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.

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

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

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=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)

Figure 5.2 Stray loss in a structural component

Copyright © 2004 by Marcel Dekker, Inc.

174 Chapter 5

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 Hy and Ex 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/(H2

/2σδ), versus the normalised plate thickness

(2b/δ) for three different cases of the tangential surface excitation.

Case 1 (H1=H and H2=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 H2

/(2σδ). It is to be

remembered that H, H1 and H2 denote peak values.

Figure 5.3 Eddy Loss in a structural plate for different surface excitations

Copyright © 2004 by Marcel Dekker, Inc.