**2.1. Modeling assumptions**

In the proposed approach, we assumed that:


method in fault diagnosis systems [9].

operation)

PCA: data treatment.

**2. WRIM modeling** 

circuits were chosen.

**2.1. Modeling assumptions** 

compared to the vacuum, the skin effect is neglected,

In the proposed approach, we assumed that:

are necessary.

Results Analysis: system diagnosis.

The following are the different steps of the approach:

**Figure 1.** Synoptic diagram of the different steps of the data treatment

The Figure 1 shows that the proposed approach is divided in four blocs: WRIM modeling: mathematical equations calculation and simulation.

Simulation :graph showing the output states of the system (healthy and faulted

In the process of faults survey and diagnosis, an accurate modeling of the machine is necessary. In this paper, three phases model based on magnetically coupled electrical

The aim of the modeling is to highlight the electrical faults influences on the different state variables of the WRIM. For that, some modeling assumptions given in the following section

the magnetic circuit is linear, and the relative permeability of iron is very large

These models are then implemented in the Matlab software. Simulation results of several variables (stator and rotor currents, shaft rotational speed, electrical power, electromagnetic torque and other variables issued from mathematical transformations) of healthy and faulted WRIM are analyzed. Comparisons of simulation results with those of other diagnostic methods are performed to show the effectiveness and importance of the PCA


#### **2.2. Differential equation system of the WRIM**

We note that the voltage vectors ([VS], [VR]), the current vectors ([IS], [IR]) and the flux vectors ([ΦS], [ΦR]) of the stator and rotor are respectively:

$$
\begin{bmatrix} V\_S \\ \end{bmatrix} = \begin{bmatrix} V\_A \\ V\_B \\ V\_C \\ \end{bmatrix}; \ \begin{bmatrix} I\_S \\ \end{bmatrix} = \begin{bmatrix} I\_A \\ I\_B \\ I\_C \\ \end{bmatrix}; \ \begin{bmatrix} \phi\_S \\ \end{bmatrix} = \begin{bmatrix} \phi\_A \\ \phi\_B \\ \phi\_C \\ \end{bmatrix}
$$

$$
\begin{bmatrix} V\_R \\ \end{bmatrix} = \begin{bmatrix} V\_a \\ V\_b \\ V\_c \\ \end{bmatrix}; \ \begin{bmatrix} I\_R \\ \end{bmatrix} = \begin{bmatrix} I\_a \\ I\_b \\ I\_c \\ \end{bmatrix}; \ \begin{bmatrix} \phi\_R \\ \end{bmatrix} = \begin{bmatrix} \phi\_a \\ \phi\_b \\ \phi\_c \\ \end{bmatrix}
$$

**Figure 2.** Equivalent electrical circuit of the WRIM

Vj, Ij and Φj (j : A, B, C for the stator phases and a, b, c, for the rotor phases) are respectively the voltages, the electrical currents and the magnetic flux of the stator and the rotor phases, θ is the angular position of the rotor relative to the stator.

The Figure 2 shows the equivalent electrical circuit of the WRIM. Each coil, for both stator and rotor, is modelised with a resistance and an inductance connected in series configuration (Figure 3).

$$
\begin{bmatrix} V\_S \ \end{bmatrix} = \begin{bmatrix} R\_S \ \end{bmatrix} \begin{bmatrix} I\_S \ \end{bmatrix} + \frac{d\begin{bmatrix} \phi\_S \ \end{bmatrix}}{dt} \tag{1}
$$

$$
\left[\begin{array}{c} V\_R \end{array}\right] = \left[\begin{array}{c} R\_R \end{array}\right] \left[\begin{array}{c} I\_R \end{array}\right] + \frac{d\left[\begin{array}{c} \phi\_R \end{array}\right]}{dt} \tag{2}
$$

$$
\begin{bmatrix} \phi\_{\rm S} \end{bmatrix} = \begin{bmatrix} I\_{\rm S} \end{bmatrix} \begin{bmatrix} I\_{\rm S} \end{bmatrix} + \begin{bmatrix} M\_{\rm SR} \end{bmatrix} \begin{bmatrix} I\_{\rm R} \end{bmatrix} \tag{3}
$$

$$
\begin{bmatrix} \phi\_{\mathcal{R}} \end{bmatrix} = \begin{bmatrix} L\_{\mathcal{R}} \ \end{bmatrix} \begin{bmatrix} I\_{\mathcal{R}} \ \end{bmatrix} + \begin{bmatrix} M\_{\mathcal{R}S} \ \end{bmatrix} \begin{bmatrix} I\_{S} \ \end{bmatrix} \tag{4}
$$

 [*RS*] and [*RR*] are the resistance matrices, [*LS*] and [*LR*] the self inductance matrices, and [*MSR*] and [*MRS*] the mutual inductances matrix between the stator and the rotor coils.

With (3) and (4), (1) and (2) become:

$$d\left[\left.\boldsymbol{V}\_{\mathcal{S}}\right\|\right] = \left[\left.\boldsymbol{R}\_{\mathcal{S}}\right\|\right] \left[\left.\boldsymbol{I}\_{\mathcal{S}}\right] + \frac{d\left\{\left[\left.\boldsymbol{L}\_{\mathcal{S}}\right\|\right] \left[\left.\boldsymbol{I}\_{\mathcal{S}}\right]\right\}}{dt} + \frac{d\left\{\left[\left.\boldsymbol{M}\_{SR}\right] \left[\left.\boldsymbol{I}\_{R}\right]\right]\right\}}{dt} \tag{5}$$

$$d\left[\left.\boldsymbol{V}\_{R}\right.\right] = \left[\left.\boldsymbol{R}\_{R}\right.\right] \left.\left.\left.\boldsymbol{I}\_{R}\right.\right] + \frac{d\left<\left[\left.\boldsymbol{L}\_{R}\right.\right] \left.\left.\left.\boldsymbol{I}\_{R}\right.\right]\right>}{dt} + \frac{d\left<\left[\left.\boldsymbol{M}\_{RS}\right.\right] \left.\left.\left.\boldsymbol{I}\_{S}\right.\right]}{dt}\right.\tag{6}$$

