transformer engineering design and practice 1_phần 7 pptx

169 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. Copyright © 2004 by Marcel Dekker, Inc. Chapter 5170 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 (H rms ), has been derived in Chapter 4 as (5.1) Hence, the total power loss in a steel plate with a permeability µ s can be given in terms of the peak value of the field (H 0 ) 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 a l 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.,a l =1). Copyright © 2004 by Marcel Dekker, Inc. Chapter 5172 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 H 1 and H 2 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 J x and magnetic field intensity H y 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 Figure 5.1 Types of excitation Copyright © 2004 by Marcel Dekker, Inc. Stray Losses in Structural Components 173 (5.4) The solution of this equation is Hy=C 1 e γz +C 2 e -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 H y =H 1 at z=+b and H y =H 2 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) Figure 5.2 Stray loss in a structural component Copyright © 2004 by Marcel Dekker, Inc. Chapter 5174 In terms of complex vectors, the (time average) power flow per unit area of the plate (in the x-y plane) can be calculated with the help of Poynting’s theorem [3]: (5.11) Substituting the values of H y and E x from equations 5.8 and 5.10, the value of eddy loss per unit area of the plate can be calculated. Figure 5.3 shows the plot of the normalized value of eddy loss, P/(H 2 /2σδ), versus the normalised plate thickness (2b/δ) for three different cases of the tangential surface excitation. Case 1 (H 1 =H and H 2 =0): As expected, the eddy loss for this case decreases with the increase in plate thickness until the thickness becomes 1 to 2 times the skin depth. This situation resembles the case in a transformer when a current carrying conductor is placed parallel to a conducting plate (mild steel tank/ pocket). For this case (see figure 5.3), the normalised active power approaches unity as the thickness and hence the ratio 2b/δ increases. This is because it becomes a case similar to an infinite half space, where the power loss equals H 2 /(2σδ). It is to be remembered that H, H 1 and H 2 denote peak values. Figure 5.3 Eddy Loss in a structural plate for different surface excitations Copyright © 2004 by Marcel Dekker, Inc. Stray Losses in Structural Components 175 The plot also shows that the active power loss is very high for a thin plate. A qualitative explanation for this phenomenon can be given with reference to figure 5.4 (a). Consider a contour C shown in the figure. By applying Ampere’s circuital law on the contour we obtain (5.12) Noting that H is only in the y direction with H 1 =H and H 2 =0, the equation simplifies to HL=I As the thickness 2b decreases, the same amount of current passes through a smaller cross section of the plate and thus through a higher resistance, resulting in more loss. Case 2 (H 1 =H 2 =H): Here, the eddy loss increases with the increase in the plate thickness. This situation arises in lead terminations/bushing mounting plates, where a current passes through holes in the metallic plates. In this case, as the thickness increases, normalized active power loss approaches the value of 2 because, for 2b/ δ >>1, the problem reduces to that of two infinite half-spaces, each excited by the peak value of field (H) on their surfaces. Therefore, the total loss adds up to 2 per-unit. As the thickness decreases, the active power loss decreases in contrast with Case 1. As shown in figure 5.4 (b), the currents in two halves of the plate are in opposite directions (as forced by the boundary conditions of H 1 and H 2 ). For a sufficiently small thickness, the effects of these two currents tend to cancel each other reducing the loss to zero. Case 3 (H 1 =-H 2 =H): Here, the eddy loss decreases with the increase in thickness. For very high thickness (much greater than the skin depth), the loss approaches the value corresponding to two infinite half-spaces, i.e., H 2 /( σδ ). As the thickness decreases, the power loss approaches very high values. For the representation Figure 5.4 Explanation for curves in Figure 5.3 Copyright © 2004 by Marcel Dekker, Inc. Chapter 5176 given in figure 5.4 (c), an explanation similar to that for Case 1 can be given. The application of Ampere’s circuital law gives double the value of current (i.e., 2HL=I) as compared to Case 1. Hence, as the thickness (2b) decreases, the current has to pass through a smaller cross section of the plate and thus through a higher resistance causing more loss. In the previous three cases, it is assumed that the incident magnetic field intensity is tangential to the surface of a structural component (e.g., bushing mounting plate). If the field is incident radially, the behavior of stray loss is different. Based on a number of 2-D FEM simulations involving a configuration in which the leakage field from the windings is radially incident on a structural component (e.g., tank or flitch plate), the typical curves are presented in figure 5.5. The figure gives the variation of loss in a structural component as the thickness is increased, for three different types of material: magnetic steel, non- magnetic steel and aluminum. The curves are similar to those given in [4] wherein a general formulation is given for the estimation of losses in a structural component for any kind of spatial distribution of the incident magnetic field. Let us now analyse the graphs of three different