A new and best approach for early detection of rotor and stator faults in induction motors coupled to variable loads

Abstract

Today, induction machines are playing, thanks to their robustness, an important role in world industries. Although they are quite reliable, they have become the target of various types of defects. Thus, for a long time, many research laboratories have been focusing their works on the theme of diagnosis in order to find the most efficient technique to predict a fault in an early stage and to avoid an unplanned stopping in the chain of production and costs ensuing. In this paper, an approach called Park’s vector product approach (PVPA) was proposed which was endowed with a dominant sensitivity in the case in which there would be rotor or stator faults. To show its high sensitivity, it was compared with the classical methods such as motor current signature analysis (MCSA) and techniques studied in recent publications such as motor square current signature analysis (MSCSA), Park’s vector square modulus (PVSM) and Park-Hilbert (P-H) (PVSM_P-H). The proposed technique was based on three main steps. First, the three-phase currents of the induction motor led to a Park’s vector. Secondly, the proposed PVPA was calculated to show the distinguishing spectral signatures of each default and specific frequencies. Finally, simulation and experimental results were presented to confirm the theoretical assumptions.

Keywords

induction motor / incipient broken bar / extended Park’s vector approach / spectral analysis / inter-turn short-circuit / Hilbert transform

Introduction

Induction machines are the backbones of many industrial processes due to their robustness and reliability. Online fault diagnostics of these machines are very important to ensure safe operation, timely maintenance, increased operation reliability, and preventive rescue especially in high power applications [ 1] mainly when it is installed in sites of wind turbines as a generator. A winding-function-based method for modeling cage rotor induction motor with inter-turn short-circuit in the stator winding is developed. Using this model, all harmonics of the magneto motive force (MMF) are taken into account [ 2]. To get closer to reality, the fact that MMF is not sinusoidal is taken into consideration in order to obtain good simulation results for the approaches studied in this paper. In addition, the diversity of defects (stator and rotor defects) of the induction machine cannot be obtained by the same mathematical model. Indeed, every failure should be modeled separately by its own model [ 3]. Numerous methods have been proposed for machine line currents. One of the most used and studied approaches which is based on sideband detection around the supply frequency is known as motor current signature analysis (MCSA) [ 4]. In recent years, there have been more advanced signal processing methods based on Park-Hilbert transform PVSM_P-H [ 5− 7]. In these years, other studies have used the line current to obtain “motor square current signature analysis” (MSCSA) [ 4]. In addition, “motor current Park’s vector square modulus” (PVSM) [ 7− 9] have also been proposed.

In this paper, a new method called “Park’s vector product approach (PVPA)” is proposed. This technique is based on the spectral analysis, of the motor current Park’s vector product. The proposed approach is compared with the three techniques studied recently [ 4, 7− 9]. In the event, there would be a diversity of defects (stator or rotor defects), for different operating states of our motor.

To discover the efficiency and the sensitivity of the proposed method inspired by Ref. [ 4] MSCSA, in this paper, the PVPA which is richer in information and more sensitive to the different defects is used. In this context, the advantages and strong points of the new diagnostic method are revealed compared to the three latest techniques recently published.

System equations

Modeling induction motor in case of stator defects

Consider a squirrel cage induction machine having three identical and symmetric phases in the stator. The rotor cage having (N_r=n) bars is viewed as n identical spaced loops and the currents distribution can be specified in terms of n+1 independent rotor currents. These currents are formed of n rotor loop currents [i_nr] plus a circulating current in one of the end rings i_e. The mesh model is based on a coupled magnetic circuits approach and the stator voltage equations in vector matrix form can be written as [ 10, 11]

(1) [{\mathit{V}}_{\mathrm{3}s}]=[{\mathit{R}}_s][{\mathit{i}}_{\mathrm{3}s}]+{\displaystyle \frac{d}{dt}}[{\mathit{\psi }}_{\mathrm{3}s}],

