As the method is based on the analysis of the external magnetic field existing in the vicinity of a rotating electrical machine, sensors for magnetic field measurement are used. They can be classified into three categories: based on the Hall Effect, those that exploit the magneto-resistive phenomenon, and those which used specific coils. As the behavior of the external magnetic field harmonic components tied to the slotting effect are considered, the corresponding frequencies are in a medium frequency range going from the hundredth to the thousandth of hertz. Consequently, a coil sensor is more convenient because it induces electromotive force (emf) and this derivative effect amplifies the medium frequencies.

The used sensor is circular of S area (S = 8,04cm²), and the coil is constituted of n^c turns (n^c = 200). Figure 1(a) presents the sensor symbol. Figure 1(b) gives the sensor frequency response, especially the modulus |z| and the phase of the sensor impedance θ_z measured with an impedance analyzer. It can be observed that a resonance appears at 559,7 kHz. This resonance is tied to the inter-turn capacitive effect combined with the inductive effects. The considered frequency range is lower (some kHz) compared to resonance frequency, meaning that this sensor is well suited for the intended application. The drawback of a coil sensor is that it does not give localized information but an average value over the sensor area, so in order to have the most localized information possible, the sensor has to be small compared to the machine size. However, use of a small sensor leads to a decrease of the sensitivity, which can be compensated by increasing n^c, but a high number n^c decreases the resonance frequency. Consequently a compromise has to be made in the choice of sensors.

The use of the proposed diagnosis method requires at least two sensors, located symmetrically about the machine axis (180° spatially shifted) and placed close to the motor frame between the end bells, roughly in the middle of the machine. The principle consists in the comparison of the delivered sensor signals according whether the machine runs under no-load or loading conditions. The analysis concerns the magnitude of the specific harmonics of the induced coil emf. One will be interested in the magnitude of the third rank harmonic for SSM and the rotor slotting harmonics for IM. The principle of the method can be described considering a load increase:

▪ if the harmonic amplitudes measured on both sides of a machine vary in the same direction, then the stator winding does not present an inter-turn short circuit fault,

Let us point out that the amplitude of the measured harmonics strongly depends on the fault severity and the location of the sensor in relation to the machine.

The stray external magnetic field results from the combination of its axial and transverse components. The axial field is in a plane that contains the machine axis; it is generated by the end overhang effects. The transverse field is located in a plane perpendicular to the machine axis. It is an image of the air-gap flux density b which is attenuated by the stator magnetic circuit (sheets package with length L) and by the external machine frame. On the other hand, the eddy currents introduce a phase change which differs according to the considered component. It is however possible to measure mainly the transversal field by choosing an adequate position of the sensor such as the effects of the axial field are minimized. This position corresponds roughly to L/2 as shown in Figure 2.

In the following, from a theoretical point of view, only the transverse field is considered. In order to identify its harmonic components, the b air-gap flux density, which results from the product between the λ air-gap permeance and the ε resulting magnetomotive force (mmf), has to be determined.

To determine b, the following assumptions are formulated:

- the iron permeability is supposed to be infinite,

Let us adopt as space references:

- the d^s axis which is confounded with the stator phase 1 axis,

Any point M in the air-gap can be located by the variables α^s in relation to d^s and α^r in relation to d^r (Figure 3). The axes d^s and d^r are distant of θ (the angular gap between the stator and rotor references).

The ε^s mmf generated by a healthy stator relatively to d^s can be expressed as: (1) ɛ s = I s ∑ h s A h s s   cos ( ω t − h s p α s )where h^s is defined by: h^s = 6k + 1 where k is an integer number which varies between −∞ to +∞. A h s s is a function that takes into account the winding coefficient tied to the rank h^s.