types of materials given in figure 5.5. Figure 5.5 Loss in different materials for radial excitation Copyright © 2004 by Marcel Dekker, Inc. Stray Losses in Structural Components 177 1) Magnetic steel: One can assume that the magnetic steel plate is saturated due to its small skin depth. Hence, the value of relative permeability corresponding to the saturation condition is taken (µ r =100). With σ =7×10 6 mho/m, we get the value of skin depth as 2.69 mm at 50 Hz. It can be seen from the graph that the power loss value reaches a maximum in about two skin depths and thereafter remains constant. This behavior is in line with the theory of eddy currents and skin depth elaborated in Chapter 4. Since eddy currents and losses are concentrated at the surface only, increasing the plate thickness beyond few skin depths does not change the effective resistance offered to the eddy currents and hence the loss remains constant (at a value which is governed by equation 4.74). 2) Aluminum: In case of aluminum with µ r =1 and σ =29×10 6 mho/m, the skin depth at 50 Hz is 13.2 mm. It can be observed from the graph that the loss first increases with thickness and then reduces. The phenomenon can be analyzed qualitatively from the supply end as an equivalent resistive-inductive circuit. For small thickness (thin plates), it becomes a case of resistance-limited behavior (as discussed in Section 4.5.1) and the effective resistance is larger compared to the inductance. Hence, the equivalent circuit behaves as a predominantly resistive circuit, for which the loss can be given as P=(V 2 /R), where V is the supply voltage. An increase of the thickness of the aluminum plate leads to a decrease of resistance, due to the increased cross section available for the eddy-currents, and hence the loss increases. This is reflected in a near-linear increase in losses with the increase of plate thickness. Upon further increase of the plate thickness, the resistance continues to decrease while the inductance gradually increases, and the circuit behavior changes gradually from that of a purely resistive one to that of a series R–L circuit. The power loss undergoes a peak, and starts to decrease as the circuit becomes more inductive. Finally, when the thickness is near or beyond the skin depth, the field and eddy currents are almost entirely governed by the inductive effects (inductance-limited behavior). The field does not penetrate any further when the plate thickness is increased. The equivalent resistance and inductance of the circuit become independent of the increase in the plate thickness. The power loss also approaches a constant value as the thickness increases significantly more than the skin depth making it a case of infinite half space. Since the product ( σ · δ ) is much higher for the aluminum plate than that for the mild steel plate, the constant (minimum) value of loss for the former is much lower (the loss is inversely proportional to the product ( σ · δ ) as per equation 4.74). The curves of aluminum and mild steel intersect at about 3 mm (point A). 3) Non-magnetic stainless steel: For the non-magnetic steel plate, the behavior is similar to that of the aluminum plate, both being non-magnetic materials. The curve is more flat as compared to aluminum as the skin depth of stainless steel is quite high. For a typical grade of stainless steel material with relative permeability of 1 and conductivity of 1.136×10 6 mho/m, the skin depth is 66.78 mm at 50 Hz. Copyright © 2004 by Marcel Dekker, Inc. Chapter 5178 Another difference is that as the thickness is increased, loss approaches a constant value higher than the aluminum plate but lower than the magnetic steel plate since the product ( σ · δ ) for stainless steel lies between that of mild steel and aluminum. The intersection point (B) of the curves for stainless steel and aluminum occurs at about 5 mm and the intersection point (C) of the curves for stainless steel and mild steel occurs at about 10 mm. The location of intersection points depends on the configuration being analyzed and the nature of the incident field. With the increase in the plate thickness, the values of losses in the mild steel (MS), aluminum (AL) and stainless steel (SS) plates stabilize to 12.2 kW/m, 1.5 kW/m and 5.7 kW/m respectively for particular values of currents in the windings. For large thickness, it becomes a case of infinite half space and the three loss values should actually be in proportion to (1/ σδ ) for the same value of tangential component of magnetic field intensity (H 0 ) on the surface of the plate (as per equation 4.74). The magnitude and nature of eddy currents induced in these three types of plates are different, which makes the value of H 0 different for these cases. Also, the value of H 0 is not constant along the surface (as observed from the FEM analysis). Hence, the losses in the three materials are not in the exact proportion of their corresponding ratios (1/ σδ ). Nevertheless, the expected trend is there; the losses follow the relationship (loss) MS >(loss) SS >(loss) AL since (1/ σδ ) MS >(l/ σδ ) SS >(1/ σδ ) AL . A few general conclusions can be drawn based on the above discussion: 1) When a plate made of non-magnetic and highly conductive material (aluminum or copper) is used in the vicinity of field due to high currents or leakage field from windings, it should have thickness at least comparable to its skin depth (13.2 mm for aluminum and 10.3 mm for copper at 50 Hz) to reduce the loss in it to a low value. For the field due to a high current, the minimum value of loss is obtained