(2) [{\mathit{\psi }}_{\mathrm{3}s}]=[{\mathit{L}}_s][{\mathit{i}}_{\mathrm{3}s}]+[{\mathit{M}}_{\mathrm{sr}}][{\mathit{i}}_{nr}],

where [R_s], [L_s] and [M_sr] depend on the number of turns per phase N_s (see Eqs. (5)–(10)), with [V_3s]=[V_s1V_s2V_s3]^T being the stator voltage matrix, [i_3s]=[i_s1i_s2i_s3]^T the stator current matrix, [Ψ_3s]=[Ψ_s1Ψ_s2Ψ_s3]^T the stator flux matrix,

(3) \left[\begin{array}{l}[{\mathit{V}}_r]\hfill \\ V_e\hfill \end{array}\right]=[\mathrm{0}]=\left[\begin{array}{l}[{\mathit{R}}_r]\cdots R_e/n\hfill \\ R_e/n\cdots R_e\hfill \end{array}\right].\left[\begin{array}{l}[{\mathit{i}}_{nr}]\hfill \\ i_e\hfill \end{array}\right]+{\displaystyle \frac{d}{dt}}\left[\begin{array}{l}[{\mathit{\psi }}_{rn}]\hfill \\ {\psi }_e\hfill \end{array}\right],

and

(4) \left[\begin{array}{l}[{\mathit{\psi }}_{rn}]\hfill \\ {\psi }_e\hfill \end{array}\right]=\left[\begin{array}{l}[{\mathit{M}}_{\mathrm{rs}}][{\mathit{i}}_{\mathrm{3}s}]\hfill \\ \mathrm{0}\hfill \end{array}\right]+\left[\begin{array}{l}[{\mathit{L}}_r]\cdots L_e/n\hfill \\ L_e/n\cdots L_e\hfill \end{array}\right].\left[\begin{array}{l}[{\mathit{i}}_{rn}]\hfill \\ i_e\hfill \end{array}\right].

with

$$\begin{gathered} \left[{\mathit{i}}_{\mathrm{r}n}\right]={\left[\begin{array}{ccccc}\hfill i_{r1}\hfill & \hfill i_{r2}\hfill & \hfill i_{r3}\hfill & \hfill ...\hfill & \hfill i_{\mathrm{r}n}\hfill \end{array}\right]}^{\mathrm{T}} \mathrm{and} \\ \left[{\mathit{\psi }}_{\mathrm{r}n}\right]={\left[\begin{array}{ccccc}\hfill {\psi }_{r1}\hfill & \hfill {\psi }_{r2}\hfill & \hfill {\psi }_{r3}\hfill & \hfill ...\hfill & \hfill {\psi }_{\mathrm{r}n}\hfill \end{array}\right]}^{\mathrm{T}}. \end{gathered}$$

where n is the number of rotor loop, N_r is the number of rotor bars, [R_r] is the rotor resistance matrix, R_e is the end ring segment resistance, [L_r] is the rotor leakage inductance matrix, and L_e is the rotor end ring leakage inductance (see Table A1 in Appendix).

Either N_s is the number of coils in healthy regime of the induction machine. A short-circuit stator leads to a decrease in the number of coils of each stator phase. The coefficient of short-circuit on the 1st, 2nd, and 3rd phase of the stator is definited as [ 3]

(5) k_{s1}={\displaystyle \frac{N_{cc1}}{N_s}},k_{s2}={\displaystyle \frac{N_{cc2}}{N_s}} \mathrm{and} k_{s3}={\displaystyle \frac{N_{cc3}}{N_S}}.

The number of turns in short-circuit is N_cc.

