# Analytical Model of Eccentric Induction Machines Using the Conformal Winding Tensor Approach

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## Abstract

**:**

## 1. Introduction

- Numerical models, mostly based on the finite elements method (FEM). They can accurately reproduce the behaviour of the eccentric IM [20], but they require detailed information about construction aspects of the IM and are computationally intensive. This problem can be alleviated using order-reduction models [21], solving the machine at some positions and performing a field reconstruction based on them or with hybrid FEM-analytical models [22,23,24].
- Analytical models, based on a network of magnetically coupled circuits [25]. Their accuracy may not be as high as FEM models, but they are much faster to build and solve, need only the most basic motor parameters [26], and can correctly reproduce the position and amplitude of the fault-related harmonic components [27].

## 2. Simple Analytical Model of the IM

- $\mathit{e}={[{e}_{{s}_{1}},{e}_{{s}_{2}},\dots ,{e}_{{s}_{{n}_{s}}},{e}_{{r}_{1}},{e}_{{r}_{2}},\dots ,{e}_{{r}_{{n}_{r}}}]}^{t}$ is the voltage vector, which represents the terminal voltages applied to the n windings;
- $\mathit{\phi}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{[{{\phi}_{s}}_{1},{{\phi}_{s}}_{2},\dots ,{{\phi}_{s}}_{{n}_{s}},{{\phi}_{r}}_{1},{{\phi}_{r}}_{2},\dots ,{{\phi}_{r}}_{{n}_{r}}]}^{t}$ is the flux linkage vector, which represents the flux linkages of the n windings;
- $\mathit{i}={[{i}_{{s}_{1}},{i}_{{s}_{2}},\dots ,{i}_{{s}_{{n}_{s}}},{i}_{{r}_{1}},{i}_{{r}_{2}},\dots ,{i}_{{r}_{{n}_{r}}}]}^{t}$ is the current vector, which represents the n winding currents;
- $\mathit{R}$ is the resistance tensor. It is a square matrix, with ${n}^{2}$ components, for which its elements are winding resistances.

## 3. Determination of the Parameters of the IM Model

#### 3.1. Resistance Matrix of the Primitive IM Network

#### 3.2. Inductance Matrix of the Primitive Network

#### 3.3. From the Primitive IM Network to the Actual One Using the Connection Matrix

## 4. Computation of the Main Inductance Matrix of the Healthy and the Eccentric IM Using the Conformal Winding Tensor Approach

- A primitive spatial network, similarly to Figure 4, is constructed by removing all interconnections between winding conductors and short circuiting each one without changing their spatial positions. For this simple network, the matrix with the partial inductances between conductors is obtained, which makes it easier to take into account the effect of IM eccentricity.
- A transformation matrix, similarly to (12), is constructed. It represents the interconnections of the conductors of each winding for each angular position of the rotor, i.e., the winding tensor.

#### 4.1. Partial Inductance Matrix of the Conductors in an Eccentric IM

- A pure static eccentricity (SE) is characterized (Figure 8) by a displacement of the axis of rotation of the rotor (${O}_{\theta}$) with respect to the geometric center of the stator (${O}_{s}$). The axis of rotation of rotor ${O}_{\theta}$ coincides with the geometric center of the rotor. It can be caused by misalignments of the mounted bearings or of the bearing plates. The rotor is not centered with the stator bore, but it rotates around its own geometric centre: that is, ${\Theta}_{r}$ = constant in Figure 7. The air gap length is non uniform, but its shape does not change when the rotor turns (Figure 8).

- A pure dynamic eccentricity (DE) is characterized (Figure 9) by a displacement of the geometric centre of the rotor (${O}_{r}$) from its axis of rotation (${O}_{\theta}$), which coincides with the axis of the stator bore (${O}_{s}$). It can be caused by a manufacturing defect, a bent shaft, bearings defects, etc. Under DE, the center of the rotor rotates along a circular path in Figure 7, with the same speed as the rotor. In this case, the position of the minimum air gap rotates with the rotor (Figure 9)
- A mixed eccentricity fault (ME) consists of the simultaneous presence of SE and DE (Figure 10). In this case, the axis of rotation (${O}_{\theta}$ in Figure 10) is displaced both from the geometric center of the stator (${O}_{s}$), as in the case of pure static eccentricity, and from the centre of the rotor (${O}_{r}$), as in the case of pure dynamic eccentricity.

