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Chapter 1
INTRODUCTION
Twelve to fifteen percentages of transformer failures are caused by winding deformations provoked by the high electrodynamics forces appearing during short circuits. These geometric vibrations lead to an increase of the winding vibration and, consequently, to an increase of the solid insulation mechanical fatigue. In this way, the isolation can be degraded and short circuit between turns will appear. On the other hand the change in the distance among the conductor implies a change in the series and shunt capacitances, and, thus, the voltage distribution in case of lightning or switching over voltage being different from, that which the transformer was designed to withstand, increasing the risk of failure. Due to the above factors the early detection of winding deformation is needed. Some techniques, such as frequency-response analysis (FRA), or the leakage reactance measurement, are widely used to detect change in transformer geometry, especially winding deformations. These are offline techniques.
Transformer tank vibration is complementary technique to frequency response analysis and leakage reactance measurement having the advantage that it can be used for online monitoring and, thus, avoiding catastrophic failures between successive maintenance outages. Vibration analysis is a key test in rotating machines, predictive maintenance programs and is widely used to detect on load tap changer failures by means of the noise signature analysis during tap regulation.
Chapter 2
FORCES IN THE TRANSFORMER
The main source of vibration is forces appearing in the winding and the core. As the model calculates the fundamental vibration (100 Hz), vibration generated by partial discharge is not taken into account. Winding vibrations are due to electrodynamic forces by the interaction of the current in a winding with leakage flux (Bd). These forces are proportional to the current squared and they have components in the axial and radial directions. Axial forces tend to compress the winding vertically. In a simple case of a two-winding transformer, radial forces tend to compress the internal winding and to expand the external one, since currents in both windings flow in opposite directions. Figure 2.1 shows the direction of radial and axial forces and their relative magnitude depending on the height (Frad F’ax) and the radius (F’rad) of the transformer winding.
Figure 2.1. Forces distribution with in windings
Since vibration depends on the square current and taking into account that current is practically sinusoidal, the winding vibration main harmonic component is that of 100Hz. Some contribution to 100Hz vibration can come due to magnetizing current harmonic or to some residual harmonic currents Fw  i2 (1)
Core vibration is caused by magnetostrication and magnetic forces. Magnetic materials suffer minute changes in their dimensions of a few parts per million, when they are submitted to a magnetic field (magnetostrication). Figure 2.2 shows the relationship between length variation (in %) and iron magnetic flux density. As can be seen, the curve represents hysteresis. Neglecting the hysteresis effect, this curve (plotted in the continuous line in figure 2.2) can be replaced by the idealized curve plotted in the dashed line in figure 2.2.The mathematical expression of the idealized curve can be approximated to a quadratic law, establishing a linear relation between the elongation and the flux density squared.
Figure 2.2. Iron magneostriction as function of induction
Taking into the account the relation between applied voltage and flux density (2) and admitting the elongation is proportional to the forces, the result is magnestostriction forces being proportional to voltage squared
U = 22* f* N* Bs (2)
Fc  U2 (3)
Magnetoristriction forces fundamental frequency is 100 Hz with harmonics being an even multiple of 50 Hz. Higher frequency harmonics are due to the nonlinear character of this phenomenon. Magnetostriction causes vibration only in the core plane in an iron homogenous mass, but the core is made of laminations and the joints between legs and yokes are overlapped. Under these conditions, flux density distribution is irregular because of small vibrations of the gap between legs and yokes laminations and by interlaminar flux in the joints. This irregularity causes other magnetostriction forces to act in a plane perpendicular to the core. Also, laminations have microscopic irregularities and friction among sheets excites other core vibration modes in a direction perpendicular to that of the lamination plane. Other kind of forces appearing tends to minimize the air gap length between legs and yokes and, the energy in the magnetic circuit. The forces are sinusoidal of 100 Hz frequency.