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7. WINDING FORCES AND PERFORMANCE UNDER SHORT-CIRCUIT
The effect of short-circuit currents on transformers, as on most other items of electrical plant, fall into two categories:
(2) Mechanical forces
It is a fairly simple matter to deal with the thermal effects of a short-circuit.
This is deemed to persist for a known period of time, EN 60076-5 specifies 2 seconds, allowing for clearance of the fault by back-up protection. During this brief time, it is a safe assumption that all the heat generated remains in the copper. Therefore knowing the mass of the copper, its initial temperature and the heat input, the temperature which it can reach can be fairly easily calculated. It simply remains to ensure that this is below a permitted maxi mum which for oil-immersed paper insulated windings is taken to be 250ºC, in accordance with Table 3 of EN 60076-5. Strictly speaking, the resistivity of the copper will change significantly between its initial temperature, which might be in the region of 115ºC, and this permitted final temperature, and there is also some change in its specific heat over this temperature range, hence, a rigorous calculation would involve an integration with respect to time of the I^2 R loss, which is increasing, plus the eddy current loss, which is decreasing, divided by the copper weight times the specific heat, which is also increasing with temperature. In reality the likely temperature rise occurring within the permitted 2 seconds will fall so far short of the specified figure that an approximate calculation based on average resistivity and specific heat will be quite adequate. Current on short-circuit will be given by the expression:
F = 1 / (ez +es) (eqn. 1)
...where F is the factor for short-circuit current as multiple of rated full-load current
ez is the per unit impedance of transformer
es is the impedance of supply, per unit, expressed on the basis of the transformer rating
The supply impedance is normally quoted in terms of system short-circuit apparent power (fault level) rather than as a percentage. This may be expressed in percentage terms on the basis of the transformer rating in MVA as follows:
es = MVA/S (eqn. 2)
where S is the system short-circuit apparent power in MVA.
An approximate expression for the temperature rise of the conductor after t seconds is then:
θ = t(1 + e/100)D2 ρF2 / dh
where θ is the temperature rise in degrees centigrade
e is the winding eddy current loss, percent
D is the current density in windings, A/mm2
ρ is the resistivity of the conductor material
d is the density of the conductor material
h is the specific heat of the conductor material
For copper the density may be taken as 8.89 g/cm^3 and the specific heat as 0.397 J/g ºC. An average resistivity value for fully cold worked material at, say, 140ºC may be taken as 0.0259 Ohm mm^2/m. Substituting these and a value of t equal to 2 second in the above expression gives (3) An indication of the typical magnitude of the temperature rise produced after 2 seconds can be gained by considering, for example, a 60 MVA, 132 kV grid transformer having an impedance of 13.5 percent. The UK 132 kV system can have a fault level of up to 5000 MVA. Using expression (2) this equates to 1.2 percent based on 60 MVA and inserting this together with the transformer impedance in expression (1) gives an short-circuit current factor of 6.8 times.
A 60 MVA ODAF transformer might, typically, have a current density of up to 6 A/mm2. The winding eddy current losses could, typically, be up to 20 percent. Placing these values in expression (3) gives…
θ = 0.0147(1 + e/100)D2 ρF2 / dh (eqn. 3)
…which is quite modest. With a hot spot temperature before the short-circuit of 125ºC (which is possible for some designs of OFAF transformer in a maximum ambient of 40ºC) the temperature at the end of the short-circuit is unlikely to exceed 155ºC, which is considerably less than the permitted maximum.
The limiting factor for this condition is the temperature reached by the insulation in contact with the copper, since copper itself will not be significantly weakened at a temperature of 250ºC. Although some damage to the paper will occur at this temperature, short-circuits are deemed to be sufficiently infrequent that the effect on insulation life is considered to be negligible. If the winding were made from aluminum, then this amount of heating of the conductor would not be considered acceptable and risk of distortion or creepage of the aluminum would be incurred, so that the limiting temperature for aluminum is restricted to 200ºC.
EN 60076-5 gives a formula for calculation of the conductor temperature following short-circuit that differs in form from the above. For copper this is given as …
where θ0 is the initial winding temperature, in degrees Celsius (ºC) J is the short-circuit current density, in A/mm2 t is the duration, in seconds The resistivity of copper is taken as 0.0224 Ohm mm2/m, which is somewhat less than the value quoted above and taken from a Copper Development Association publication.
If the values used in the example above are inserted into the EN 60076-5 formula, this gives a conductor temperature rise following a 2 second fault of 22.7ºC or some 22 percent less than the value calculated in the example. This difference is not entirely attributable to the lower resistivity assumed in the standard.