By applying the fundamental principle of dynamics to the rotor, the mechanical motion equation is [13]:

$$f\_t \frac{d\Omega}{dt} + f\_v \Omega = \mathbb{C}\_{em} - \mathbb{C}\_r \tag{7}$$

$$
\Omega = \frac{d\theta}{dt} \tag{8}
$$

with:

$$C\_{em} = \frac{1}{2} \left[ \begin{array}{c} \end{array} \right]^t \* \frac{d\left( \begin{array}{c} \begin{bmatrix} L \ \end{bmatrix} \end{array} \right)}{d\theta} \* \begin{bmatrix} I \end{bmatrix} \tag{9}$$

is the total inertia brought to the rotor shaft, the shaft rotational speed, [*I*]*=*[*IA IB IC Ia Ib Ic*]*<sup>t</sup>* the current vectors, the viscous friction torque, the electromagnetic torque, the load torque, the angular position of the rotor relative to the stator and [*L*] the inductance matrix of the machine.

Introducing the cyclic inductances of the stator and the rotor <sup>3</sup> <sup>2</sup> *SC S L L* and <sup>3</sup> <sup>2</sup> *RC R L L* (*LS* is

the self inductance of each phase of the stator and *LR* is the self inductance of each phase of the rotor), the mutual inductances between the stator and the rotor coils *MSR* and pole pair number *p*, the inductance matrix of the WRIM can be written as follow:

$$\begin{bmatrix} L\_{\rm SC} & 0 & 0 & M\_{\rm SR}f\_1 & M\_{\rm SR}f\_2 & M\_{\rm SR}f\_3 \\ 0 & L\_{\rm SC} & 0 & M\_{\rm SR}f\_3 & M\_{\rm SR}f\_1 & M\_{\rm SR}f\_2 \\ 0 & 0 & L\_{\rm SC} & M\_{\rm SR}f\_2 & M\_{\rm SR}f\_3 & M\_{\rm SR}f\_1 \\ M\_{\rm SR}f\_1 & M\_{\rm SR}f\_3 & M\_{\rm SR}f\_2 & L\_{\rm RC} & 0 & 0 \\ M\_{\rm SR}f\_2 & M\_{\rm SR}f\_1 & M\_{\rm SR}f\_3 & 0 & L\_{\rm RC} & 0 \\ M\_{\rm SR}f\_3 & M\_{\rm SR}f\_2 & M\_{\rm SR}f\_1 & 0 & 0 & L\_{\rm RC} \end{bmatrix} \tag{10}$$

$$f\_1 = \cos(p\theta) \tag{11}$$

$$f\_2 = \cos(p\theta + \frac{2\pi}{3})\tag{12}$$

$$f\_3 = \cos(p\theta - \frac{2\pi}{3})\tag{13}$$

By choosing the stator and rotor currents, the shaft rotational speed and the angular position of the rotor relative to the stator as state variables, the differential equation system modeling the WRIM is given by:

<sup>1</sup> [ ] [ ] ([ ] [ ][ ]) *X A U BX* (14)

with:

72 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

configuration (Figure 3).

**Figure 3.** Equivalent electrical circuit of the WRIM coils

With (3) and (4), (1) and (2) become:

equation is [13]:

with:

The Figure 2 shows the equivalent electrical circuit of the WRIM. Each coil, for both stator and rotor, is modelised with a resistance and an inductance connected in series

*S SS*

*R RR*

*V RI*

and [*MRS*] the mutual inductances matrix between the stator and the rotor coils.

*S SS*

*R RR*

*V RI*

*V RI*

*S S S SR R LI M I*

*R R R RS S LI M I*

[*RS*] and [*RR*] are the resistance matrices, [*LS*] and [*LR*] the self inductance matrices, and [*MSR*]

By applying the fundamental principle of dynamics to the rotor, the mechanical motion

*t v em r <sup>d</sup> J f CC dt*

> *d dt*

<sup>1</sup> \* \* <sup>2</sup> *t*

*d L CI I d*

*em*

*S S SR R*

*R R RS S*

*dL I dM I*

*dt dt* (5)

*dL I dM I*

*dt dt* (6)

(7)

(8)

(9)

*V RI*

*S*

*R*

*dt*

(1)

(2)

(3)

(4)

*dt*

*d*

*d*

$$\begin{aligned} [X] &= [I\_A \quad I\_B \quad I\_C \quad I\_a \quad I\_b \quad I\_c \quad \Omega \quad \theta]^\dagger \\ [A] &= \begin{bmatrix} [L] & 0 & 0 \\ 0 & I\_t & 0 \\ 0 & 0 & 1 \end{bmatrix}; \ [II] = \begin{bmatrix} [V] \\ -C\_r \\ 0 \end{bmatrix}; \\ [V] &= [V\_A \quad V\_B \quad V\_C \quad V\_a \quad V\_b \quad V\_c]^\dagger; \\ [B] &= \begin{bmatrix} [R] + \Omega \frac{d[L]}{d\theta} & 0 & 0 \\ -\frac{1}{2}[I]^\dagger \frac{d[L]}{d\theta} & f\_v & 0 \\ 0 & -1 & 0 \end{bmatrix} \end{aligned}$$

This model of the WRIM will be used to simulate both the healthy and the faulted configuration of the stator and the rotor.
