transformer engineering design and practice 1_phần 7

5 Stray Losses in Structural Components The previous chapter covered the theory and fundamentals of eddy currents. It also covered in detail, the estimation and reduction of stray losses in windings, viz., eddy loss and circulating current loss. This chapter covers estimation of remaining stray losses, which predominantly consist of stray losses in structural components. Various countermeasures required for the reduction of these stray losses and elimination of hot spots are discussed. The stray loss problem becomes increasingly important with growing transformer ratings. Ratings of generator transformers and interconnecting auto-transformers are steadily increasing over last few decades. Stray losses of such large units can be appreciably high, which can result in higher temperature rise, affecting their life. This problem is particularly severe in the case of large auto-transformers, where actual impedance on equivalent two-winding rating is higher giving a very high value of stray leakage field. In the case of large generator transformers and furnace transformers, stray loss due to high current carrying leads can become excessive, causing hot spots. To become competitive in the global marketplace it is necessary to optimize material cost, which usually leads to reduction in overall size of the transformer as a result of reduction in electrical and magnetic clearances. This has the effect of further increasing stray losses if effective shielding measures are not implemented. Size of a large power transformer is also limited by transportation constraints. Hence, the magnitude of stray field incident on the structural parts increases much faster with growing rating of transformers. It is very important for a transformer designer to know and estimate accurately all the stray loss components because each kW of load loss may be capitalized by users from US$750 to US$2500. In large transformers, a reduction of stray loss by even 3 to 5 kW can give a competitive advantage. 169 Copyright © 2004 by Marcel Dekker, Inc. 170 Chapter 5 Stray losses in structural components may form a large part (>20%) of the total load loss if not evaluated and controlled properly. A major part of stray losses occurs in structural parts with a large area (e.g., tank). Due to inadequate shielding of these parts, stray losses may increase the load loss of the transformer substantially, impairing its efficiency. It is important to note that the stray loss in some clamping elements with smaller area (e.g., flitch plate) is lower, but the incident field on them can be quite high leading to unacceptable local high temperature rise seriously affecting the life of the transformer. Till 1980, a lot of work was done in the area of stray loss evaluation by analytical methods. These methods have certain limitations and cannot be applied to complex geometries. With the fast development of numerical methods such as Finite Element Method (FEM), calculation of eddy loss in various metallic components of the transformer is now easier and less complicated. Some of the complex 3-D problems when solved by using 2-D formulations (with major approximations) lead to significant inaccuracies. Developments of commercial 3-D FEM software packages since 1990 have enabled designers to simulate the complex electromagnetic structure of transformers for control of stray loss and elimination of hot spots. However, FEM analysis may require considerable amount of time and efforts. Hence, wherever possible, a transformer designer would prefer fast analysis with sufficient accuracy so as to enable him to decide on various countermeasures for stray loss reduction. It may be preferable, for regular design use, to calculate some of the stray loss components by analytical/hybrid (analytically numerical) methods or by some formulae derived on the basis of one-time detailed analysis. Thus, the method of calculation of stray losses should be judiciously selected; wherever possible, the designer should be given equations/curves or analytical computer programs providing a quick and reasonably accurate calculation. Computation of stray losses is not a simple task because the transformer is a highly asymmetrical and three-dimensional structure. The computation is complicated by - magnetic non-linearity - difficulty in quick and accurate computation of stray field and its effects - inability in isolating exact stray loss components from tested load loss values - limitations of experimental verification methods for large power transformers Stray losses in various clamping structures (frame, flitch plate, etc.) and the tank due to the leakage field emanating from windings and due to the field of high current carrying leads are discussed in this chapter. The methods used for estimation of these losses are compared. The effectiveness of various methods used for stray loss control is discussed. Some interesting phenomena observed during