The number of useful coils for the three stator phases is then given by [ 3]

(6) \begin{array}{l}{N}_{\mathrm{1}}=N_{\mathrm{s}}-N_{\mathrm{c}\mathrm{c}\mathrm{1}}=(\mathrm{1}-k_{\mathrm{s}\mathrm{1}})N_{\mathrm{s}}=f_{\mathrm{s}\mathrm{1}}{N}_{\mathrm{s}},\hfill \\ N_{\mathrm{2}}=N_{\mathrm{s}}-N_{\mathrm{c}\mathrm{c}\mathrm{2}}=(\mathrm{1}-k_{\mathrm{s}\mathrm{2}})N_{\mathrm{s}}=f_{\mathrm{s}\mathrm{2}}{N}_{\mathrm{s}},\hfill \\ N_{\mathrm{3}}=N_{\mathrm{s}}-N_{\mathrm{c}\mathrm{c}\mathrm{3}}=(\mathrm{1}-k_{\mathrm{s}\mathrm{3}})N_{\mathrm{s}}=f_{\mathrm{s}\mathrm{3}}{N}_{\mathrm{s}}\hfill \end{array}

The matrixes [R_s], [L_sf], [M_ss], [M_sr] et [M_rs] depend on three coefficients f_sa, f_sb, f_sc.

The inductors are given by [ 3]

(7) [{\mathit{L}}_{\mathrm{s}}]=[{\mathit{M}}_{\mathrm{s}\mathrm{s}}]+[{\mathit{L}}_{\mathrm{s}\mathrm{f}}],

(8) [{\mathit{L}}_{\mathrm{sf}}]=\left[\begin{array}{lll}{f}_{s\mathrm{1}}^{\mathrm{2}}{L}_{\mathrm{sf}}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& f_{s2}^2{L}_{\mathrm{sf}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& f_{s3}^2{L}_{\mathrm{sf}}\end{array}\right],

(9) [{\mathit{M}}_{\mathrm{ss}}]=M_s\left[\begin{array}{lll}{f}_{s1}^{\mathrm{2}}& -f_{s1}{f}_{s2}/\mathrm{2}& -f_{s1}{f}_{s3}/\mathrm{2}\\ -f_{s1}{f}_{s2}/\mathrm{2}& f_{s2}^{\mathrm{2}}& -f_{s3}{f}_{s2}/\mathrm{2}\\ -f_{s1}{f}_{s3}/\mathrm{2}& -f_{s3}{f}_{s2}/\mathrm{2}& f_{s3}^2\end{array}\right].

The matrix of stator resistances [R_s]is given by [ 3]

(10) [{\mathit{R}}_s]=R_s\left[\begin{array}{lll}{f}_{s1}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& f_{s2}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& f_{s3}\end{array}\right],

where [M_sr] is the mutual matrix inductances between the stator and the rotor, and [R_s] and [R_r] are the stator and rotor resistance matrixes, respectively.

The space distribution of the mutual inductance is not sinusoidal. This implies that the mutual inductances matrix presents harmonics with respect to the electrical angle θ. Consequently, this matrix can be resolved into its Fourier series [ 10]

(11) \begin{array}{c}\hfill \left[{\mathit{M}}_{\mathrm{sr}}\right]={\displaystyle \sum _{h=\mathrm{1}}^{\infty }{M}_{\mathrm{sr}h}}\times \left[\begin{array}{ccc}\hfill \mathrm{...}\hfill & \hfill f_{\mathrm{s1}}\cos \left(h\left(\theta +{\phi }_h+ka\right)\right)\hfill & \hfill \mathrm{...}\hfill \\ \hfill \mathrm{...}\hfill & \hfill f_{\mathrm{s2}}\cos \left(h\left(\theta +{\phi }_h+ka\right)-{\displaystyle \frac{\mathrm{2}{\xi }_h\mathrm{\pi }}{\mathrm{3}}}\right)\hfill & \hfill \mathrm{...}\hfill \\ \hfill \mathrm{...}\hfill & \hfill f_{\mathrm{s3}}\cos \left(h\left(\theta +{\phi }_h+ka\right)+{\displaystyle \frac{\mathrm{2}{\xi }_h\mathrm{\pi }}{\mathrm{3}}}\right)\hfill & \hfill \mathrm{...}\hfill \end{array}\right]\hfill \\ \hfill \left[{\mathit{M}}_{\mathrm{sr}}\right]={\left[{\mathit{M}}_{\mathrm{rs}}\right]}^{\mathrm{T}},\hfill \end{array}

where φ_h is the initial phase angle, k = 0, 1, …, N_r−1, h=1, h=(6v±1)_v=1,2,3,… and

(12) {\xi }_h=\{\begin{array}{l}+\mathrm{1}\hspace{1em}h\in F,\\ -\mathrm{1}\hspace{1em}h\in B.\end{array}

F= {1,7,13,19,…} is the set of forward harmonic components, B={5,11,17,…} is the set of backward harmonic components, a = 2πp/N_r is the electrical angle of a rotor loop [ 10].

