transformer engineering design and practice 1_phần 9 pdf

277 7 Surge Phenomena in Transformers For designing the insulation of a transformer suitable for all kinds of overvoltages, the voltage stresses within the windings need to be determined. For this purpose, voltage distributions within the transformer windings for the specific test voltages are calculated. For AC test voltages of power frequency, the voltage distribution is linear with respect to the number of turns and can be calculated exactly. For the calculation of the impulse voltage distribution in the windings, they are required to be simulated in terms of an equivalent circuit consisting of lumped R, L and C elements. There are a number of accurate methods described in the literature for computation of winding response to impulse voltages, some of which are discussed in this chapter. Electric stresses in the insulation within and outside the windings are obtained by analytical or numerical techniques which are described in the next chapter. 7.1 Initial Voltage Distribution When a step voltage impinges on the transformer winding terminals, the initial distribution in the winding depends on the capacitances between turns, between windings, and those between windings and ground. The winding inductances have no effect on the initial voltage distribution since the magnetic field requires a finite time to build up (current in an inductance cannot be established instantaneously). Thus, the inductances practically do not carry any current, and the voltage distribution is predominantly decided by the capacitances in the network, and the problem can be considered as entirely electrostatic without any appreciable error. In other words, the presence of series capacitances between winding sections causes the transformer to respond to abrupt impulses as a network of capacitances for all frequencies above its lower natural frequencies of oscillations. When the applied voltage is maintained for a sufficient time (50 to 100 microseconds), Copyright © 2004 by Marcel Dekker, Inc. Chapter 7278 appreciable currents begin to flow in the inductances eventually leading to the uniform voltage distribution. Since there is difference between the initial and final voltage distributions, as shown in figure 7.1, a transient phenomenon takes place during which the voltage distribution readjusts itself from the initial to final value. During this transient period, there is continual interchange of energy between electric and magnetic fields. On account of a low damping factor of the transformer windings, the transient is oscillatory. The voltage at any point in the winding oscillates about the final voltage value, reaching a maximum as shown by curve c. It is obvious that the strength of the transformer windings to lightning voltages can be significantly increased if the difference between the initial and final distributions can be minimized. This not only reduces the excessive stresses at the line end but also mitigates the oscillations thereby keeping voltage to ground at any point in the winding insignificantly higher than the final voltage distribution. The differential equation governing the initial voltage distribution u 0 =u(x,0), for the representation of a winding shown in figure 7.2 (and ignoring inductive effects), is [1] (7.1) In figure 7.2, L s , c g and c s denote self inductance per unit length, shunt capacitance per unit length to ground and series capacitance per unit length between adjacent turns respectively. Figure 7.1 Impulse voltage distribution Copyright © 2004 by Marcel Dekker, Inc. Surge Phenomena in Transformers 279 Solution of the above equation is given by µ 0 =A 1 e kx +A 2 e -kx (7.2) where (7.3) The constants of integration A 1 and A 2 can be obtained from the boundary conditions at the line and neutral ends of the winding. For the solidly grounded neutral, we have µ 0 =0 for x=0. Putting these values in equation 7.2 we get A 1 +A 2 =0 or A 1 =-A 2 Whereas at the line end, x=L (L is the winding axial length) and u 0 =U (amplitude of the step impulse voltage) giving (7.4) Substituting the above expression in equation 7.2 we get (7.5) The initial voltage gradient at the line end of the winding is given by Figure 7.2 Representation of a transformer winding Copyright © 2004 by Marcel Dekker, Inc. Chapter 7280 (7.6) The initial voltage gradient is maximum at the line end. Since kL>3 in practice, coth giving the initial gradient at the line end for a unit amplitude surge (U=1)as (7.7) The uniform gradient for the unit amplitude surge is 1/L. (7.8) where C G and C S are the total ground capacitance and series capacitance of the transformer winding respectively. The ratio has been denoted by the distribution constant α . Thus, the maximum initial gradient at the line end is α times the uniform gradient. The higher the value of ground capacitance, the higher are the values of α and voltage stress at the line end. For the isolated neutral condition, the boundary conditions, give the following expression for the initial voltage distribution: (7.9) For the isolated neutral condition, the maximum initial gradient at the line end can be written as (7.10) For a unit amplitude surge and ( α =kL)>3, Hence, the initial gradient becomes (7.11) Copyright © 2004 by Marcel