transformer engineering design and practice 1_phần 5

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. 77 Copyright © 2004 by Marcel Dekker, Inc. 78 Chapter 3 Figure 3.1 Leakage field in a transformer 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), Copyright © 2004 by Marcel Dekker, Inc. Impedance Characteristics 79 Figure 3.2 (a) Leakage field with equivalent height (b) Magnetomotive force or flux density diagram (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 Copyright © 2004 by Marcel Dekker, Inc. 80 Chapter 3 Figure 3.3 (a) Flux tube (b) MMF diagram (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) Copyright © 2004 by Marcel Dekker, Inc. Impedance Characteristics 81 where A is the area of flux tube given by A=π(ID+2x)dx (3.6) Substituting equations 3.4 and 3.6 in equation 3.5, (3.7) Hence, the total flux linkages of the flux tube are given by (3.8) After integration and a few arithmetic operations, we get (3.9) The last term in square bracket can be neglected without introducing an appreciable error to arrive at a simple formula for the regular design use. (3.10) The term can be taken to be approximately equal to the mean diameter (Dm) of the flux tube (for large diameters of windings/gaps with comparatively lower values of their radial depths). (3.11) Now, let (3.12) which corresponds to the area of Ampere-Turn Diagram. The leakage inductance of a transformer with n flux tubes can now be given as (3.13) Copyright © 2004 by Marcel Dekker, Inc. ... - tailieumienphi.vn