Incipient rotor faults simulation

The partial or the total breakage fault in the bar is modeled by the assumed linear change of the value of the resistance R_bf, named fault bar resistance. The variation of the resistance R_bf as a function of time is modeled by Eq. (13) and represented by two cases of the resistance time ratio named [ 12].

(13) R_{\mathrm{b}\mathrm{f}\mathrm{k}}=R_{\mathrm{b}}[\mathrm{1}+\alpha \pm (t-t_{\mathrm{0}})],

where α=tanγ which is the angle resistance rate.

The first case assumes that the broken bar is totally broken at t₀=1s with α=∞.

The second case characterizes the slowly and partially broken bar situation with α=5.

The fault is considered in the kth bar and the fault matrix [R_F](Eq. (14)) is added to the rotor sub-matrix [ 12]

(14) $$\begin{gathered} [{\mathit{R}}_F]=\left[\begin{array}{llllll}\mathrm{0}& \mathrm{...}& \mathrm{0}& \mathrm{...}& \mathrm{...}& \mathrm{0}\\ .& \mathrm{...}& .& .& \mathrm{...}& .\\ .& \mathrm{...}& \mathrm{0} \\ .& \mathrm{0} \\ .& \mathrm{...}& \mathrm{0} \\ .\\ \mathrm{0}& \mathrm{0}& R_{{\mathrm{bF}}_k}& -R_{{\mathrm{bF}}_k}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& -R_{{\mathrm{bF}}_k}& R_{{\mathrm{bF}}_k}& \mathrm{0}& \mathrm{0}\\ . \\ .& .& \mathrm{0}& \mathrm{0}& \\ & .\\ .& .& .& .& .& .\end{array}\mathrm{..}\right]. \end{gathered}$$

Proposed approach diagnosis

In a balanced three-phase induction motor, the sum of stator currents is zero. Therefore, only two currents are sufficient for processing and the third one can be obtained from the other two phases [ 6, 13]. A suitable representation is the use of Park’s vector as a function of main phase variables ( i_{\mathrm{1}},i_{\mathrm{2}},i_{\mathrm{3}} ) given. The current Park’s vector components ( i_d , i_q ) are expressed as [ 6, 11]

(15) \{\begin{array}{l}{i}_d={\displaystyle \frac{\mathrm{2}}{\sqrt{\mathrm{6}}}}{i}_{\mathrm{1}}-{\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{6}}}}{i}_{\mathrm{2}}-{\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{6}}}}{i}_{\mathrm{3}},\hfill \\ i_q={\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{2}}}}{i}_{\mathrm{2}}-{\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{2}}}}{i}_{\mathrm{3}}.\hfill \end{array}

Considering a healthy induction motor and a sinusoidal balanced supply voltage [ 4], stator currents can be expressed as [ 6, 11]

(16) \{\begin{array}{l}{i}_{\mathrm{1}}=i_m\cos (\omega t),\hfill \\ i_{\mathrm{2}}=i_m\cos \left(\omega t-{\displaystyle \frac{\mathrm{2}\pi }{\mathrm{3}}}\right),\hfill \\ i_{\mathrm{3}}=i_m\cos \left(\omega t-{\displaystyle \frac{\mathrm{4}\pi }{\mathrm{3}}}\right).\hfill \end{array}

In an ideal condition, only when fundamental harmonics exist (with no unwanted harmonics such as odd harmonics in stator current due to asymmetry in motor structure and/or power supply), do i_d and i_q in Eq. (15) have physical meanings, and are simplified in a steady-state as [ 6]