Dekker, Inc. Surge Phenomena in Transformers 281 Hence, the value of maximum initial gradient at the line end is the same for the grounded and isolated neutral conditions for abrupt impulses or very steep wave fronts. The initial voltage distribution for various values of a is plotted in figure 7.3 for the grounded and isolated neutral conditions. The total series capacitance (C S ) and ground capacitance (C G ) of the transformer winding predominantly decide the initial stresses in it for steep fronted voltage surges. The total series capacitance consists of capacitance between turns and capacitance between disks/ sections of the winding, whereas the total ground capacitance includes the capacitance between the winding and core/tank/other windings. Thus, the initial voltage distribution is characterized by the distribution constant, (7.12) This parameter indicates the degree of deviation of the initial voltage distribution from the final linear voltage distribution which is decided solely by winding inductances. The higher the value of α , the higher are the deviation and amplitudes of oscillations which occur between the initial and final voltage distributions. For a conventional continuous disk winding, the value of α may be in the range of 5 to 30. Any change in the transformer design, which decreases the distribution constant of the winding, results in a more uniform voltage distribution and reduces the voltage stresses between different parts of the winding. The initial voltage distribution of the winding can be made closer to the ideal linear distribution ( α =0) by increasing its series capacitance and/ or reducing its capacitance to ground. If the ground capacitance is reduced, more current flows through the series capacitances, tending to make the voltage across the various winding sections more uniform. The (ideal) uniform initial impulse voltage distribution will be achieved if no current flows through the (shunt) ground capacitances. Usually, it is very difficult and less cost-effective to reduce the Figure 7.3 Initial voltage distribution Copyright © 2004 by Marcel Dekker, Inc. Chapter 7282 ground capacitances. Insulation gaps between windings predominantly decide the ground capacitances. These capacitances depend on the radial gap and circumferential area between the windings. These geometrical quantities get usually fixed from optimum electrical design considerations. Hence, any attempt to decrease the distribution constant α by decreasing the ground capacitance is definitely limited. The more cost-effective way is to increase the winding series capacitance by using different types of windings as described in the subsequent sections. 7.2 Capacitance Calculations In order to estimate the voltage distribution within a transformer winding subjected to impulse overvoltages, the knowledge of its effective series and ground capacitances is essential. The calculation of ground capacitance between a winding and ground or between two windings is straightforward. The capacitance between two concentric windings (or between the innermost winding and core) is given by (7.13) where D m is mean diameter of the gap between two windings, t oil and t solid are thicknesses of oil and solid insulations between two windings respectively, and H is height of windings (if the heights of two windings are unequal, average height is taken in the calculation). Capacitance between a cylindrical conductor and ground plane is given by (appendix B, equation B30) (7.14) where R and H are radius and length of the cylindrical conductor respectively and s is distance of center of the cylindrical conductor from the plane. Hence, the capacitance between a winding and tank can be given as (7.15) In this case, R and H represent the radius and height of the winding respectively and s is the distance of the winding axis from the plane. The capacitance between the outermost windings of two phases is half the value given by above equation Copyright © 2004 by Marcel Dekker, Inc. Surge Phenomena in Transformers 283 7.15, with s equal to half the value of distance between the axes of the two windings (refer to equation B28). 7.3 Capacitance of Windings 7.3.1 Development of winding methods for better impulse response In the initial days of transformer technology development for higher voltages, use of electrostatic shields was quite common (see figure 7.4). A non-resonating transformer with electrostatic shields was reported in [2,3,4]. It is a very effective shielding method in which the effect of the ground capacitance of individual section is neutralized by the corresponding capacitance to the shield. Thus, the currents in the shunt (ground) capacitances are supplied from the shields and none of them have to flow through the series capacitances of the winding. If the series capacitances along the windings are made equal, the uniform initial voltage distribution can be achieved. The electrostatic shield is at the line terminal potential and hence requires to be insulated from the winding and tank along its height. As the voltage ratings and corresponding dielectric test levels increased, transformer designers found it increasingly difficult and cumbersome to design the shields. The shields were found to be less cost-effective since extra space and material were