Experimental investigation and comparative study of inter-turn short-circuits and unbalanced voltage supply in induction machines

Abstract

A transient model for an induction machine with stator winding turn faults on a single phase is derived using reference frame transformation theory. The negative sequence component and the 3rd harmonic are often considered as accurate indicators. However, small unbalance in the supply voltage and/or in the machine structure that exists in any real system engenders the same harmonics components. In this case, it is too difficult to distinguish between the current harmonics due to the supply voltage and those originated by inter-turn short-circuit faults. For that, to have the correct diagnosis and to increase the sensitivity and the reliability of the diagnostic system, it is crucial to provide the relationship between the inter-turn short-circuits in the stator winding and the supply voltage imbalance through an accurate mathematical model and via a series of experimental essays.

Keywords

induction machines / fault indicator / inter-turn short-circuit fault / unbalance supply voltage

Introduction

The squirrel cage induction machine is the most important part of an assembly line in industry. It is unrivalled because of its ruggedness, simplicity, and relatively low cost. Its role in industry increased after the development of adjustable speed drives and its integration in energy conversion. However, abnormal operation may occur, sometimes due to the faults arising in the induction machine. Therefore, it would be crucial to detect any fault in an early stage, eliminating the hazards of severe motor faults [1]. Approximately 30% to 40% of the induction machine faults are stator faults. An inter-turn short-circuit fault is one of the most difficult failures to detect. Depending on the type of motor protection, the motor may continue to run, but the heating in the short-circuited turns will soon cause severe faults like phase-to-phase or coil-to-ground faults. That is why, the incipient inter-turn short-circuit in the stator winding is a very important fault and a valuable research subject [2,3].

Many sophisticated methods such as Park’s vector approach, negative sequence impedance, axial flux monitoring or the motor current signature analysis (MCSA) method [4-7] have been proposed for electrical machine inter-turn fault detection and localisation.

Certain current components are reported to be effective inter-turn fault indicators. The first component is -f_s caused directly by the fault and the second is 3f_s caused by the consequent speed ripple [8-11]. These two components were proven to be very good fault indicators, suitably correlated to the inter-turn fault severity for stator winding. Unfortunately, other working conditions such as small unbalances in the voltage supply system, which are inevitable in practice, produce similar effects. Therefore, it is virtually impossible to distinguish between inter-turn short-circuits fault and unbalanced voltage supply.

An accurate mathematical model under inter-turn fault and unbalanced voltage supply is elaborated and the relationship between the two is proved and verified using this model and a series of experimental essays.

Induction motor modelling

Healthy machine

Some assumptions must be made to achieve a model simple enough to be useful. First, the state of operation remains far from magnetic saturation. Second, the magnetic permeability of iron is considered to be infinite and the air-gap is very small and smooth. Next, the space magnetic motive force (MMF) and flux profiles are considered to be sinusoidally distributed and higher harmonics are negligible. And, finally, the voltage and flux equations of the induction machine can be written as [11]

(1)[{\mathit{V}}_s]=[{\mathit{R}}_s]\cdot [{\mathit{I}}_s]+{\displaystyle \frac{d[{\mathbf{\Phi }}_s]}{dt}},

(2)[{\mathit{V}}_r]=[{\mathit{R}}_r]\cdot [{\mathit{I}}_r]+{\displaystyle \frac{d[{\mathbf{\Phi }}_r]}{dt}},

where

(3)[{\mathbf{\Phi }}_s]=[{\mathit{L}}_{ss}]\cdot [{\mathit{I}}_s]+[{\mathit{M}}_{sr}]\cdot [{\mathit{I}}_r],

(4)[{\mathbf{\Phi }}_r]={[{\mathit{M}}_{sr}]}^T[{\mathit{I}}_s]+[{\mathit{L}}_{rr}]\cdot [{\mathit{I}}_r],

and

[{\mathit{I}}_s]=[i_{sa}{i}_{sb}{i}_{sc}]{}^T,

[{\mathit{I}}_r]=[i_{ra}{i}_{rb}{i}_{rc}]{}^T,

[{\mathit{V}}_s]=[v_{sa}{v}_{sb}{v}_{sc}]{}^T,

[{\mathit{R}}_s]=R_s{\mathit{E}}_{\mathrm{3}\times \mathrm{3}} is the stator matrix resistance, and the stator inductance matrix is given by