(17) \{\begin{array}{l}{i}_q={\displaystyle \frac{\sqrt{\mathrm{6}}}{\mathrm{2}}}{i}_{\mathrm{m}}\cos (\omega t),\hfill \\ i_d={\displaystyle \frac{\sqrt{\mathrm{6}}}{\mathrm{2}}}{i}_{\mathrm{m}}\sin (\omega t),\hfill \end{array}

where i_m is maximum current Park’s vector value (amperes) in the supply phase, ω is the angular supply frequency (radians per second, f_s= 50 Hz) and, t is the time (in seconds). Given these conditions, the current’s Park’s vector modulus is constant [ 6].

In the case of incipient rotor faults, considering primary and secondary harmonics only [ 6]

(18) $$\begin{gathered} \{\begin{array}{l}{i}_q={\displaystyle \frac{\sqrt{\mathrm{6}}}{\mathrm{2}}}{i}_m\cos (\omega t)+{\displaystyle \frac{\sqrt{\mathrm{6}}}{\mathrm{2}}}{i}_{h\mathrm{1}}\cos ((\mathrm{1}-\mathrm{2}s)\omega t_{\mathrm{1}}) \\ +{\displaystyle \frac{\sqrt{\mathrm{6}}}{\mathrm{2}}}{i}_{h\mathrm{2}}\cos ((\mathrm{1}+\mathrm{2}s)\omega t),\hfill \\ i_d={\displaystyle \frac{\sqrt{\mathrm{6}}}{\mathrm{2}}}{i}_{\mathrm{m}}\sin (\omega t)+{\displaystyle \frac{\sqrt{\mathrm{6}}}{\mathrm{2}}}{i}_{h\mathrm{1}}\sin ((\mathrm{1}-\mathrm{2}s)\omega t_{\mathrm{1}}) \\ +{\displaystyle \frac{\sqrt{\mathrm{6}}}{\mathrm{2}}}{i}_{h\mathrm{2}}\sin ((\mathrm{1}+\mathrm{2}s)\omega t_{\mathrm{2}}),\hfill \end{array} \end{gathered}$$

where i_h1 is the maximum value of the current Park’s vector lower sideband component (amperes), i_h2 is the maximum value of the current Park’s vector upper sideband component (amperes), and s is the slip [ 4].

The proposed approach in this paper is mainly based on the product of i_d with i_q current Park (see Fig.A1 in Appendix). After mathematical calculations, the final expression of the approach is obtained.

(19) $$\begin{gathered} i_q(t)\cdot i_d(t)=\left({\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}{i}_m^{\mathrm{2}}+{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{i}_{h\mathrm{1}}{i}_{h\mathrm{2}}\right)\sin (\mathrm{2}\omega t) \\ +{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{i}_{\mathrm{m}}{i}_{h\mathrm{1}}\sin ((\mathrm{1}-s)\mathrm{2}\omega t)+{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{i}_{\mathrm{m}}{i}_{h\mathrm{2}}\sin ((\mathrm{1}+s)\mathrm{2}\omega t) \\ +{\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}{i}_{h\mathrm{1}}^{\mathrm{2}}\sin ((\mathrm{1}-\mathrm{2}s)\mathrm{2}\omega t)+{\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}{i}_{h\mathrm{2}}^{\mathrm{2}}\sin ((\mathrm{1}+\mathrm{2}s)\mathrm{2}\omega t). \end{gathered}$$

Equation (19) shows the 2f_s frequency component, and sideband components at frequencies (1–s)2f_s and (1+s)2f_s. The spectrum of this current Park’s vector product, presented in Fig. 1, contains the same additional components at frequencies 2f_s, (1–s)2f_s and (1+s)2f_s which allows the cracked bars to be easily revealed.