#### 4.2. Simplified Formulation of the Partial Inductance between Conductors in Case of Rotor Eccentricity with the Conformal Winding Tensor Approach

## 5. Numerical Validation

## 6. Experimental Validation

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Commercial IM

## Appendix B. Computer Features

## Appendix C. Current Clamp

## References

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**Figure 1.**Reference frame of the IM with the axes of all stator windings (${s}_{1},{s}_{2},\dots ,{s}_{{n}_{s}}$) rigidly connected to the stator iron and conductors and those of the rotor windings (${r}_{1},{r}_{2},\dots ,{r}_{{n}_{r}}$) rigidly connected to the rotor iron and conductors.

**Figure 2.**Dynamical model that implements (2) and (4) in Simulink. This model has three input ports: (1) the voltage vector $\mathit{e}$, (2) the applied shaft torque T, and (3) the moment of inertia of the rotor J. It has also three output ports: (1) the current vector $\mathit{i}$, (2) the rotor speed $\dot{\theta}$, and (3) the rotor angular position $\theta $.

**Figure 3.**IM network. The three stator phases (

**left**) have a delta connection, and each one has resistance ${R}_{s}$ and leakage inductance ${L}_{\sigma s}$. The rotor cage (

**right**) has ${n}_{b}$ bars. Each rotor loop consists in two consecutive bars, each one with resistance ${R}_{b}$ and leakage inductance ${L}_{\sigma b}$. The bars are connected trough end ring segments, each one with resistance ${R}_{e}$ and leakage inductance ${L}_{\sigma e}$. The self and mutual inductances of the windings are not represented in this circuit.

**Figure 4.**Primitive IM network, found by removing all interconnections between the windings and short circuiting each. The arrows show the mutual impedances between stator windings and cage bars. The end ring segments do not couple with the other windings through mutual impedances.

**Figure 6.**Elementary conductors placed in the air gap that constitute the primitive spatial network of the IM. These N conductors are considered to be disconnected, and their currents are considered to be independent variables.

**Figure 7.**Air gap length $g\left(\phi \right)$ of an eccentric machine as a function of the angular coordinate $\phi $, which depends on the position of the rotor centre ${O}_{r}$ with respect to the stator centre ${O}_{s}$.

**Figure 8.**Pure static eccentricity. Relative position of a rotor conductor, A, and a stator conductor, B, when the rotor turns an angle ${\theta}_{r}\left(t\right)$ (

**right**) from the initial line (

**left**), in the case of SE. The minimum air gap length is always located at the position of the stator conductor B.

**Figure 9.**Pure dynamic eccentricity. Relative position of a rotor conductor, A, and a stator conductor, B, when the rotor turns an angle ${\theta}_{r}\left(t\right)$ (

**right**) from the initial line (

**left**), in the case of DE. The minimum air-gap length is always located at the position of the rotor conductor A.

**Figure 10.**Position of the rotor centre (${O}_{r}$), the stator centre (${O}_{s}$), and the axis of rotation (${O}_{\theta}$) in case of an IM witth ME eccentricity (${\delta}_{r}$), as a geometric combination of static (${\delta}_{s}$) and dynamic (${\delta}_{d}$) eccentricity. ${\theta}_{r}$ represents the angle of rotation of the rotor.

**Figure 11.**The Moebius transformation of an eccentric IM with non-uniform air gap (

**left**) gives a non-eccentric IM with a uniform air gap (

**right**), but with modified conductor angular positions and with a different rotor radius.

**Figure 12.**Moebius transformation of the elementary conductors placed in the air gap that constitute the primitive spatial network of the IM, given in Figure 6. The original set of N equally spaced elementary conductors placed in the non-uniform air gap of the eccentric IM (

**left**), becomes a set of N elementary conductors with non-uniform spacing in the smooth air gap of the transformed IM (

**right**).

**Figure 13.**Number and direction of the conductors per air gap interval of a stator winding (

**top**) and a rotor bar (

**bottom**).

**Figure 14.**Mutual inductance between an elementary conductor placed at the origin and an elementary conductor placed at a given angular coordinate $\phi $ for the IM given in Appendix A, without eccentricity. This corresponds to the first column of matrix ${\mathit{L}}_{\mathit{c}\mathit{\mu}}$ (13).

**Figure 15.**FEM simulation of the IM of Appendix A for a mixed eccentricity with a static eccentricity degree of 40% (${\delta}_{s}$ = 0.4) and a dynamic eccentricity degree of 20% (${\delta}_{d}$ = 0.2), with only the first stator phase fed with a 1 A constant current.

**Figure 16.**FEM simulation of the IM of Appendix A for a mixed eccentricity with a static eccentricity degree of 40% (${\delta}_{s}$ = 0.4) and a dynamic eccentricity degree of 20% (${\delta}_{d}$ = 0.2), with only the first rotor phase fed with a 1 A constant current.

