Mathematical Problems in Engineering

Volume 2016, Article ID 5284815, 13 pages

http://dx.doi.org/10.1155/2016/5284815

## Impact of Different Static Air-Gap Eccentricity Forms on Rotor UMP of Turbogenerator

Department of Mechanical Engineering, North China Electric Power University, Baoding 071003, China

Received 11 May 2016; Accepted 16 June 2016

Academic Editor: Zhike Peng

Copyright © 2016 Yu-Ling He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Theoretical analysis and numerical FEM calculations, together with segmental experiment studies, are used to study the impact of the static air-gap eccentricity forms on the rotor unbalanced magnetic pull (UMP) of turbogenerator. The universal expression of the magnetic flux density under different forms of SAGE is firstly deduced, based on which the detailed UMP formulas for the normal condition and three SAGE cases are obtained, respectively. Then the exciting characteristics of the UMP for each SAGE form to generate vibrations are analyzed. Finally, numerical FEM calculations and segmental experiments are carried out to investigate the effect of SAGE forms on the rotor UMP, taking the SDF-9 type non-salient-pole fault simulating generator as the object. It is shown that, no matter what kind of SAGE occurs, amplitude increments at each even harmonic component of the UMP and the rotor vibration, especially the 2nd harmonic component, will be brought in. Meanwhile, the UMP keeps directing to the very position where the minimum radial air-gap is. Among the different SAGE forms, the rotor offset has the most sensitive effect on the rotor UMP and vibration, while the stator ellipse deformation has the weakest impact.

#### 1. Introduction

Due to the assembly quality, the bearing damage, and the performing environments, most generators are running under an air-gap eccentricity condition [1]. The air-gap eccentricity, which is also usually named as rotor eccentricity by many scholars, appears as the air-gap is larger on one side but meanwhile smaller on the other side. For example, the bearing offset or the stator core deformation can cause a typical static air-gap eccentricity [2]. A very tiny air-gap eccentricity will not bring in serious impact on the generator’s regular performing. However, when the eccentricity degree is more than 10% of the total air-gap length, severe vibrations, stator core deformations, and even winding damage will be caused [3]. Therefore, accurate monitoring and timely control on this kind of fault is of significance.

In most cases, the air-gap eccentricity is usually mentioned in the radial direction and can be divided into three types, that is, the static air-gap eccentricity (SAGE), the dynamic air-gap eccentricity (DAGE), and the mixed air-gap eccentricity (MAGE) [4, 5]. In this paper, we mainly focus on the static one.

By far, achievements of the monitoring and diagnosis on the air-gap eccentricity fault primarily focus on the stator current [6] and voltage [7, 8], the rotor current [8] and the shaft voltage [9], the inductance variation of the windings [10, 11], and the rotor UMP and vibration analysis [12–14]. The inductance variation is mainly based on the winding function theory [15] and the improved winding function theory [16, 17] and needs a large amount of calculation, while the current and the voltage analysis is actually based on the harmonic changes of the magnetic flux density [6]. The direct analysis on the spectrum of the stator and rotor current or voltage obtained via Fourier transform can sometimes hardly exactly identify the eccentricity due to the inconspicuous amplitude change comparing with the noise signal magnitude, especially when the capacity of the generator is not so large or the eccentricity is not so severe. To overcome this disadvantage, scholars developed an improved method based on the search coil [18]. However, this method needs to install extra components, that is, the search coil, in the generator, which requires a higher cost and is not welcomed by the practical performers in the power plant because it has to stop the generator for the specific installation work. Comparatively, the method based on the rotor vibration is more convenient because most generators have already installed vibration sensors before the generator is put into operation. Even if extra vibration sensors are needed to be installed in addition, there is no need to stop the generator but just to fix the sensors at the bearing block. Therefore, many researchers have paid much attention to the rotor UMP [19, 20] which is the essential cause of the rotor vibration and the rotor dynamics with UMP [21, 22].

However, people have only considered the eccentricity condition caused by the rotor side such as the rotor offset or asymmetry, while other potential causes are mostly ignored. That is why air-gap eccentricity is usually also named as rotor eccentricity by many scholars. Taking SAGE as an example, scholars primarily focus on the rotor offset form [23, 24], while the stator deformation conditions are not taken into account. Actually, since the stator has radial vibrations at ( is the electrical frequency) even in normal condition [25], the hollow stator core which is composed of lots of fan-shape silicon steel sheets and fixed by double-screw bolts and frames will probably have a concave deformation or a convex deformation after a long suffering from the continuous vibration and the magnetic pull [26]. Specifically, due to the slot harmonic effect on the magnetic pull [27] and the elliptical rotating magnetomotive force caused by the asymmetric 3-phase currents [28], the stator will be dragged into an ellipse form after a long period performance [27, 28]. Then another problem appears. What is the effect of different eccentricity forms on the rotor UMP? Which kind of air-gap eccentricity has the most sensible impact on the rotor UMP?

The intent of this paper is to investigate the impact of SAGE forms, including rotor offset, stator concave deformation, stator convex deformation, stator ellipse deformation, and mixed SAGE composed of rotor offset and stator deformation, on the rotor UMP characteristics. Also, the sensitivity of these SAGE forms acting on the UMP is studied.

#### 2. Theoretical Analysis

##### 2.1. Magnetic Flux Density (MFD) Analysis

According to [2], the air-gap magnetomotive force (MMF) in normal condition indicated in Figure 1 can be expressed aswhere , , and are, respectively, the rotor MMF, the stator MMF, and the composite MMF at the fundamental frequency, is the rotor mechanical angular frequency, is the rotating frequency of rotor (for turbogenerator, equals the electrical frequency , and hereafter, we write as and as for short), is the mechanical angle to indicate the circumferential location of the air-gap, and is the internal power angle of the generator.