[{\mathit{L}}_{ss}]=\left(\begin{array}{lll}{L}_{saa}& M_{sab}& M_{sac}\\ M_{sba}& L_{sbb}& M_{sbc}\\ M_{sca}& M_{scb}& L_{scc}\end{array}\right),

where L_saa = L_sbb = L_scc = l_s is the stator self inductance, and M_sab = M_sac = M_sbc = M_ba = M_ca = M_cb = M_s is the stator phase mutual inductance.

[{\mathit{L}}_{rr}]=\left(\begin{array}{lll}{L}_{raa}& M_{rab}& M_{rac}\\ M_{rba}& L_{rbb}& M_{rbc}\\ M_{rca}& M_{rcb}& L_{rcc}\end{array}\right),

where L_raa = L_rbb = L_rcc = l_r is the stator self inductance, M_rab = M_rac = M_rbc = M_ra = M_rb = M_rc = M_r is the rotor phase mutual inductance.

[{\mathit{M}}_{sr}] is the mutual inductance matrix between the stator and rotor, which is the function of θ, the spatial position of the rotor.

[{\mathit{M}}_{sr}]=M_{sr}\left[\begin{array}{lll}\cos (\theta )& \cos \left(\theta +{\displaystyle \frac{\mathrm{2}\pi }{\mathrm{3}}}\right)& \cos \left(\theta -{\displaystyle \frac{\mathrm{2}\pi }{\mathrm{3}}}\right)\\ \cos \left(\theta -{\displaystyle \frac{\mathrm{2}\pi }{\mathrm{3}}}\right)& \cos (\theta )& \cos \left(\theta +{\displaystyle \frac{\mathrm{2}\pi }{\mathrm{3}}}\right)\\ \cos \left(\theta +{\displaystyle \frac{\mathrm{2}\pi }{\mathrm{3}}}\right)& \cos \left(\theta -{\displaystyle \frac{\mathrm{2}\pi }{\mathrm{3}}}\right)& \cos (\theta )\end{array}\right].

The torque and mechanical equations are given by

(5)T_e=[{\mathit{I}}_s]{}^T{\displaystyle \frac{\partial [{\mathit{M}}_{sr}]}{\partial {\theta }_m}}[{\mathit{I}}_r],

(6){\displaystyle \frac{d\omega }{dt}}={\displaystyle \frac{p}{J}}(T_e-T_L),

(7){\displaystyle \frac{d\theta }{dt}}=\omega ,

where θ_m is the mechanical angle; \omega /p, the mechanical speed; T_L, the load torque; and J, the rotor inertia. Park transformation is often used to facilitate the solution of difficult differential equations with time varying coefficients. The Park transformation equation is of the form of

(8)[{\mathit{X}}_{rodq}]=\left[\begin{array}{lll}{X}_{ro}& X_{rd}& X_{rq}\end{array}\right]{}^T={[\mathit{P}({\theta }_r)]}^{-\mathrm{1}}\cdot [{\mathit{X}}_{\mathrm{3}r}],

where [{\mathit{X}}_{\mathrm{3}r}] represents the current or voltage rotor matrix, and the transformation matrix [\mathit{P}({\theta }_r)] is defined as

(9)[\mathit{P}({\theta }_r)]=\sqrt{{\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}\cdot }\left(\begin{array}{lll}{\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{2}}}}& cos({\theta }_r)& -\sin ({\theta }_r)\\ {\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{2}}}}& cos\left({\theta }_r-{\displaystyle \frac{\mathrm{2}\pi }{\mathrm{3}}}\right)& -\sin \left({\theta }_r-{\displaystyle \frac{\mathrm{2}\pi }{\mathrm{3}}}\right)\\ {\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{2}}}}& \cos \left({\theta }_r+{\displaystyle \frac{\mathrm{2}\pi }{\mathrm{3}}}\right)& -\sin \left({\theta }_r+{\displaystyle \frac{\mathrm{2}\pi }{\mathrm{3}}}\right)\end{array}\right),

where θ_r is the angular displacement between Park reference and the first phase of the rotor.