**Figure 17.**Mutual inductance between the first stator and the first rotor phase (${L}_{{s}_{1}{r}_{1}}$,

**top row**) and self-inductances of the first rotor phase (${L}_{{r}_{1}{r}_{1}}$,

**middle row**) and of the first stator phase (${L}_{{s}_{1}{s}_{1}}$,

**bottom row**), for the IM of Appendix A, with three different degrees of static (${\delta}_{s}$) and dynamic (${\delta}_{d}$) eccentricity: ( ${\delta}_{s}$ = 0.2, ${\delta}_{d}$ = 0.2), (${\delta}_{s}$ = 0.2, ${\delta}_{d}$ = 0.4), and (${\delta}_{s}$ = 0.4, ${\delta}_{d}$ = 0.2). The case of healthy machine (${\delta}_{s}$ = 0.0, ${\delta}_{d}$ = 0.0) has also been included for comparative purposes. The first column presents the results obtained with the conformal winding tensor approach, the second column contains the results obtained with FEA, and the third column contains the errors between both approaches.

**Figure 18.**Mutual inductance between the first stator and the first rotor phase (${L}_{{s}_{1}{r}_{1}}$,

**top row**) and self-inductances of the first rotor phase (${L}_{{r}_{1}{r}_{1}}$,

**middle row**) and of the first stator phase (${L}_{{r}_{1}{r}_{1}}$,

**bottom row**) for the IM of Appendix A, with three different degrees of static (${\delta}_{s}$) eccentricity: ${\delta}_{s}$ = 0.2, ${\delta}_{s}$ = 0.4, and ${\delta}_{s}$ = 0.6. The case of healthy machine ${\delta}_{s}$ = 0.0 has also been included for comparative purposes. The first column presents the results obtained with the conformal winding tensor approach, the second column presents the results obtained with FEA, and the third column presents the errors between both approaches.

**Figure 19.**Mutual inductance between the first stator and the first rotor phase (${L}_{{s}_{1}{r}_{1}}$,

**top row**) and self-inductances of the first rotor phase (${L}_{{r}_{1}{r}_{1}}$,

**middle row**) and of the first stator phase (${L}_{{s}_{1}{s}_{1}}$,

**bottom row**) for the IM of Appendix A with three different degrees of dynamic (${\delta}_{d}$) eccentricity: ${\delta}_{d}$ = 0.2, ${\delta}_{d}$ = 0.4, and ${\delta}_{d}$ = 0.6. The case of healthy machine ${\delta}_{d}$ = 0.0 has also been included for comparative purposes. The first column presents the results obtained with the conformal winding tensor approach, the second column presents the results obtained with FEA, and the third column presents the errors between both approaches.

**Figure 20.**Mutual inductance between the first stator and the first rotor phase (${L}_{{s}_{1}{r}_{1}}$,

**top row**) and self-inductances of the first rotor phase (${L}_{{r}_{1}{r}_{1}}$,

**middle row**) and of the first stator phase (${L}_{{s}_{1}{s}_{1}}$,

**bottom row**), for the IM of Appendix A with three different types of eccentricity: mixed eccentricity (${\delta}_{s}$ = 0.4, ${\delta}_{d}$ = 0.2) in the first column, pure static eccentricity (${\delta}_{s}=0.6$) in the second column, and pure dynamic eccentricity (${\delta}_{d}=0.6$) in the third column. The results obtained with the conformal winding approach and with FEA have been plotted together for comparison purposes.

**Figure 21.**Test rig used for the experimental validation of the proposed approach. Two motors of the same type as the simulated one, labelled as (1) and (2) in the

**right**,

**bottom**part of the Figure, have been experimentally tested. To avoid the influence of the coupling on the eccentricity measurement, both motors have been tested uncoupled and powered directly form the mains, as shown in the schema (

**left**). The current has been recorded using a Chauvin Arnoux MN60 current probe (see Appendix B) and a Yokogawa DL750 ScopeCorder (

**right**,

**top**), at a rate of 10 kHz during an acquisition time of 100 s, to achieve a 0.01 Hz resolution in the current spectrum. The registered data have been stored and processed with the computer platform given in Appendix C.

**Figure 22.**Spectra of the currents of the two tested motors, both of them of the type described in Appendix A. These spectra show the fault harmonics of an incipient-mixed eccentricity fault (marked in the figure), with a low level (around −50 dB), which may be produced by inherent and unavoidable manufacturing defects.