Mechanical short-circuit forces are more complex. First there is a radial force which is a mutual repulsion between LV and HV windings. This tends to crush the LV winding inwards and burst the HV winding outwards. Resisting the crushing of the LV winding is relatively easy since the core lies immediately beneath and it is only necessary to ensure that there is ample support, in the form of the number and width of axial strips, to transmit the force to the core.
The outwards bursting force in the HV winding is resisted by the tension in the copper, coupled with the friction force produced by the large number of HV turns which resists their slackening off. This is usually referred to as the 'capstan effect.' Since the tensile strength of the copper is quite adequate in these circumstances, the outward bursting force in the HV winding does not normally represent too serious a problem either. An exception is any outer winding having a small number of turns, particularly if these are wound in a simple helix.
This can be the case with an outer tapping winding or sometimes the HV winding of a large system transformer where the voltage is low in relation to the rating. Such a transformer will probably have a large frame size, a high volts per turn and hence relatively few turns on both LV and HV. In these situations it is important to ensure that adequate measures are taken to resist the bursting forces under short-circuit. These might involve fitting a tube of insulation material over the winding or simply securing the ends by means of taping, not forgetting the ends of any tapping sections if included. Another alternative is to provide 'keeper sticks' over the outer surface of the coil which are threaded through the interturn spacers. Such an arrangement is shown in FIG. 85 in which keeper sticks are used over the helical winding of a large reactor.
Secondly, there may also be a very substantial axial force under short circuit. This has two components. The first results from the fact that two conductors running in parallel and carrying current in the same direction are drawn together, producing a compressive force. This force arises as a result of the flux produced by the conductors themselves. However, the conductors of each winding are also acted upon by the leakage flux arising from the conductors of the other winding. As will be seen by reference to FIG. 86(a), the radial component of this leakage flux, which gives rise to the axial force, will act in one direction at the top of the leg and the other direction at the bottom. Since the current is in the same direction at both top and bottom this produces axial forces in opposite directions which, if the primary and secondary windings are balanced so that the leakage flux pattern is symmetrical, will cancel out as far as the resultant force on the winding as a whole is concerned. Any initial magnetic unbalance between primary and secondary windings, that is axial displacement between their magnetic centers ( FIG. 86(b)) will result in the forces in each half of the winding being unequal, with the result that there is a net axial force tending to increase the displacement even further.
In very large transformers the designer aims to achieve as close a balance as possible between primary and secondary windings in order to limit these axial forces and he will certainly aim to ensure that primary and secondary windings as a whole are balanced, but complete balance of all elements of the winding cannot be achieved entirely for a number of reasons. One is the problem of tappings. Putting these in a separate layer so that there are no gaps in the main body of the HV when taps are not in circuit helps to some extent.
However, there will be some unbalance unless each tap occupies the full winding length in the separate layer. One way of doing this would be to use a multistart helical tapping winding but, as mentioned above, simple helical windings placed outside the HV winding would be very difficult to brace against the outward bursting force. In addition spreading the tapping turns throughout the full length of the layer would create problems if the HV line lead were taken from the centre of the winding. Another factor which makes it difficult to obtain complete magnetic balance is the dimensional accuracy and stability of the materials used. Paper insulation and pressboard in a large winding can shrink axially by several centimeters during dry out and assembly of the windings. Although the manufacturer can assess the degree of shrink age expected fairly accurately, and will attempt to ensure that it is evenly distributed, it is difficult to do this with sufficient precision to ensure complete balance.
Furthermore, shrinkage of insulation continues to occur in service and, although the design of the transformer should ensure that the windings remain in compression, it is more difficult to ensure that such shrinkage will be uniform. With careful design and manufacture the degree of unbalance will be small. Nevertheless it must be remembered that short-circuit forces are proportional to current squared and that the current in question is the initial peak asymmetrical current and not the r.m.s. value. Considering the 60 MVA transformer of the previous example for which the r.m.s. short-circuit current was calculated as 6.8 times full-load current. EN 60076-5, lists in Table 4 values of asymmetry factor, k√2, against X/R ratio for the circuit. These are reproduced in Table 2. For most grid transformer circuits this is likely to fall into the greater than or equal to 14 category, so that k√2 is 2.55. Thus the first peak of the current is 2.55 x 6.8 x 17.34 times full-load current. Force is proportional to the square of this, that is over 300 times that occurring under normal full load current conditions.
Table 2 Values of factor k√2
Expressing the above in general terms, the first peak of the short-circuit current will be:
Isc =[ k√2 MVA 106 / √ 3 V (ez + es)] A
where k √2 is the asymmetry factor
MVA is the transformer rating in volt-amperes
V is the transformer rated voltage in volts
Axial forces under short-circuit are resisted by transmitting them to the core.