Transforming the above sets of rotor variables described in Eqs. (1) and (2) to the Park reference frame, Eq. (10) can be obtained.

(10)\left(\begin{array}{l}{V}_{sa}\hfill \\ V_{sb}\hfill \\ V_{sc}\hfill \\ 0\hfill \\ 0\hfill \end{array}\right)=\left(\begin{array}{ccccc}{R}_s& 0& 0& 0& 0\\ 0& R_s& 0& 0& 0\\ 0& 0& R_s& 0& 0\\ 0& 0& 0& R_r& 0\\ 0& 0& 0& 0& R_r\end{array}\right)\left(\begin{array}{l}{i}_{sa}\hfill \\ i_{sb}\hfill \\ i_{sc}\hfill \\ i_{rd}\hfill \\ i_{rq}\hfill \end{array}\right)+\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -{\displaystyle \frac{d{\theta }_r}{dt}}\\ 0& 0& 0& {\displaystyle \frac{d{\theta }_r}{dt}}& 0\end{array}\right)\left[{\mathit{L}}_T\right]\left(\begin{array}{l}{i}_{sa}\hfill \\ i_{sb}\hfill \\ i_{sc}\hfill \\ i_{rd}\hfill \\ i_{rq}\hfill \end{array}\right)+\left[{\mathit{L}}_T\right]{\displaystyle \frac{d}{dt}}\left(\begin{array}{l}{i}_{sa}\hfill \\ i_{sb}\hfill \\ i_{sc}\hfill \\ i_{rd}\hfill \\ i_{rq}\hfill \end{array}\right),

where

[{\mathit{L}}_T]=\left(\begin{array}{lllll}{l}_s& M_s& M_s& M_{sr}& \mathrm{0}\\ M_s& l_s& M_s& -{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{M}_{sr}& {\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{M}_{sr}\\ M_s& M_s& l_s& -{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{M}_{sr}& -{\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{M}_{sr}\\ M_{sr}& -{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{M}_{sr}& -{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{M}_{sr}& l_r& \mathrm{0}\\ \mathrm{0}& {\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{M}_{sr}& -{\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{M}_{sr}& \mathrm{0}& l_r\end{array}\right)

and the electromagnetic torque given by Eq. (5) becomes

(11)T_e={[{\mathit{I}}_s]}^T{\displaystyle \frac{\partial }{\partial \theta }}[{\mathit{M}}_{sr}][\mathit{P}({\theta }_r)][{\mathit{i}}_{rodq}],

(12)T_e=\sqrt{{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}}{M}_{sr}\left[-i_{sa}{i}_{rq}+i_{sb}\left({\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{i}_{rd}-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{i}_{rq}\right)+i_{sc}\left(-{\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{i}_{rd}-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{i}_{rq}\right)\right].

Stator short-circuit model of induction machine

A short-circuit results in the creation of one or more additional windings. To take into account the existence of an additional short-circuited winding, it is necessary to define two parameters essential for the comprehension of the model introducing the inter-turn short-circuits [10,11]:

The angle θ_cc which is a real angle between the inter-turn short-circuit stator winding and the first stator phase axis (phase a). This parameter allows the localization of the faulty winding and can take only three values 0, 2π/3 or 4π/3 corresponding respectively to a short circuit on the stator phases a, b or c.

The parameter η_cc makes it possible to quantify the number of short-circuited turns. This parameter can be written as

{\eta }_{cc}={\displaystyle \frac{Numberofinter-turnsshort-circuitwindings}{Totalnumberofinter-turnsinonephase}}.