In the case of inter-turn shorts-circuit in the phase windings of induction motors, this fault will create a flux of short-circuit current, which produces a negative MMF [ 14, 15]. This distorts the net air-gap MMF, introducing harmonics in the stator current at certain characteristic fault frequencies induced in the current Park’s vector [ 14, 15]

(20) $$\begin{gathered} \begin{array}{l}{i}_q={\displaystyle \frac{\sqrt{\mathrm{6}}}{\mathrm{2}}}{i}_{\mathrm{m}}\cos (\omega t)+{\displaystyle \frac{\sqrt{\mathrm{6}}}{\mathrm{2}}}{i}_{h\mathrm{1}}\cos ((\omega -{\omega }_{\mathrm{r}})t) \\ +{\displaystyle \frac{\sqrt{\mathrm{6}}}{\mathrm{2}}}{i}_{h\mathrm{2}}\cos ((\omega +{\omega }_{\mathrm{r}})t),\hfill \\ i_d={\displaystyle \frac{\sqrt{\mathrm{6}}}{\mathrm{2}}}{i}_{\mathrm{m}}\sin (\omega t-\alpha )+{\displaystyle \frac{\sqrt{\mathrm{6}}}{\mathrm{2}}}{i}_{h\mathrm{1}}\sin ((\omega -{\omega }_{\mathrm{r}})t)+{\displaystyle \frac{\sqrt{\mathrm{6}}}{\mathrm{2}}}{i}_{h\mathrm{2}}\cos ((\omega +{\omega }_{\mathrm{r}})t) \\ \mathrm{.}\hfill \end{array} \end{gathered}$$

In this situation, the PVPA is given by

(21) $$\begin{gathered} i_q(t)\cdot i_d(t)={\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}{i}_m^{\mathrm{2}}\sin (\mathrm{2}\omega t)+{\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}{i}_{h\mathrm{1}}^{\mathrm{2}}\sin (\mathrm{2}(\omega -{\omega }_r)t) \\ +{\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}{i}_{h\mathrm{2}}^{\mathrm{2}}\sin (\mathrm{2}(\omega +{\omega }_r)t)-{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{i}_m{i}_{h\mathrm{1}}\sin ((\mathrm{2}\omega -{\omega }_r)t) \\ +{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{i}_m{i}_{h\mathrm{2}}\sin ((\mathrm{2}\omega +{\omega }_r)t) \\ +{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{i}_{h\mathrm{1}}{i}_{h\mathrm{2}}\sin (\mathrm{2}{\omega }_{r}t)+{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{i}_m(i_{h\mathrm{1}}-i_{h\mathrm{2}})\sin ({\omega }_{r}t). \end{gathered}$$

Equation (21) has a component at frequency 2f_s, side-band components at 2f_s +f_r and 2f_s-f_r and additional components at f_r, 2f_r and 2f_s +2f_r, 2f_s-2f_r. Figure 2 presents the spectrum of the PVPA for this kind of fault. Thus, even for a motor with inter-turn shorts-circuit in the phase windings, there are new frequency components that can be used to discover this type of fault, with f_r=(1-s)/ f_s/p is a rotor frequency. p is the number of pole pairs.

Recent diagnosis approaches used

Laboratories have focused their research on finding a more sensitive approach to the diagnosis to detect the lowest abnormal variation in the motor, which offers the possibility of predicting defects to avoid considerable economic losses in the industries of high level production.

Motor square current signature analysis (MSCSA)

Proposed in Ref. [ 4], this approach is based on the stator current. The MCSA approach uses the spectrum analysis of one motor stator phase current. Considering a healthy motor or faulty motor, the square phase current is expressed by Ref. [ 4]

(22) \mathrm{MSCSA}=i_{\mathrm{1}}^{\mathrm{2}}(t).

In Eq. (22), a new motor square current signature analysis MSCSA fault diagnosis methodology is presented.

The proposed technique is based on three main steps. First, the induction motor current is measured. Next, the square of the current is computed. Finally, a frequency analysis of the square current is performed [ 4].

Motor current PVSM

Park transformation is used to transform stator currents from the three-phase system (1-2-3) to the two-phase system (d-q) [ 5]. The “motor current Park’s vector square modulus” is presented by

(23) \mathrm{PVSM}={\left|i_{d_s}+j{i}_q\right|}^{\mathrm{2}}=i_d^{\mathrm{2}}+i_q^{\mathrm{2}}.

