Xem mẫu
4
Eddy Currents and Winding Stray Losses
The load loss of a transformer consists of losses due to ohmic resistance of windings (I2R 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 (I2R 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
127 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=εE (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.
Eddy Currents and Winding Stray Losses 131
γ=α+jβ (4.26)
Substituting the value of Ex from equation 4.25 in equation 4.24 we get
(4.27)
which gives
(4.28)
(4.29)
If the field Ex is incident on a surface of a conductor at z=0 and gets attenuated inside the conductor (z>0), then only the plus sign has to be taken for γ (which is consistent for the case considered).
(4.30)
(4.31)
Substituting ω=2π f we get
(4.32)
Hence,
(4.33)
The electric field intensity (having a component only along the x axis and traveling/penetrating inside the conductor in +z direction) expressed in the complex exponential notation in equation 4.25 becomes
Ex=Expe-γz (4.34)
which in time domain can be written as
Ex=Expe-αzcos(ωt-βz) (4.35) Substituting the values of α and β from equation 4.33 we get
(4.36)
The conductor surface is represented by z=0. Let z>0 and z<0 represent the regions corresponding to the conductor and perfect loss-free dielectric medium
Copyright © 2004 by Marcel Dekker, Inc.
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