The stator and rotor equations with turn faults can be expressed as

(13)\{\begin{array}{l}\left[{\mathit{V}}_s\right]=\left[{\mathit{R}}_s\right]\cdot \left[{\mathit{I}}_s\right]+{\displaystyle \frac{d}{dt}}\left[{\mathit{\phi }}_s\right],\hfill \\ 0=R_{cc}{i}_{cc}+{\displaystyle \frac{d}{dt}}{\phi }_{cc},\hfill \\ 0=\left[{\mathit{R}}_r\right]\cdot \left[{\mathit{I}}_r\right]+{\displaystyle \frac{d}{dt}}\left[{\mathit{\phi }}_r\right],\hfill \end{array}

where

[{\mathit{\phi }}_s]=[{\mathit{L}}_s]\cdot [{\mathit{I}}_s]+[{\mathit{M}}_{sr}]\cdot [{\mathit{I}}_r]+[{\mathit{M}}_{scc}]i_{cc},

[{\mathit{\phi }}_r]=[{\mathit{M}}_{rs}]\cdot [{\mathit{I}}_s]+[{\mathit{L}}_r]\cdot [{\mathit{I}}_r]+[{\mathit{M}}_{rcc}]i_{cc},

{\phi }_{cc}={[{\mathit{M}}_{scc}]}^T[{\mathit{I}}_s]+{[{\mathit{M}}_{rcc}]}^T[{\mathit{I}}_r]+L_{cc}{i}_{cc},

(14)$$\begin{gathered} [{\mathit{M}}_{scc}]={\eta }_{cc}{l}_{sp}[\cos ({\theta }_{cc}) \\ \cos \left({\theta }_{cc}-{\displaystyle \frac{\mathrm{2}\pi }{\mathrm{3}}}\right) \\ \cos \left({\theta }_{cc}+{\displaystyle \frac{\mathrm{2}\pi }{\mathrm{3}}}\right)], \end{gathered}$$

(15)$$\begin{gathered} [{\mathit{M}}_{rcc}]={\eta }_{cc}{l}_{sp}[\cos ({\theta }_{cc}-\theta ) \\ \cos \left({\theta }_{cc}-\theta -{\displaystyle \frac{\mathrm{2}\pi }{\mathrm{3}}}\right) \\ \cos \left({\theta }_{cc}-\theta +{\displaystyle \frac{\mathrm{2}\pi }{\mathrm{3}}}\right)]. \end{gathered}$$

When the short circuit occurs in the phase s_a, θ_cc can be given as θ_cc = 0.

Applying Park transformation on the rotor equations, Eq. (13) becomes

(16)\left(\begin{array}{c}{V}_{sa}\\ V_{sb}\\ V_{sc}\\ 0\\ 0\\ 0\end{array}\right)=\left(\begin{array}{cccccc}{R}_s& 0& 0& 0& 0& 0\\ 0& R_s& 0& 0& 0& 0\\ 0& 0& R_s& 0& 0& 0\\ 0& 0& 0& R_{cc}& 0& 0\\ 0& 0& 0& 0& R_r& 0\\ 0& 0& 0& 0& 0& R_r\end{array}\right)\cdot \left(\begin{array}{c}{i}_{sa}\\ i_{sb}\\ i_{sc}\\ i_{cc}\\ i_{rd}\\ i_{rq}\end{array}\right)+\left[{\mathit{L}}_{Tcc}\right]\cdot {\displaystyle \frac{d}{dt}}\left(\begin{array}{c}{i}_{sa}\\ i_{sb}\\ i_{sc}\\ i_{cc}\\ i_{rd}\\ i_{rq}\end{array}\right)+\left(\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -{\displaystyle \frac{d{\theta }_r}{dt}}\\ 0& 0& 0& 0& {\displaystyle \frac{d{\theta }_r}{dt}}& 0\end{array}\right)\cdot \left[{\mathit{L}}_{Tcc}\right]\cdot \left(\begin{array}{c}{i}_{sa}\\ i_{sb}\\ i_{sc}\\ i_{cc}\\ i_{rd}\\ i_{rq}\end{array}\right),

where [{\mathit{L}}_{Tcc}] represents the total inductance matrix including the short-circuit equation.

