformation** 

*magH(t*) representing amplitude demodulation and the phase

demodulation can be computed according to (9), (10).

monitoring and diagnostics in real time.

problem with the phase demodulation computation.

**3.2. The demodulation using space transformation** 

motor vector control. A three phase *ia,ib,ic*, *ia(t)=Il cos(*

Phase determination

*<-,*  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,

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

(BSFA) or Zoom. Dynamic signal analyzers are equipped with this function (zoom mode)

above mentioned problems with higher order harmonics. In addition the modulating

The Hilbert transform (Bendat, 1989) is a well-known tool which enables to create an artificial complex signal *H(t*)= *x(t)+jy(t)*, called analytical signal, from a real input signal *x(t).*  The real part *x(t)* of the analytical signal *H(t)* is the original signal – stator current, the imaginary part *jy(t)* represents the Hilbert transform of a real part *x(t).* The absolute value

( ) arctan( ( ) / ( )) , *t y t x t in*

The amplitude demodulation computation is quite easy and can be used for continual

The space (Park) transform, based on the physical motor model, is used primarily for the

From the 3 possible choices of *Ks: Ks =1* - amplitude invariance, *Ks =sqrt(2/3)*- power invariance, *Ks*=2/3 is used. In this case the additional recomputation coefficient between

current vector *i= Ks (ia +a ib +a2 ic), a= e j2π/3* projected to the complex *d-q* plane (11).

phase currents and transformed currents does not have to be used.

*>* and if the phase overlaps these limits, it is necessary to unwrap it and moreover to remove the phase trend component which increases by*2π* every revolution with the increase of the common phase carrier signal (Randall, 1987). The computation of phase demodulation is a little difficult comparing to amplitude demodulation, but generally there is usually no

*f* increasing which is often called Band Selectable Fourier Analysis

2 2 *magH t x t y t* () () () (9)

 

*(t)* [rad] from a complex number position holds only in the range

*(t*) representing phase

(10)

*lt)* system is expressed in one space

*f* = 1mHz. However, this analysis has a little practical sense, because of

Various simulations have been performed. The main aim of the simulation was the verification of (5), (6) for the IM current MCS - *aAPL, aAPH* computation, namely the influence of angle *ϕ* on the sizes of *aAPL, aAPH*. The verification of the equality of MCDA fault indicators (lower window in Fig. 5) with input data values *Ispa, Ira* also has been performed.

As it was previously derived in the section 2, the IM current of healthy motor *ia=Il cos(ωlt)* changes at dynamic rotor faults - rotor broken bars and dynamic eccentricity to

$$i\_a = \left(I\_l + I\_{spa}\cos\alpha\_{sp}t + I\_{ru}\cos\alpha\_r t\right)\cos\left(\alpha\_l t - I\_{spp}\cos\left(\alpha\_{sp}t + \varphi\right) - I\_{rp}\cos\left(\alpha\_r t\right)\right) \tag{12}$$

Input data for the simulation result from (12). Simulation values of *Il*, *Ispa*, *Ispp*, *Ira*, *Irp*, *ϕ* start from the measurement, but various values can also be used. Other data processing is the same as in experiments. Hilbert transform, (9), (10) was used for the IM current amplitude and phase demodulation. The values for *aAPL, aAPH* (2nd window in Fig.5) were compared to the values computed from (3),(6). Full identity with the theory was found.

Rotor Cage Fault Detection in Induction Motors by Motor Current Demodulation Analysis 533

The first IM was SIEMENS type 1LA7083-2AA10, 1.1 kW, two-pole, rated revolutions 2850 min-1, *Inom*=2.4A, *nrb* =23, air gap dimension=0.25 mm, health motor, 1 interrupted rotor bar and 2 contiguous interrupted rotor bars, both 3 rotors were balanced with factory set-up dynamic eccentricity (setting the exact value of dynamic eccentricity is at so small air gap very difficult). The 2nd IM was SIEMENS type 1LA 7083-4AA10, 0.75 kW, four-pole motor,

Various motors and fault rotors used in experiments were manufactured directly at Siemens Electromotor. Motors were tested at 25%, 50%, 75% and 85% of the full load according to the motor load record from Siemens. The changes in the broken bar fault indicator *Ispa* was also

The experiments were based on Bruel&Kjaer PULSE 20 bits dynamic signal analyzer (DSA) based on the frequency filtration and decimation principle. All channels are sampled simultaneously. FFT analyzer was set on the base band mode, frequency span 100Hz, 400 frequency lines, *Δf* =0.25Hz, Hanning window, continuous RMS exponential averaging with 75% overlapping. For the experiment of *Ispa* changes at very low load the measuring time

To find out the possible differences in both introduced demodulation methods, both Hilbert

The experiments results were verified by 16 channels PC measurement system based on two 8 channels, 24-bit DSA NI 4472B from National Instruments setting in the lowest possible

To obtain the maximal measurement accuracy the possible errors in Digital Signal Processing (DSP) should be avoided. Sampling theorem with full agreement between the sampling frequency and surveyed analog frequency band should be strictly kept. At the violation of sampling theorem, signal frequencies higher than Nyquist frequency *fN* are tilted - masked to the basic frequency region from *0-fN* and they can create there aliasing frequencies or interfere with regular frequencies, changing their amplitudes. Masking can come through many higher bands of the sampling frequency. Unlike dynamic signal analyzers, simple PC cards and scopes are usually not equipped with anti aliasing filters.

The measurement acquisition time *T* should be optimally set. Spectral frequency resolution

A great DSP error, both in frequency and magnitude, occurs if the analog frequency of the

low analog sideband frequency *fl-fsp* is nearer to the discrete spectral frequency than the high sideband frequency *fl+fsp*, the *aAPL* can be higher than *aAPH* and vice versa. The optimal

*/2)=2/* *f =1/T*.

*f*. In the case of a rectangular window, the

*=0.636=-3.92 dB*, representing 36% fault in

*))=0.848=-1.43 dB.* If the

*f* = 0.25Hz. In the case of very

*f* = 0.125Hz can be used for the accurate

 *-1/(3*

*Inom* = 1.8A, *nrb* =26, balanced rotor and rotor with 2 contiguous interrupted rotor bars.

tested in the low load range from no load to 20% of full load.

and space transform were simultaneously evaluated in the real time.

*f* is a reciprocal value of the acquisition window *T,* 

magnitude! In the case of Hanning window the decline is *(3/*

acquisition time should be longer than 1sec. e.g. 4sec. with

*fsp* detection and for the decrease of DSP errors probability.

low load, the minimal acquisition time 8 sec with

examined signal is exactly in the centre of

spectral magnitude decline is *sinc*(

T= 32s, *Δf* =0.03125Hz was used.

frequency range 1kHz.

The simulation was also used for the case where angle *ϕ* is positive and *aAPL< aAPH*. But this IM state is not stable and can come only in IM dynamic regime.

Simulation results are depicted in Fig.5. Note that the time course of amplitude demodulated current follows the envelope of IM current – compare the window 3 to the window 1. Time course of amplitude and phase demodulation (windows 3, 4) shows small ripple at the beginning - t=0, given by nonsequenced modulating current in time window.

**Figure 5.** The demodulation analysis of stator current, 4-poles motor 0.75 kW, 2 broken bars, great inertia - *ϕ* <0, low dynamic eccentricity. Simulation results.