The top and bottom core frames incorporate pads which bear on the ends of the windings, these pads distributing the load by means of heavy-section pressboard or compressed laminated-wood platforms. The top and bottom core frames, in turn, are linked together by steel tie bars which have the dual function of resisting axial short-circuit forces and ensuring that when the core and coils are lifted via the top core frames on the assembly, the load is supported from the lower frames. These tie bars can be seen in FIG. 7 which shows a completed core before fitting of the windings.
Calculation of forces
The precise magnitude of the short-circuit forces depends very much upon the leakage flux pattern, and the leakage flux pattern also determines such important parameters as the leakage reactance and the magnitude of the stray losses. Manufacturers nowadays have computer programs based on finite element analysis which enable them to accurately determine the leakage flux throughout the windings. These computer programs can be very simply extended for the calculation of short-circuit forces to enable manufacturers to accurately design for these. Occasionally, however, it might be necessary to make a longhand calculation and in this case the following, which is based on an ERA Report Ref. Q/T134, 'The measurement and Calculation of Axial Electromagnetic Forces in Concentric Transformer Windings,' by M. Waters, B.Sc., F.I.E.E., and a paper with the same title published in the Proceedings of the Institution of Electrical Engineers, Vol. 10, Part II, No.79, February, 1954, will be of assistance.
The calculations are based on the first peak of short-circuit current derived in expression (4) above. The limb current Imax corresponding to this value is used in force calculations.
The impedance voltage ez is dependent upon the tapping position, and to calculate the forces accurately it is necessary to use the value of impedance corresponding to the tapping position being considered. For normal tapping arrangements the change in the percentage impedance due to tappings is of the order of 10 percent, and if this is neglected the force may be in error by an amount up to +-20 percent.
For preliminary calculations, or if a margin of safety is required, the mini mum percentage impedance which may be obtained on any tapping should be used, and in the case of tapping arrangements shown in column one of Table 3 this corresponds to the tapping giving the best balance of ampere-turns along the length of the limb. However, in large transformers, where a good ampere turn balance is essential to keep the forces within practical limits, the change in percentage impedance is small and can usually be neglected.
When calculating forces the magnetizing current of the transformer is neglected, and the primary and secondary windings are assumed to have equal and opposite ampere-turns. All forces are proportional to the square of the ampere-turns, with any given arrangement of windings.
It has been suggested by other authors that the mechanical strength of a power transformer should be defined as the ratio of the r.m.s. value of the symmetrical short-circuit current to the rated full-load current. The corresponding stresses which the transformer must withstand are based upon the peak value of the short-circuit current assuming an asymmetry referred to earlier. A transformer designed to withstand the current given by Eq. (4) would thus have a strength of i/(ez x es).
It will be appreciated that the strength of a transformer for a single fault may be considerably greater than that for a series of faults, since weakening of the windings and axial displacement may be progressive. Moreover, a transformer will have a mechanical strength equal only to the strength of the weakest component in a complex structure. Progressive weakening also implies a short-circuit 'life' in addition to a short-circuit strength. The problem of relating system conditions to short-circuit strength is a complex one and insufficient is yet known about it for definite conclusions to be drawn.
Radial electromagnetic forces
These forces are relatively easy to calculate since the axial field producing them is accurately represented by the simple two-dimensional picture used for reactance calculations. They produce a hoop stress in the outer winding, and a compressive stress in the inner winding.
Table 3 Arrangement of tappings R
The mean hoop stress s_mean in the conductors of the outer winding at the peak of the first half-wave of short-circuit current, assuming an asymmetry factor of 2.55 and a supply impedance es is given by:
where Wcu is I^2Rdc loss in the winding in kW at rated full load and at 75ºC h is the axial height of the windings in mm.
Normally this stress increases with the kVA per limb but it is important only for ratings above about 10 MVA per limb. Fully annealed copper has a very low mechanical strength and a great deal of the strength of a copper conductor depends upon the cold working it receives after annealing, due to coiling, wrapping, etc. It has been suggested that 0.054 kN/mm^2 represents the maxi-mum permissible stress in the copper, if undue permanent set in the outer winding is to be avoided. For very large transformers, some increase in strength may be obtained by lightly cold working the copper or by some form of mechanical restraint. Ordinary high-conductivity copper when lightly cold worked softens very slowly at transformer temperatures and retains adequate strength during the life of an oil-filled transformer.
The radial electromagnetic force is greatest for the inner conductor and decreases linearly to zero for the outermost conductor. The internal stress relationship in a disc coil is such that considerable leveling up takes place and it is usually considered that the mean stress as given in Eq. (5) may be used in calculations.