[{\mathit{L}}_{Tcc}]=\left(\begin{array}{llllll}{l}_s& M_s& M_s& {\eta }_{cc}{l}_s& M_{sr}& \mathrm{0}\\ M_s& l_s& M_s& {\eta }_{cc}{M}_s& -{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{M}_{sr}& {\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{M}_{sr}\\ M_s& M_s& l_s& {\eta }_{cc}{M}_s& -{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{M}_{sr}& -{\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{M}_{sr}\\ {\eta }_{cc}{l}_s& {\eta }_{cc}{M}_s& {\eta }_{cc}{M}_s& {\eta }_{cc}^{\mathrm{2}}{l}_s& {\eta }_{cc}{M}_{sr}& \mathrm{0}\\ M_{sr}& -{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{M}_{sr}& -{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{M}_{sr}& {\eta }_{cc}{M}_{sr}& l_r& \mathrm{0}\\ \mathrm{0}& {\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{M}_{sr}& -{\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{M}_{sr}& \mathrm{0}& \mathrm{0}& l_r\end{array}\right).

On adding the first and third rows of Eq. (16), and rearranging terms, the machine equations can be expressed as

(17)$$\begin{gathered} \left(\begin{array}{l}{V}_{sa}\hfill \\ V_{sb}\hfill \\ V_{sc}\hfill \\ \mathrm{0}\hfill \\ \mathrm{0}\hfill \end{array}\right)=\left(\begin{array}{lllll}{R}_s& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& R_s& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& R_s& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& R_r& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& R_r\end{array}\right)\cdot \left(\begin{array}{l}{i}_{\mathrm{s}a}+{\eta }_{cc}{i}_{cc}\hfill \\ i_{\mathrm{s}b}\hfill \\ i_{\mathrm{s}c}\hfill \\ i_{\mathrm{r}d}\hfill \\ i_{\mathrm{r}q}\hfill \end{array}\right)+ \\ +\left(\begin{array}{lllll}\mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& -{\displaystyle \frac{d{\theta }_r}{dt}}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& {\displaystyle \frac{d{\theta }_r}{dt}}& \mathrm{0}\end{array}\right)\cdot [L_{\mathrm{T}cc\mathrm{2}}]\cdot \left(\begin{array}{l}{i}_{\mathrm{s}a}+{\eta }_{cc}{i}_{cc}\hfill \\ i_{\mathrm{s}b}\hfill \\ i_{\mathrm{s}c}\hfill \\ i_{\mathrm{r}d}\hfill \\ i_{\mathrm{r}q}\hfill \end{array}\right)+[L_{\mathrm{T}cc\mathrm{2}}]\cdot {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(\begin{array}{l}{i}_{\mathrm{s}a}+{\eta }_{cc}{i}_{cc}\hfill \\ i_{\mathrm{s}b}\hfill \\ i_{\mathrm{s}c}\hfill \\ i_{\mathrm{r}d}\hfill \\ i_{\mathrm{r}q}\hfill \end{array}\right), \end{gathered}$$

[{\mathit{L}}_{Tcc\mathrm{2}}] represents the total inductance matrix after the summation.

{[{\mathit{L}}_{Tcc\mathrm{2}}]}_{}=\left(\begin{array}{lllll}(\mathrm{1}+{\eta }_{cc})l_s& (\mathrm{1}+{\eta }_{cc})m_s& (\mathrm{1}+{\eta }_{cc})m_s& (\mathrm{1}+{\eta }_{cc})M_{sr}& \mathrm{0}\\ m_s& l_s& m_s& -{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{M}_{sr}& {\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{M}_{sr}\\ m_s& m_s& l_s& -{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{M}_{sr}& -{\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{M}_{sr}\\ (\mathrm{1}+{\eta }_{cc})M_{sr}& -{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{M}_{sr}& -{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{M}_{sr}& L_r& \mathrm{0}\\ \mathrm{0}& {\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{M}_{sr}& -{\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{M}_{sr}& \mathrm{0}& L_r\end{array}\right).

