transformer engineering design and practice 1_phần 6 doc

127

4

Eddy Currents and Winding Stray

Losses

The load loss of a transformer consists of losses due to ohmic resistance of

windings (I

2

R losses) and some additional losses. These additional losses are

generally known as stray losses, which occur due to leakage field of windings and

field of high current carrying leads/bus-bars. The stray losses in the windings are

further classified as eddy loss and circulating current loss. The other stray losses

occur in structural steel parts. There is always some amount of leakage field in all

types of transformers, and in large power transformers (limited in size due to

transport and space restrictions) the stray field strength increases with growing

rating much faster than in smaller transformers. The stray flux impinging on

conducting parts (winding conductors and structural components) gives rise to

eddy currents in them. The stray losses in windings can be substantially high in

large transformers if conductor dimensions and transposition methods are not

chosen properly.

Today’s designer faces challenges like higher loss capitalization and optimum

performance requirements. In addition, there could be constraints on dimensions

and weight of the transformer which is to be designed. If the designer lowers

current density to reduce the DC resistance copper loss (I2

R loss), the eddy loss in

windings increases due to increase in conductor dimensions. Hence, the winding

conductor is usually subdivided with a proper transposition method to minimize

the stray losses in windings.

In order to accurately estimate and control the stray losses in windings and

structural parts, in-depth understanding of the fundamentals of eddy currents

starting from basics of electromagnetic fields is desirable. The fundamentals are

described in first few sections of this chapter. The eddy loss and circulating

current loss in windings are analyzed in subsequent sections. Methods for

Copyright © 2004 by Marcel Dekker, Inc.

128 Chapter 4

evaluation and control of these two losses are also described. Remaining

components of stray losses, mostly the losses in structural components, are dealt

with in Chapter 5.

4.1 Field Equations

The differential forms of Maxwell’s equations, valid for static as well as time

dependent fields and also valid for free space as well as material bodies are:

(4.1)

(4.2)

(4.3)

(4.4)

where H=magnetic field strength (A/m)

E=electric field strength (V/m)

B=flux density (wb/m2

)

J=current density (A/m2

)

D=electric flux density (C/m2

)

ρ=volume charge density (C/m3

)

There are three constitutive relations,

J=σE (4.5)

B=µ H (4.6)

D=ε E (4.7)

where µ=permeability of material (henrys/m)

ε=permittivity of material (farads/m)

σ=conductivity (mhos/m)

The ratio of the conduction current density (J) to the displacement current density

(∂D/∂t) is given by the ratio σ/(jωε), which is very high even for a poor metallic

conductor at very high frequencies (where ω is frequency in rad/sec). Since our

analysis is for the (smaller) power frequency, the displacement current density is

Copyright © 2004 by Marcel Dekker, Inc.

Eddy Currents and Winding Stray Losses 129

neglected for the analysis of eddy currents in conducting parts in transformers

(copper, aluminum, steel, etc.). Hence, equation 4.2 gets simplified to

(4.8)

The principle of conservation of charge gives the point form of the continuity

equation,

(4.9)

In the absence of free electric charges in the present analysis of eddy currents in a

conductor we get

(4.10)

To get the solution, the first-order differential equations 4.1 and 4.8 involving both

H and E are combined to give a second-order equation in H or E as follows.

Taking curl of both sides of equation 4.8 and using equation 4.5 we get

For a constant value of conductivity (σ), using vector algebra the equation can be

simplified as

(4.11)

Using equation 4.6, for linear magnetic characteristics (constant µ) equation 4.3

can be rewritten as

(4.12)

which gives

(4.13)

Using equations 4.1 and 4.13, equation 4.11 gets simplified to

(4.14)

or

(4.15)

Equation 4.15 is a well-known diffusion equation. Now, in the frequency domain,

equation 4.1 can be written as follows:

(4.16)

Copyright © 2004 by Marcel Dekker, Inc.

130 Chapter 4

In above equation, term jω appears because the partial derivative of a sinusoidal

field quantity with respect to time is equivalent to multiplying the corresponding

phasor by jω. Using equation 4.6 we get

(4.17)

Taking curl of both sides of the equation,

(4.18)

Using equation 4.8 we get

(4.19)

Following the steps similar to those used for arriving at the diffusion equation

4.15 and using the fact that (since no free

electric charges are present) we get

(4.20)

Substituting the value of J from equation 4.5,

(4.21)

Now, let us assume that the vector field E has component only along the x axis.

(4.22)

The expansion of the operator ∇ leads to the second-order partial differential

equation,

(4.23)

Suppose, if we further assume that Ex is a function of z only (does not vary with x

and y), then equation 4.23 reduces to the ordinary differential equation

(4.24)

We can write the solution of equation 4.24 as

(4.25)

where Exp is the amplitude factor and γ is the propagation constant, which can be

given in terms of the attenuation constant α and phase constant β as

Copyright © 2004 by Marcel Dekker, Inc.