**Figure 23.**Spectra of the simulated currents of the motor described in Appendix A, for different degrees of mixed eccentricity (static eccentricity ${\delta}_{s}$, dynamic eccentricity ${\delta}_{d}$). Top: healthy motor (${\delta}_{s}=0$, ${\delta}_{d}=0$). Below, from top to bottom, increasing mixed eccentricity faults (${\delta}_{s}=0.05$, ${\delta}_{d}=0.05$), (${\delta}_{s}=0.1$, ${\delta}_{d}=0.05$), (${\delta}_{s}=0.05$, ${\delta}_{d}=0.1$), and (${\delta}_{s}=0.1$, ${\delta}_{d}=0.1$). This last spectrum displays the fault harmonics closest to the measured ones in Figure 22, which is compatible with a degree of mixed eccentricity fault with (${\delta}_{s}=0.1$, ${\delta}_{d}=0.1$).

Degree of Eccentricity | ||||
---|---|---|---|---|

Static ${\mathit{\delta}}_{\mathit{s}}$ | Dynamic ${\mathit{\delta}}_{\mathit{d}}$ | Error ${\mathit{L}}_{{\mathit{s}}_{1}{\mathit{r}}_{1}}$ | Error ${\mathit{L}}_{{\mathit{r}}_{1}{\mathit{r}}_{1}}$ | Error ${\mathit{L}}_{{\mathit{s}}_{1}{\mathit{s}}_{1}}$ |

0.0 | 0.0 | 1.10 × 10^{−6} | 2.84 × 10^{−9} | 3.69 × 10^{−5} |

0.0 | 0.2 | 1.39 × 10^{−6} | 3.45 × 10^{−9} | 3.86 × 10^{−5} |

0.0 | 0.4 | 1.84 × 10^{−6} | 5.20 × 10^{−9} | 6.80 × 10^{−5} |

0.0 | 0.6 | 2.65 × 10^{−6} | 2.60 × 10^{−9} | 2.46 × 10^{−4} |

0.2 | 0.0 | 1.13 × 10^{−6} | 4.54 × 10^{−9} | 3.85 × 10^{−5} |

0.2 | 0.2 | 1.48 × 10^{−6} | 5.20 × 10^{−9} | 4.24 × 10^{−5} |

0.2 | 0.4 | 2.06 × 10^{−6} | 6.83 × 10^{−9} | 1.20 × 10^{−4} |

0.4 | 0.0 | 1.29 × 10^{−6} | 7.94 × 10^{−9} | 4.38 × 10^{−5} |

0.4 | 0.2 | 1.85 × 10^{−6} | 9.27 × 10^{−9} | 8.27 × 10^{−5} |

0.6 | 0.0 | 1.82 × 10^{−6} | 1.48 × 10^{−8} | 5.29 × 10^{−5} |

**Table 2.**Amplitude of the fault harmonics corresponding to the experimental tests and the simulated motor conditions.

Motor | Eccentricity Degree | Amplitude of the Fault Harmonics | ||
---|---|---|---|---|

Static ${\mathit{\delta}}_{\mathit{s}}$ | Dynamic ${\mathit{\delta}}_{\mathit{d}}$ | ${\mathit{f}}_{1}-{\mathit{f}}_{\mathit{r}}=25.2$ Hz | ${\mathit{f}}_{1}+{\mathit{f}}_{\mathit{r}}=75$ Hz | |

Motor 1 | Unknown | Unknown | −50.11 dB | −48.94 dB |

Motor 2 | Unknown | Unknown | −51.17 dB | −54.49 dB |

Simulated | 0 | 0 | <−100 dB | <−100 dB |

0.05 | 0.05 | −62.31 dB | −62.86 dB | |

0.1 | 0.05 | −56.32 dB | −56.68 dB | |

0.05 | 0.1 | −56.32 dB | −56.68 dB | |

0.1 | 0.1 | −50.29 dB | −50.55 dB |

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**MDPI and ACS Style**

Terron-Santiago, C.; Martinez-Roman, J.; Puche-Panadero, R.; Sapena-Bano, A.; Burriel-Valencia, J.; Pineda-Sanchez, M.
Analytical Model of Eccentric Induction Machines Using the Conformal Winding Tensor Approach. *Sensors* **2022**, *22*, 3150.
https://doi.org/10.3390/s22093150

**AMA Style**

Terron-Santiago C, Martinez-Roman J, Puche-Panadero R, Sapena-Bano A, Burriel-Valencia J, Pineda-Sanchez M.
Analytical Model of Eccentric Induction Machines Using the Conformal Winding Tensor Approach. *Sensors*. 2022; 22(9):3150.
https://doi.org/10.3390/s22093150

**Chicago/Turabian Style**

Terron-Santiago, Carla, Javier Martinez-Roman, Ruben Puche-Panadero, Angel Sapena-Bano, Jordi Burriel-Valencia, and Manuel Pineda-Sanchez.
2022. "Analytical Model of Eccentric Induction Machines Using the Conformal Winding Tensor Approach" *Sensors* 22, no. 9: 3150.
https://doi.org/10.3390/s22093150