transformer engineering design and practice 1_phần 5 docx

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3

Impedance Characteristics

The leakage impedance of a transformer is one of the most important

specifications that has significant impact on its overall design. Leakage

impedance, which consists of resistive and reactive components, has been

introduced and explained in Chapter 1. This chapter focuses on the reactive

component (leakage reactance), whereas Chapters 4 and 5 deal with the resistive

component. The load loss (and hence the effective AC resistance) and leakage

impedance are derived from the results of short circuit test. The leakage reactance

is then calculated from the impedance and resistance (Section 1.5 of Chapter 1).

Since the resistance of a transformer is generally quite less as compared to its

reactance, the latter is almost equal to the leakage impedance. Material cost of the

transformer varies with the change in specified impedance value. Generally, a

particular value of impedance results into a minimum transformer cost. It will be

expensive to design the transformer with impedance below or above this value. If

the impedance is too low, short circuit currents and forces are quite high, which

necessitate use of lower current density thereby increasing the material content.

On the other hand, if the impedance required is too high, it increases the eddy loss

in windings and stray loss in structural parts appreciably resulting into much

higher load loss and winding/oil temperature rise; which again will force the

designer to increase the copper content and/or use extra cooling arrangement. The

percentage impedance, which is specified by transformer users, can be as low as

2% for small distribution transformers and as high as 20% for large power

transformers. Impedance values outside this range are generally specified for

special applications.

Copyright © 2004 by Marcel Dekker, Inc.

78 Chapter 3

3.1 Reactance Calculation

3.1.1 Concentric primary and secondary windings

Transformer is a three-dimensional electromagnetic structure with the leakage

field appreciably different in the core window cross section (figure 3.1 (a)) as

compared to that in the cross section perpendicular to the window (figure 3.1 (b)).

For reactance ( impedance) calculations, however, values can be estimated

reasonably close to test values by considering only the window cross section. A

high level of accuracy of 3-D calculations may not be necessary since the

tolerance on reactance values is generally in the range of ±7.5% or ±10%.

For uniformly distributed ampere-turns along LV and HV windings (having

equal heights), the leakage field is predominantly axial, except at the winding

ends, where there is fringing (since the leakage flux finds a shorter path to return

via yoke or limb). The typical leakage field pattern shown in figure 3.1 (a) can be

replaced by parallel flux lines of equal length (height) as shown in figure 3.2 (a).

The equivalent height (Heq) is obtained by dividing winding height (Hw) by the

Rogowski factor KR (<1.0),

Figure 3.1 Leakage field in a transformer

Copyright © 2004 by Marcel Dekker, Inc.

Impedance Characteristics 79

(3.1)

The leakage magnetomotive (mmf) distribution across the cross section of

windings is of trapezoidal form as shown in figure 3.2 (b). The mmf at any point

depends on the ampere-turns enclosed by a flux contour at that point; it increases

linearly with the ampere-turns from a value of zero at the inside diameter of LV

winding to the maximum value of one per-unit (total ampere-turns of LV or HV

winding) at the outside diameter. In the gap (Tg) between LV and HV windings,

since flux contour at any point encloses full LV (or HV) ampere-turns, the mmf is

of constant value. The mmf starts reducing linearly from the maximum value at

the inside diameter of the HV winding and approaches zero at its outside diameter.

The core is assumed to have infinite permeability requiring no magnetizing mmf,

and hence the primary and secondary mmfs exactly balance each other. The flux

density distribution is of the same form as that of the mmf distribution. Since the

core is assumed to have zero reluctance, no mmf is expended in the return path

through it for any contour of flux. Hence, for a closed contour of flux at a distance

x from the inside diameter of LV winding, it can be written that

Figure 3.2 (a) Leakage field with equivalent height

(b) Magnetomotive force or flux density diagram

Copyright © 2004 by Marcel Dekker, Inc.

80 Chapter 3

(3.2)

or

(3.3)

For deriving the formula for reactance, let us derive a general expression for the

flux linkages of a flux tube having radial depth R and height Heq. The ampere-turns

enclosed by a flux contour at the inside diameter (ID) and outside diameter (OD)

of this flux tube are a(NI) and b(NI) respectively as shown in figure 3.3, where NI

are the rated ampere-turns. The general formulation is useful when a winding is

split radially into a number of sections separated by gaps. The r.m.s. value of flux

density at a distance x from the ID of this flux tube can now be inferred from

equation 3.3 as

(3.4)

The flux linkages of an incremental flux tube of width dx placed at x are

(3.5)

Figure 3.3 (a) Flux tube

(b) MMF diagram

Copyright © 2004 by Marcel Dekker, Inc.