required for insulating shields from other electrodes inside the transformer. Subsequent development of interleaved windings phased out completely the use of electrostatic shielding method. The principle of electrostatic shielding method is being made use of in the form of static end rings at the line end and static rings within the winding which improve the voltage distribution and reduce the stresses locally. Figure 7.4 Electrostatic shields Copyright © 2004 by Marcel Dekker, Inc. Chapter 7284 In order to understand the effectiveness of an interleaved winding, let us first analyze a continuous (disk) winding shown in figure 7.5. The total series capacitance of the continuous winding is an equivalent of all the turn-to-turn and disk-to-disk capacitances. Although the capacitance between two adjacent turns is quite high, all the turn-to-turn capacitances are in series, which results in a much smaller capacitance for the entire winding. Similarly, all the disk-to-disk capacitances which are also in series, add up to a small value. With the increase in voltage class of the winding, the insulation between turns and between disks has to be increased which further worsens the total series capacitance. The inherent disadvantage of low series capacitance of the continuous winding was overcome by electrostatic shielding as explained earlier till the advent of the interleaved winding. The original interleaved winding was introduced and patented by G.F.Stearn in 1950 [5]. A simple disposition of turns in some particular ways increases the series capacitance of the interleaved winding to such an extent that a near uniform initial voltage distribution can be obtained. A typical interleaved winding is shown in figure 7.6. Figure 7.5 Continuous winding Figure 7.6 Interleaved winding Copyright © 2004 by Marcel Dekker, Inc. Surge Phenomena in Transformers 285 In an interleaved winding, two consecutive electrical turns are separated physically by a turn which is electrically much farther along the winding. It is wound as a conventional continuous disk winding but with two conductors. The radial position of the two conductors is interchanged (cross-over between conductors) at the inside diameter and appropriate conductors are joined at the outside diameter, thus forming a single circuit two-disk coil. The advantage is obvious since it does not require any additional space as in the case of complete electrostatic shielding or part electrostatic shielding (static ring). In interleaved windings, not only the series capacitance is increased significantly but the ground capacitance is also somewhat reduced because of the improvement in the winding space factor. This is because the insulation within the winding in the axial direction can be reduced (due to improvement in the voltage distribution), which reduces the winding height and hence the ground capacitance. Therefore, the distribution constant ( α ) is reduced significantly lowering stresses between various parts of the winding. It can be seen from figure 7.6 that the normal working voltage between adjacent turns in an interleaved winding is equal to voltage per turn times the turns per disk. Hence, one may feel that a much higher amount of turn insulation may be required, thus questioning the effectiveness of the interleaved winding. However, due to a significant improvement in the voltage distribution, stresses between turns are reduced by a great extent so that % safety margins for the impulse stress and normal working stress can be made of the same order. Hence, the turn-to-turn insulation is used in more effective way [6]. Since the voltage distribution is more uniform, the number of special insulation components (e.g., disk angle rings) along the winding height reduces. When a winding has more than one conductor per turn, the conductors are also interleaved as shown in figure 7.7 (a winding with 6 turns per disk and two parallel conductors per turn) to get maximum benefit from the method of interleaving. Figure 7.7 Interleaving with 2-parallel conductors per turn Copyright © 2004 by Marcel Dekker, Inc. Chapter 7286 In [7], improved surge characteristics of interleaved windings are explained based on transmission line like representation of the disks with surge impedance, without recourse to the hypothesis of increased series capacitance. There can be two types of interleaved windings as regards the crossover connections at the inside diameter as shown in figure 7.8. When steep impulse waves such as chopped waves or front-of-waves enter an interleaved winding, a high oscillatory voltage occurs locally between turns at the center of the radial build of the disk. This phenomenon is analyzed in [8,9] for these two types of crossovers in the interleaved windings. 