In the reference frame related to d^r, the ε^r rotor mmf expression, function of α^r, depends on the kind of machine. As α^r = α^s − θ, this variable change makes it possible to define ε^0r which represents the mmf generated by the rotor but defined relative to d^s. Consequently, ε is given by: (2) ɛ = ɛ s + ɛ 0 r

The λ air-gap permeance expressions, different according to the kind of machine, defined relative to d^s, are deduced from the general Equation (3) [15]: (3) λ = ∑ k s = − ∞ + ∞ ∑ k r = − ∞ + ∞ Λ k s k r cos       [ ( k s N s + k r N r ) p α s − p k r N r θ ]

This relationship concerns an IM taking into account the interaction between the stator and the rotor teeth [16]. N^s and N^r are the stator and rotor per pole pair tooth numbers. Λ_k^sk^r is a coefficient which depends on the ranks k^s and k^r and the air-gap geometry. Let s denote the slip, θ is given by: (4) θ = ( 1 − s ) ω t / p + θ 0

For the SSM, it suffices to adapt the parameters taking into account that N^r = 2 and s = 0.

As b = λε, the calculus developments lead to define b in the reference frame related to d^s as follows: (5) b = ∑ K , H b K , Hwith: (6) b K , H = b ^ K , H   cos ( K   ω t − H α s − φ K , H )

An elementary component b_K,H is characterized by two parameters: K and H. When only one indicator appears in the variable, it corresponds to K. A flux density component is thus characterized by its pulsation Kω and by its pole pair number H.

The transverse external flux density is deduced from b through an attenuation coefficient as previously mentioned. This coefficient, denoted C K , H x, is a function of K, H and the distance x of a point outside of the machine from its axis. Let us point out that for a given K, C K , H x decreases when H increases [17]. A flux density component of Kω angular frequency, denoted b K x, is obtained by summation of components having the same frequency rank K but different pole pair number H. According Equations (5) and (6), at comes: (7) b K x = ∑ H b K , H xwith: (8) b K , H x = b ^ K , H x   cos ( K   ω t − H α s − φ K , H x )and: (9) b ^ K , H x = C K , H x b ^ K , H φ K , H x results from φ_K,H by taking into account the phase change possibly introduced by the eddy currents.

The measurements are performed with a flux sensor presented in section 2.1 placed in α^s = β₀. The Kω component of the flux linked by the coil sensor results from: (10) Ψ K x = ∫ S b K x d S

The result of this integration depends of the sensor parameters (n^c, S), but also H and x. Introducing these parameters in the coefficient K H x, Ψ K x is expressed by: (11) Ψ K x = ∑ H C K , H x K H x b ^ K , H   cos   ( K   ω   t − H β 0 − φ K , H x )

Among the components which constitute Ψ K x only few of them, relative to low pole number (low H), have a significant contribution [18]. The other components will be absorbed by the ferromagnetic parts of the machine. The e^x delivered by the sensor is given by: (12) e x = ∑ K e ^ K x   sin   ( K   ω t − φ K , H x )with: (13) e ^ K x = − K   ω ( ∑ H C K , H x K H x b ^ K , H e − j ( H β 0 + φ K ,   H x ) ) φ K x = Arg ( ∑ H C K , H x K H x b ^ K , H e − j ( H β 0 + φ K ,   H x ) ) }

Let us consider a SSM where the rotor winding is energized by a J DC current. One assumes that d^r is confounded with a north rotor pole axis. ε^r can be written as follows: (17) ɛ r = J ∑ h r A h r r   cos   ( h r p α r )where h^r takes all odd values varying between 1 and +∞. One can deduce that: (18) ɛ 0 r = J ∑ h r A h r r cos     ( h r ω t − h r p α s + h r p θ 0 )