three-phase and single-phase load loss tests are also reported. Copyright © 2004 by Marcel Dekker, Inc. Stray Losses in Structural Components 171 5.1 Factors Influencing Stray Losses With the increase in ratings of transformers, the proportion of stray losses in the load loss may increase significantly. These losses in structural components may exceed the stray losses in windings in large power transformers (especially autotransformers). A major portion of these stray losses occurs in structural components with a large area (e.g., tank) and core clamping elements (e.g., frames). The high magnitude of stray flux usually does not permit designers to disregard the non-linear magnetic characteristics of steel elements. Stray losses in structural steel components depend in a very complicated manner on the parameters such as the magnitude of stray flux, frequency, resistivity, type of excitation, etc. In the absence of hysteresis and non-linearity of magnetic characteristics, the expression for the eddy loss per unit surface area of a plate, subjected to (on one of its surfaces) a magnetic field of r.m.s. value (Hrms), has been derived in Chapter 4 as (5.1) Hence, the total power loss in a steel plate with a permeability µs can be given in terms of the peak value of the field (H0) as (5.2) This equation assumes a constant permeability. It is necessary to take into account the non-linear magnetic saturation effect in structural steel parts because their surfaces are often saturated due to the skin effect. Non-linearity of magnetic characteristics can be taken into account by a linearization coefficient as explained in Section 4.4. Thus, the total power loss with the consideration of non-linear characteristics can be given by (5.3) The term al in the above equation is the linearization coefficient. Equation 5.3 is applicable to a simple geometry of a plate excited by a tangential field on one of its sides. It assumes that the plate thickness is sufficiently larger than the depth of penetration (skin depth) so that it becomes a case of infinite half space. For magnetic steel, as discussed in Section 4.4, the linearization coefficient has been taken as 1.4 in [1]. For a non-magnetic steel, the value of the coefficient is 1(i.e.,al=1). Copyright © 2004 by Marcel Dekker, Inc. 172 Chapter 5 Figure 5.1 Types of excitation 5.1.1 Type of surface excitation In transformers, there are predominantly two kinds of surface excitation as shown in figure 5.1. In case (a), the incident field is tangential (e.g., bushing mounting plate). In this case, the incident tangential field is directly proportional to the source current since the strength of the magnetic field (H) on the plate surface can be determined approximately by the principle of superposition [2]. In case (b), for estimation of stray losses in the tank due to a leakage field incident on it, only the normal (radial) component of the incident field (φ) can be considered as proportional to the source current. The relationship between the source current and the tangential field component is much more complicated. In many analytical formulations, the loss is calculated based on the tangential components (two orthogonal components in the plane of plate), which need to be evaluated from the normal component of the incident field with the help of Maxwell’s equations. The estimated values of these two tangential field components can be used to find the resultant tangential component and thereafter the tank loss as per equation 5.3. Let us use the theory of eddy currents described in Chapter 4 to analyze the effect of different types of excitation on the stray loss magnitude and distribution. Consider a structural component as shown in figure 5.2 (similar to that of a winding conductor of figure 4.5) which is placed in an alternating magnetic field in the y direction having peak amplitudes of H1 and H2 at its two surfaces. The structural component can be assumed to be infinitely long in the x direction. Further, it can be assumed that the current density Jx and magnetic field intensity Hy are functions of z only. Proceeding in a way similar to that in Section 4.3 and assuming that the structural component has linear magnetic characteristics, the diffusion equation is given by Copyright © 2004 by Marcel Dekker, Inc. Stray Losses in Structural Components 173 Figure 5.2 Stray loss in a structural component (5.4) The solution of this equation is Hy=C1eγz+C2e-γz (5.5) where γ is propagation constant given by equation 4.39, viz. γ=(1+j)/δ, δ being the depth of penetration or skin depth. Now, for the present case the boundary conditions are Hy=H1 at z=+b and Hy=H2 at z=-b (5.6) Using these boundary conditions, we can get expressions for the constants as (5.7) Substituting these values of constants back into equation 5.5 we get (5.8) Since ∇×H=J and J=σE, and only the y component of H and x component of J are non-zero we get (5.9) (5.10) Copyright © 2004 by Marcel Dekker, Inc. ... - tailieumienphi.vn