7.3.2 Turn-to-turn and disk-to-disk capacitances For the calculation of series capacitances of different types of windings, the calculations of turn-to-turn and disk-to-disk capacitances are essential. The turn- to-turn capacitance is given by (7.16) where D m is average diameter of winding, w is bare width of conductor in axial direction, t p is total paper insulation thickness (both sides), ε 0 is permittivity of the free space, and ε p is relative permittivity of paper insulation. The term t p is added to the conductor width to account for fringing effects. Similarly, the total axial capacitance between two consecutive disks based on geometrical considerations only is given by (7.17) where R is winding radial depth, t s and ε s are thickness and relative permittivity of solid insulation (radial spacer between disks) respectively, and k is fraction of circumferential space occupied by oil. The term t s is added to R to take into account fringing effects. Figure 7.8 Two types of crossovers in interleaved winding Copyright © 2004 by Marcel Dekker, Inc. [...]... 7 .92 we get (7 .93 ) Replacing y(t) and in equation 7 .90 by their values from equations 7 .91 and 7 .92 respectively and simplifying we get (7 .94 ) where α2=C-1Γ, α1=C-1G, β2=-C-1(Γk-GC-1(Gk-GC-1Ck)-ΓC-1Ck) Also, rearranging equation 7 .91 we have y(t)=X1(t)+β0x(t) (7 .95 ) Equations 7 .93 , 7 .94 and 7 .95 can be written in the matrix form as (7 .96 ) (7 .97 ) Comparing equations 7 .96 and 7 .97 with equations 7.88 and. .. terms in both time and space, it includes both standing and traveling waves In the standing wave approach, the expression of assumed solution is put in equation 7.62 which after simplification becomes (7.64) Copyright © 2004 by Marcel Dekker, Inc Surge Phenomena in Transformers 301 (7.65) and (7.66) It can be seen from the equations 7.65 and 7.66 that both space frequency (ψ: number of standing wave cycles... Transformers 299 The mutual inductance between two thin wire, coaxial coil loops (A and B) of radii RA and RB with a distance S between them is given in SI units as [15, 18, 19] (7.56) where (7.57) and NA and NB are the turns in sections A and B respectively, whereas K(k) and E(k) are the complete elliptic integrals of the first and second kinds respectively The formula is applicable for thin circular filaments... Surge Phenomena in Transformers 295 If Csh denotes the capacitance between a shield turn and adjacent disk turn, the energy between a shield turn i and touching adjacent disk turns is (7.43) Using the expressions from equations 7.41 and 7.42 we get (7.44) Similarly for the second disk, voltages of ith turn and ith shield are given by (7.45) (7.46) The energy between a shield turn i and touching adjacent... incidence matrix Qc(n×2n) is (7. 79) Copyright © 2004 by Marcel Dekker, Inc Surge Phenomena in Transformers 3 09 Figure 7. 19 Numbering of conductive branches We can get from equation 7.76, (7.80) 7.7.3 Formation of G matrix Similarly, the branch conductance matrix (Gb) and incidence conductance matrix QG of the order (n×n) for figure 7. 19 can be given as (7.81) (7.82) and (7.83) 7.7.4 Formation of matrix... by Marcel Dekker, Inc  292 Chapter 7 The total energy stored by the capacitance between the first disk and SER is (7.33) Substituting the values of V1(x) and V2(x) from equations 7.30 and 7.31, and simplifying we get (7.34) Thus, the resultant capacitance, CSER, between SER and the first disk can be given by the equation (7.35) (7.36) Thus, the resultant capacitance between SER and the first disk is... table 7.2 The input voltage is assumed to be the standard full wave defined by x(t)=x0(e-βt-e-δt) (7 .99 ) For the standard (1/50) microsecond wave (which rises to its maximum value at 1 microsecond and decays to half the maximum value in 50 microseconds), when/is expressed in microseconds the values of the constants are x0=1.0167, β=0.01423 and δ=6.0 691 The voltages calculated for various nodes are plotted... the initial distribution of the standing wave analysis [3] In other words, the high frequency components form a standing potential distribution and the low frequency components form a traveling wave; the splitting of incoming surge into two parts is the characteristics of the traveling wave theory In [32], the standing wave and traveling wave approaches are compared and correlated The traveling component... voltage stresses are the result of electric and magnetic fields which appear in the winding under surge conditions and are function of location and time By representing the transformer winding as a network of elements, the field problem is effectively converted into a circuit problem An equivalent network for a multi-winding transformer has been reported in [ 19] in which the conventional ladder network... vector of node voltages with the inclusion of input node The relationship between the nodal matrices and branch matrices is defined by (7.76) where Qc, QG and QL are the incidence matrices for capacitive, conductive and inductive elements, and Cb, Gb and Lb are the branch matrices of capacitive, conductive and inductive elements of the network respectively Copyright © 2004 by Marcel Dekker, Inc 308 Chapter . between disks as shown in figure 7 .11 . Figure 7 .11 Static end ring (SER) and static ring (SR) Copyright © 2004 by Marcel Dekker, Inc. Surge Phenomena in Transformers 2 91 By providing a large equipotential. calculation [10 ,11 ,12 ]. The corresponding representation of capacitances for this accurate method of calculation is shown in figure 7 .9. The total series capacitance of the winding is given by [10 ,13 ] (7 .18 ) where. capacitance. (7 . 19 ) Figure 7 .10 Disk-pair of a continuous winding Copyright © 2004 by Marcel Dekker, Inc. Surge Phenomena in Transformers 2 89 Now, the voltages across the first, second and third inter-disk