Dividing the first row of Eq. (17) by (1+η_cc), the machine equations become

(18)$$\begin{gathered} \left(\begin{array}{c}{\displaystyle \frac{V_{sa}}{1+{\eta }_{cc}}}\\ V_{sb}\\ V_{sc}\\ 0\\ 0\end{array}\right)=\left(\begin{array}{ccccc}{\displaystyle \frac{R_s}{1+{\eta }_{cc}}}& 0& 0& 0& 0\\ 0& R_s& 0& 0& 0\\ 0& 0& R_s& 0& 0\\ 0& 0& 0& R_r& 0\\ 0& 0& 0& 0& R_r\end{array}\right)\cdot \left(\begin{array}{l}{i}_{sa}+{\eta }_{cc}{i}_{cc}\hfill \\ i_{sb}\hfill \\ i_{sc}\hfill \\ i_{rd}\hfill \\ i_{rq}\hfill \end{array}\right)+ \\ +\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -{\displaystyle \frac{d{\theta }_r}{dt}}\\ 0& 0& 0& {\displaystyle \frac{d{\theta }_r}{dt}}& 0\end{array}\right)\cdot \left[{\mathit{L}}_T\right]\cdot \left(\begin{array}{l}{i}_{sa}+{\eta }_{cc}{i}_{cc}\hfill \\ i_{sb}\hfill \\ i_{sc}\hfill \\ i_{rd}\hfill \\ i_{rq}\hfill \end{array}\right)+\left[{\mathit{L}}_T\right]\cdot {\displaystyle \frac{d}{dt}}\left(\begin{array}{l}{i}_{sa}+{\eta }_{cc}{i}_{cc}\hfill \\ i_{sb}\hfill \\ i_{sc}\hfill \\ i_{rd}\hfill \\ i_{rq}\hfill \end{array}\right). \end{gathered}$$

The last system of equations is complemented with the electromagnetic torque Γ_e that is obtained from the magnetic coenergy W_co.

(19){\mathrm{\Gamma }}_e={[{\mathit{i}}_{sabccc}]}^T\cdot {\displaystyle \frac{\partial }{\partial \theta }}[{\mathit{M}}_{sr}]\cdot [{\mathit{i}}_{rabc}],

(20){\mathrm{\Gamma }}_e={[{\mathit{i}}_{sabccc}]}^T\cdot {\displaystyle \frac{\partial }{\partial \theta }}[{\mathit{M}}_{sr}]\cdot [\mathit{P}({\theta }_r)]\cdot [{\mathit{i}}_{rodq}],

(21){\mathrm{\Gamma }}_e=\left[\begin{array}{llll}{i}_{sa}& i_{sb}& i_{sc}& i_{cc}\end{array}\right]\cdot \sqrt{{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}}{M}_{sr}\cdot \left(\begin{array}{ll}\mathrm{0}& -\mathrm{1}\\ {\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}& {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\\ -{\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}& {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\\ \mathrm{0}& -{\eta }_{cc}\end{array}\right)\cdot \left(\begin{array}{c}{i}_{rd}\\ i_{rq}\end{array}\right)\mathrm{.}

(22){\mathrm{\Gamma }}_e=\sqrt{{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}}{M}_{sr}\left[-(i_{sa}+{\eta }_{cc}{i}_{cc})i_{rq}+\left({\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{i}_{rd}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{i}_{rq}\right)i_{sb}+\left(-{\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{i}_{rd}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{i}_{rq}\right)i_{sc}\right].

Hence, from Eqs. (10), (18), (12) and (22), the perfect similitude between the voltage supply and the inter-turn short-circuits can be noticed, with a reduction in the supply voltage of faulty phase and the value of its resistance by ratio (1+η_cc).

Experimental results

Both of the unbalance supply voltage and inter-turn short-circuit fault were simulated. The experimental burden was developed, making it possible to study the electric motors under varying conditions of load and different levels of faults on the stator windings of the tested motor. The tested motors used in the experimental investigation of the occurrence of both inter-turn stator winding faults and unbalance supply voltage was a three-phase, 50 Hz, 4 poles, 1.1 kW, squirrel cage induction machine, rated at 400 V, 2.95 A and 1450 r/min. The experimental setup for testing is shown in Fig. 1.