Considering Equations (3) and (4) and taking the particularities that characterize the SSM into account, leads us to express λ as: (19) λ = ∑ k s = − ∞ + ∞ ∑ k r = − ∞ + ∞ Λ k s k r cos     [ 2 k r ω t − ( k s N s + 2 k r ) p α s + 2 k r p θ 0 ]K and H, which participate in the b_K,H definition given by Equation (6), are defined as: (20) K = h r + 2 k r H = p ( h sr + k s N s + 2 k r ) }where h^sr includes all the values taken by h^s and h^r: h^sr = h^s ∪ h^r. One can also deduce that: (21) b qsc s = ∑ K sc , H sc b ^ s c K sc , H sc   cos   ( K sc ω t − H sc α s − φ s c K sc , H sc )with: (22) K sc = 1 + 2 k ′ r H sc = h + p k ′ s N s + 2 p k ′ r }    k^′s and k^′r are equivalent to k^s and k^r. Consequently they vary from −∞ to +∞.

The resulting flux density appears, after attenuation, at the level of the external transverse field. That results in the presence of harmonics at Kω and K_scω angular frequencies in the signal delivered by the flux sensor. Considering the K [Equation (20)] and K_sc [Equation (22)] values, it is possible to make the following remarks:

- K_sc does not bring new frequencies, that means that with the traditional method of diagnosis, the failure presence will be appreciated through the variation of the amplitudes of existing lines in the healthy machine spectrum.

These properties make difficult the diagnosis by analysis of the changes in the amplitudes of the measured components. In the following the properties relating to the dissymmetry generated by the fault will be exploited.

The measurements are carried out in two diametrically opposed positions: position 1 in α^s = β₀ and position 2 in α^s = β₀ + π. The principle of the method consists in the analysis of the variations induced by the fault at the level of particular harmonic components measured in the both positions. The corresponding flux density components must have significant amplitude in the external magnetic field. Consequently, taking into account the magnitude of the air-gap flux density and considering the attenuation of these components in the stator core, one will be interested in the components defined by the lowest values of h^s, h^r, k^s, k^r, k^′s, k^′r, leading to the smallest as possible value of H.

The respect of these constraints leads to consider the spectral line at ω angular frequency (K = 1) of magnitude b̂_1,1. However, this component corresponds to the fundamental of the air-gap flux density which generates the main energetic effects: b̂_1,1 ≫ b̂_1,H with H different from 1. It results that b̂₁ will be not very sensitive to the components defined for |H|>1. Consequently, the analysis will be focused on the spectral lines related to K = 3. If one considers a four pole machine (p = 2), the flux density components that contribute to this harmonic are defined as follows:

- for healthy machine: h^r = 1, k^s = 0, k^r = 1, h^sr = 1 in Equation (20), which gives K = 3 and H = 3 p = 6;

Let us consider b 3 * x the flux density component at 3ω angular frequency on the outside of the machine. This component is the sum of a term relating to the healthy machine (of amplitude b ^ 3 x) and a term relating to the faulty machine (of amplitude b ^ sc 3 x). In position 1, the resulting component can be expressed as: (23) b 3 * x ( 1 ) = b ^ 3 x   cos ( 3 ω t − 6 β 0 + φ 3 x ) + b ^ sc 3 x ( cos 3 ω t + β 0 + φ sc 3 x )

In position 2, it becomes: (24) b 3 * x ( 2 ) = b ^ 3 x cos ( 3 ω t − 6 β 0 + φ 3 x ) − b ^ sc 3 x cos ( 3 ω t + β 0 + φ sc 3 x )

The difference between the both positions is tied to the sign which precedes the component due to the fault. For a healthy machine such as b ^     sc 3     x = 0, the components defined by (23) and (24) present the same amplitude. In the presence of the fault, the amplitude of the component in one of the positions increases whereas the other one decreases. However, from practical considerations, one can observe differences between measurements for the both positions, even in the healthy case: b ^ 3 x ( 1 ) in position 1 is different from b ^ 3 x ( 2 ) in position 2. Two external causes can be responsible for this dissymmetry:

▪ positions 1 and 2 cannot be perfectly symmetrical from each other.

To solve this problem, the method can be extended thanks to the exploitation of a test in load. The method can be refined as following.