for a thickness of [5], (5.13) For aluminum (with δ =13.2 mm), we get the value of t min as 20.7 mm at 50 Hz. The ratio t min / δ corresponding to the minimum loss value is 1.57. This agrees with the graph of figure 5.3 corresponding to Case 1 (assuming that tangential field value H 2 ≅0 which is a reasonable assumption for a thickness 50% more than δ ), in which the minimum loss is obtained for the normalized thickness of 1.57. For the case of radial incident field also (figure 5.5), the loss reaches a minimum value at the thickness of about 20 mm. For t<(0.5× δ ), the loss becomes substantial and may lead to overheating of the plate. Hence, if aluminum or copper is used as an electromagnetic (eddy current) shield, then it should have sufficient thickness to eliminate its overheating and minimize the stray loss in the structural component Copyright © 2004 by Marcel Dekker, Inc. [...]... from the interwinding gap as compared to autotransformers Hence, there are more possibilities of hot spots being generated in these parts in generator transformers However, the stray loss magnitude may be of the same order in generator transformers and autotransformers due to more leakage field in autotransformers on equivalent two winding basis In large transformers, the radially incident flux may... magnetic field equations for 2-D Cartesian and Axisymmetric problems is presented, and usefulness of this analogy for numerical calculations has been elaborated The relation between finite element and finite difference methods is also clarified Results of measurement of flux densities and eddy currents on a 150 MVA experimental transformer are reported In [ 17] , a 2D finite element formulation based... large transformers If the stack height of the first step of the core is less than about 12 mm, slitting may have to be done for the next step also The use of a laminated flitch plate for large generator transformers and autotransformers is preferable since it also acts as a magnetic shunt (as described in Section 5.5) The evaluation of exact stray loss in the core poses a challenge to transformer designers... seriously affecting the transformer life There are a variety of flitch plate designs being used in power transformers as shown in figure 5.8 For small transformers, mild steel flitch plate without any slots is generally used because the incident field is not large enough to cause hot spots As the incident field increases in larger transformers, a plate with slots at the top and bottom ends can be used... of the furnace transformer Therefore, for transformers having very high currents on the LV side, magnetic clearances and material of termination structures have to be judiciously selected In most of the cases, it requires 3-D analysis in the absence of data of previous proven designs It should be noted that time-harmonic 2-D and 3-D formulations find the timeharmonic magnetic field in and around current-carrying... current sheet, and the field at any point on the tank is calculated by superimposition of the fields due to all windings The tank loss is calculated using the estimated value of the radial field at each point The analytical formulation in [ 37] determines the field in air without the presence of tank, from the construction of the transformer and the currents in windings Based on this field and the coefficient... intricate formulations, which approximate the three-dimensional transformer geometry to simplify the calculations Transformer designers prefer fast interactive design with sufficient accuracy to enable them to decide the method for reducing tank stray losses Reluctance Network Method [1] can fulfill the requirements of very fast estimation and control of the tank stray loss It is based on a three-dimensional... bolted joint between the tank and cover The currents induced in the tank and cover due to leakage and high current fields, are forced to complete their path through flange bolts The bolts are overheated if the induced currents flowing in the tank and cover are large The flow of these induced currents through the bolts can be avoided by completely isolating them from the tank and cover This results in a... shunts S1 and S3 shielding the curb joint Copyright © 2004 by Marcel Dekker, Inc Stray Losses in Structural Components 1 97 Figure 5. 17 Leakage field plot for bell tank When high current leads pass nearby the curb joint of the tank, excessive heating of bolts may occur resulting in deterioration of gaskets In such cases, either adequate number and size of external links connecting the tank and cover... to 1 According to IEC 61 378 Part-1, 19 97, Transformers for industrial applications, the winding eddy losses are assumed to depend on frequency with the exponent of 2, whereas stray losses in structural parts are assumed to vary with frequency with the exponent of 0.8 The frequency conversion factors for various stray loss components are reported and analyzed in [10] For a transformer subjected to a . been taken as 1. 4 in [1] . For a non-magnetic steel, the value of the coefficient is 1( i.e.,a l =1) . Copyright © 2004 by Marcel Dekker, Inc. Chapter 5 17 2 5 .1. 1 Type of surface excitation In transformers,. watts 1 No slots 12 0 2 1 slot throughout 92 3 3 slots throughout 45 4 7 slots throughout 32 5 1 slot of 400mm length 10 0 6 3 slots of 400mm length 52 7 7 slots of 400mm length 45 Table 5 .1 Loss. material with relative permeability of 1 and conductivity of 1. 136 10 6 mho/m, the skin depth is 66 .78 mm at 50 Hz. Copyright © 2004 by Marcel Dekker, Inc. Chapter 5 17 8 Another difference is that as