There must be an analysis of the spectrum PVSM for each operating condition.

Park-Hilbert method PVSM_P-H

This method is proposed by Refs. [ 7− 9] based on the spectral analysis, via the FFT, of a new generated signal called Park-Hilbert Method: PVSM_P-H which represents the Park’s vector square modulus computed starting from the analytical signals obtained from the three-phase current by the Hilbert transform [ 7− 9]. Mathematically, the Hilbert transform of a real signal, such as stator current, is defined by

(24) i(t)\stackrel{\mathrm{TH}}{\Rightarrow }\tilde{i}(t)={\tilde{i}}_{\mathrm{Re}}(t)+j{\tilde{i}}_{\mathrm{Im}}(t)

and is called the analytic signal.

The amplitude modulation of the real stator current signal is given by Refs. [ 7− 9],

(25) \left|\tilde{i}(t)\right|=\sqrt{{\tilde{i}}_{\mathrm{Re}}{(t)}^{\mathrm{2}}+j{\tilde{i}}_{\mathrm{Im}}{(t)}^{\mathrm{2}}}.

Accordingly, the method is based on:

1) Determination of modulus for each stator current

\left|{\tilde{i}}_{\mathrm{1}}(t)\right|,\left|{\tilde{i}}_{\mathrm{2}}(t)\right| \mathrm{and} \left|{\tilde{i}}_{\mathrm{3}}(t)\right|.

2) Calculation of the components i_dP-H and i_qP-H, vector Park

(26) \{\begin{array}{l}{i}_{dP-\mathrm{H}}={\displaystyle \frac{\sqrt{\mathrm{2}}}{\sqrt{\mathrm{3}}}}\left|{\tilde{i}}_{\mathrm{1}}(t)\right|-{\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{6}}}}{\left|{\tilde{i}}_{\mathrm{2}}(t)\right|}_{\mathrm{s}\mathrm{b}}-{\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{6}}}}\left|{\tilde{i}}_{\mathrm{3}}(t)\right|,\hfill \\ i_{qP-\mathrm{H}}={\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{2}}}}\left|{\tilde{i}}_{\mathrm{2}}(t)\right|-{\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{2}}}}\left|{\tilde{i}}_{\mathrm{3}}(t)\right|.\hfill \end{array}

3) Determination of the “Park-Hilbert Method”

(27) {\mathrm{PVSM}}_{P-H}=i_{dP-H}^{\mathrm{2}}+i_{qP-H}^{\mathrm{2}}.

4) Analysis of the spectrum “Park-Hilbert Method”: PVSM_P-H for each operating condition.

Interpretations of simulation results

The motor used for the simulation in this paper is three-phase, 3 kW, 50 Hz, 2 poles, a rotor squirrel cage of 28 bars and 360 turns in series per phase (see Figs.A2 and A3 in Appendix).

Figures 1 to 5 show the evolution of signatures and their amplitudes and their specific frequencies of the different methods as a function of the state of the motor:

1) Healthy motor.

2) Rotoric fault: Broken bar,

Partially broken bar.

3) Statoric fault: N_cci=36 turns into short-circuit, N_cci=18 turns into short-circuit.

4) Motor under different operating conditions: at low load (25%) with s= 0.01, at mid-load (50%) with s = 0.021, at full load (100%) with s = 0.039.

Figure 1 demonstrates clearly the specific frequencies of signatures for each type of default given by the PVPA (Eqs. (19) and (21)) in the theoretical part of this paper, which confirms the accuracy of the mathematical calculations in this paper.

On the other hand, Figs. 2 to 5 displays the signatures of recent technical characteristics already used above and explains some types of faults.

To demonstrate the sensitivity of PVPA, a comparative study is made of its sensitivity with other recently used technical approaches. The results are presented in Figs. 6 and 7. In all cases, the signatures with the biggest amplitude have been chosen for each method.