➢ Under healthy conditions:

When the machine is loaded, the stator current increases but, at given supply voltage, the air-gap flux density stays practically identical to that obtained at no-load. As the external elements responsible of the attenuation act in the same way in positions 1 and 2 for the no-load and for the load tests, the amplitudes b ^ 3 x ( 1 ) and B ^ 3 x ( 2 ) keep similar values or at least evolve in the same way when the machine is loaded.

➢ Under faulty conditions:

In loading conditions, b ^ sc 3 x varies taking into account the model chosen to characterize the short-circuit (the loading current contributes to the definition of the fictitious current which circulates in the damaged coil). Consequently, according to (23) and (24), the magnitude of the component at 3ω angular frequency in positions 1 and 2 will evolve in opposed directions. The variations of the lines at 3ω are thus an indicator of defect. The advantage of the method is that it does not require any knowledge of a healthy state to detect the fault.

The tests are carried out on a SSM characterized by: 7.5 kVA, 50 Hz, p = 2, 230 / 400 V, N^s = 18. This machine has been rewound so that the terminals of the different stator elementary sections are extracted from the winding and are brought back to a connector block which is fixed above the machine as indicated in Figure 7. So it is possible to produce short-circuits between elementary coils.

The tests carried out when the SSM is connected to the grid, consist in measuring and analyzing the emf delivered by the sensors in order to validate the suggested procedure when an elementary coil is short-circuited (what corresponds to 16, 6% of the whole winding of a phase). In order to avoid damaging the winding, the short-circuit current is limited by an external resistance R.

Verification have also be done in order to make sure that the currents in the wires going up from the machine towards the connector block do not disturb the measurements. Positions 1 and 2 correspond respectively to β₀ and β₀ + π (see paragraph 5.2) β₀ are chosen “arbitrarily”.

Figure 8 presents the sensor emf spectra without [Figure 8(a)] and with [Figure 8(b)] fault at no-load operation in the both positions. As discussed in the theoretical part, it can be observed that a difference appears between the amplitudes of the spectrum lines in positions 1 and 2 for the faulty machine but also in healthy case. It can be also observed that this difference concerning the line at 150 Hz is higher in the presence of the defect [Figure 8(b)], with inversion of the predominant line (for the healthy machine, the line in position 2 is higher than this relating to position 1, and inversely in the presence of the defect). However, the inversion of the dominating line as it was expected does not appear for the lines at 50 Hz. The loading operation is interesting because it makes it possible to amplify the effect of the fault characterized in theory by the term b ^ sc 3 x in Equations (23) and (24), while keeping on the same level the influence of the natural dissymmetry due to the design of the machine.

The measured spectra under loading conditions for the healthy machine and the faulty one are given respectively in Figures 9(a,b). It can be observed in Figure 9(a) that the differences between the amplitudes measured in positions 1 and 2 for the lines at 50 Hz and 150 Hz still exist under loading conditions. The difference is relatively important for the amplitudes of the lines related to the fundamental (50 Hz). Figure 9(b), relating to the faulty machine, shows that a small difference characterizes the amplitudes of the lines at 150 Hz for the both positions. Consequently, the analysis of the difference between the two positions is not enough to detect the fault because it depends on a high number of parameters: inaccurate position of the sensor, value of the short-circuit current, number of short-circuit turns, state of load, machine’s design, etc.. In order to minimize these effects, the suggested method consists in combining the tests at no-load and in load operating conditions.

Figure 10(a,b) shows the variations of the harmonic at 150 Hz relative to the healthy machine when the state (marked with arrow) varies from no-load to loading condition. It can be observed that the magnitudes vary in the same way (decrease) for the two positions.

For the faulty machine the variations of the 150 Hz harmonic are given in Figures 11(a,b). It can be seen that the magnitudes in the both positions vary in an opposite way when the machine is loaded. The results are in accordance with the theory, what validates the diagnosis procedure. This procedure has been also test on an interior permanent magnet synchronous machine. The same results were found.