The induction motor whose stator winding were modified in order to have accessible several tapings that can be used to introduce inter-turn short-circuits, with different number of turns and different locations along the stator windings. The role of the current spectra analysis is to filter the harmonic components and to perform a reduction of the large amount of spectral information to a suitable level. Testing was done for different loads on a balanced sinusoidal system and then on an unbalanced one by inserting a series of resistors in one of the phases supplying the motor. That leads to a decrease of 5%, 15%, and 30% in the voltage amplitude of the concerned phase. For the calculation of the voltage unbalance factor (VUF), the most widely used expression as per the International Electrotechnical Commission was adopted [17]

VUF={\displaystyle \frac{V_n}{V_p}}\times \mathrm{100}\%,

where V_pand V_n are the amplitudes of the positive and negative sequence voltages, respectively. For the different values of decrease in the amplitude of the faulted phase voltage (15% and 30%), a VUF of 5.2% and 11.1% can be obtained, respectively. The motor current measurements are made by using a current probe PSY30 (its accuracy 1%, 0-100 kHz) and recorded by a LeCroyWaveRunner6050 oscilloscope which includes an industry-leading signal acquisition path, and provides a 5-GS/s analog-to-digital converter (ADC) on every channel and 1 MB of standard memory, which avoids under sampling or aliasing caused by slower ADCs or shorter memories. The chosen sampling frequency for data acquisition is 25 kHz and a length of 1 s. After the acquisition, the data are processed using the MATLAB software package to compute the fast Fourier transform (FFT) of the stator current spectrum.

A series of tests are accomplished to confirm the assumption. The motor was initially tested in the absence of faults in order to verify the intensity of harmonics amplitude before the faults.

Figure 2 illustrates harmonic components under unbalanced supply voltage when the motor is healthy. It is remarked that the 3rd harmonic and the negative sequence component appear in the spectra of the line current under stator voltage dissymmetry in the first stator phase.

Figure 3 depicts the spectra of the Park vector of stator currents with 10% of winding turns short-circuit in the first phase. It clearly shows why the negative sequence components and the third harmonic in the line current cannot be used as an indicator of this default in presence of a stator voltage dissymmetry. Figure 4 demonstrates the spectra of Park vector of voltage phase supply, which shows that it is not possible to have a symmetrical voltage supply. The presence of all harmonics given by

(23)f_{\sup ply}=Kf,K=\mathrm{1},\mathrm{3},\mathrm{5},\mathrm{7},\mathrm{9},\cdots

can be noticed.

Figure 5 displays the spectra of Park vector of Stator currents of healthy machine with unbalanced voltage supply. It confirms the existence of the frequency components in the line currents (150 Hz, 250 Hz, 350 Hz, …) due to the voltage supply dissymmetry previously discussed. It can be observed that the amplitude of harmonics increases proportionally with the augmentation of the unbalanced supply voltage. Figure 6 exhibits the spectra of the Park vector of stator currents with 2%, 10% and 20% of winding turns short-circuit in the first phase and with unbalanced supply voltage. It is seen clearly that the variation of the third sequence amplitude is not influenced by the level of the inter-turn short-circuit fault.

This harmonics (+3) can mask or cancel the effect of inter-turn short-circuit fault. If the level of unbalance supply voltage becomes more severe, the prediction of the short-circuit fault is relatively impossible.

Figures 7 and 8 contain a plot of the negative sequence harmonic amplitude variation for both healthy motor under unbalanced supply voltage and inter-turn short-circuit fault as a function of the load. From Figs. 7 and 8, the same variation can be noticed.

Conclusions

A transient model for an induction machine with stator winding turn faults on a single phase is derived using reference frame transformation theory. This model is used to give a comparative study between the inter-turn short-circuits and voltage supply dissymmetry. It is clear that the inter-turn short-circuit is only a reduction of the voltage in the faulty phase by the ratio (1+η_cc), plus a variation in its resistance by the same ratio. This resemblance cannot give a clear diagnosis when there is an unbalance in the voltage supply.

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