To make a good comparison between these approaches, the specific frequencies of signatures must be chosen whose amplitude is the biggest.

The frequencies of a rotor or stator faults signatures are given in Table 1, respectively.

Figure 6, clearly shows that PVPA is absolutely more sensitive than other recent techniques for different regimes of motor operation and for all types of faults rotor:

1)Totally broken bar.

2)Incipient crack in a rotor bar (partially broken bar).

Figure 7 indicates that PVPA is also more sensitive than all other diagnosis techniques for different regimes of motor operation with 36 turns into short-circuit in phase 1.

In addition, Fig. 7 suggests that PVPA is more sensitive than MSCSA, PVSM and MCSA approach except for PVSM_P-H diagnosis technique which is a recent technique elaborated by Refs. [ 7− 9] to specialize in the detection of inter-turn short-circuit.

PVPA is a little less sensitive than PVSM_P-H diagnosis technique only in the case in which a micro-inter-turn short-circuits exists between stator windings (18 turns into short-circuit in phase 1) (Fig. 7 ) but PVSM_P-H is mediocre in the diagnosis of rotor faults (Fig. 8) (see Figs.A2 and A3 in Appendix).

Therefore, Table 2 clearly shows that PVPA is more sensitive and more efficient than other techniques for different diagnosis of induction motor.

Experimental validation of ranking results

To validate the sensitivity ranking results of the simulation, the simulation results need to be compared with the experimental results using a different motor (p = 2) and (p = 1). The verification of the concordance between the simulation results and the experimental ones can give a clearer idea of the work.

The motor used in the laboratory of Biskra University, Algeria (Laboratoire de Génie Electrique de Biskra LGEB), is a three-phase squirrel-cage induction motor, 3 kW, 4-poles (p= 2), Y connection, 200 turns per phase and with 28 rotor bars (see Table A2 in Appendix).

Figures 8 to 13 clearly show the characteristic signatures of different faults, their amplitudes and their specific frequencies for all methods of diagnosis studied above, depending on the motor state. Motor is running at different loads 80% (s=0.032), 60% (s=0.021) and 20% (s=0.0081) of full load.

1) Healthy motor.

2) Faulty stator: N_cci=10 turns short-circuited (see Fig. 14).

3) Faulty rotor: two broken bars (two bars pierced by a drilling machine).

Figures 8 to 13 demonstrate that there is a good concordance between the simulation results and the experimental results obtained in the laboratory which clearly show the fidelity of the model used compared to the real motor, especially the fault sensitivity curves given by Fig. 13 (see Figs. A4 and A5 in Appendix).

The final results of the simulation ranking given in Table 2 and those of the experimental ranking in Table 3 generally offer the best sensitivity to the proposed PVPA.

These experimental results clearly confirm the accuracy and precision of the simulation results and the conclusion.

Conclusions

This study shows that the model developed for the simulation of defects in the stator (inter-turn short-circuit) or in the rotor (partial or total breakage fault in the bar) has a great flexibility in choosing the type of defect:

1)Statoric defect by specifying the number of inter-turn short-circuit.

2)Rotoric defect by determining the rate of breaking bar.

It appears so faithful to the real machines used that a good concordance is obtained between the simulation results with experimental results, thanks to the flexibility of this model, which makes it possible to find a new, simple and very sensitive approach for different types of defects.

The theoretical calculations of this approach have been justified by the simulation results which prove the correctness of the mathematical results.

In addition, a theoretical explanation of the best techniques recently published with the simulation of their spectral signatures, and characteristics for each default order to compare them with those of the proposed approach have been given in this paper.

According to the results, it is found that the advantage of the proposed method shows a strong sensitivity to different types of defects so it dominates the other recent techniques. In addition, it is a simple and economic method.

Finally, it can be concluded that the proposed approach is endowed with high sensitivity which dominates the well-known and very recent methods MCSA, MSCSA, PVSM and PVSM_P-H. This opens an unprecedented perspective, in the diagnostic domain of electrical machines.

References

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