The same assumption is often made for multilayer windings, when the construction is such that the spacing strips between layers are able to transmit the pressure effectively from one layer to the next. If this is not so then the stress in the layer next to the duct is twice the mean value.
Inner windings tend to become crushed against the core, and it is common practice to support the winding from the core and to treat the winding as a continuous beam with equidistant supports, ignoring the slight increase of strength due to curvature. The mean radial load per mm length of conductor of a disc coil is:
where Ac is the cross-sectional area of the conductor upon which the force is required, mm^2 Dw is the mean diameter of winding, mm U is the rated kVA per limb f is the frequency, Hz ˆ s is the peak value of mean hoop stress, kN/mm^2 , from Eq. (5) d1 is the equivalent duct width, mm Dm is the mean diameter of transformer (i.e. of HV and LV windings), mm N is the number of turns in the winding
Equation (6) gives the total load per millimeter length upon a turn or conductor occupying the full radial thickness of the winding. In a multilayer winding with k layers the value for the layer next to the duct would be (2k - 1)/k times this value, for the second layer (2k - 3)/k, and so on.
Where the stresses cannot be transferred directly to the core, the winding itself must be strong enough to withstand the external pressure. Some work has been carried out on this problem, but no method of calculation proved by tests has yet emerged. It has been proposed, however, to treat the inner winding as a cylinder under external pressure, and although not yet firmly established by tests, this method shows promise of being useful to transformer designers.
Axial electromagnetic forces
Forces in the axial direction can cause failure by producing collapse of the winding, fracture of the end rings or clamping system, and bending of the conductors between spacers; or by compressing the insulation to such an extent that slackness occurs which can lead to displacement of spacers and subsequent failure.
Measurement of axial forces
A simple method is available, developed by ERA Technology Limited (formerly the Electrical Research Association), for measuring the total axial force upon the whole or part of a concentric winding. This method does not indicate how the force is distributed round the circumference of the winding but this is only a minor disadvantage.
If the axial flux linked with each coil of a disc winding at a given current is plotted against the axial position, the curve represents, to a scale which can be calculated, the axial compression curve of the winding. From such a curve the total axial force upon the whole or any part of a winding may be read off directly.
The flux density of the radial component of leakage field is proportional to the rate of change of axial flux with distance along the winding. The curve of axial flux plotted against distance thus represents the integration of the radial flux density and gives the compression curve of the winding if the points of zero compression are marked.
The voltage per turn is a measure of the axial flux, and in practice the volt age of each disc coil is measured, and the voltage per turn plotted against the mid-point of the coil on a diagram with the winding length as abscissa. The method can only be applied to a continuous disc winding by piercing the insulation at each crossover.
The test is most conveniently carried out with the transformer short-circuited as for the copper-loss test.
The scale of force at 50 Hz is given by 1 15 750 V (r.m.s.) r.m.s. ampere-turns per mm k = N N (peak)
To convert the measured voltages to forces under short-circuit conditions the values must be multiplied by (2.55Isc/It ) 2
where Isc is the symmetrical short circuit current and It
the current at which the test is carried out.
To obtain the compression curve it is necessary to know the points of zero compression, and these have to be determined by inspection. This is not difficult since each arrangement of windings produces zero points in well-defined positions.
A simple mutual inductance potentiometer can be used instead of a voltmeter, and a circuit of this type is described in ERA Report, Ref. Q/T 113, the balance being independent of current and frequency.
FIG. 87 shows typical axial compression curves obtained on a transformer having untapped windings of equal heights. There are no forces tending to separate the turns in the axial direction. The ordinates represent the forces between coils at all points, due to the current in the windings. Since the slope of the curve represents the force developed per coil it will be seen that only in the end coils are there any appreciable forces. The dotted curve, which is the sum of the axial compression forces for the inner and outer curves, has a maximum value given by:
in terms of U the rated kVA per limb and h the axial height of the windings in millimeters. This is the force at the peak of the first half cycle of fault current, assuming an asymmetry factor of 2.55.
The results shown in FIG. 87 and other similar figures appearing later in this section were obtained on a three-phase transformer constructed so that the voltage across each disc coil in both inner and outer coil stacks could easily be measured. To ensure very accurate ampere-turn balance along the whole length of the windings, primary and secondary windings consisted of disc coils identical in all respects except diameter, and spacing sectors common to both windings were used so that each disc coil was in exactly the same axial position as the corresponding coil in the other winding.
It will be noted that the forces in a transformer winding depend only upon its proportions and on the total ampere-turns, and not upon its physical size.
Thus, model transformers are suitable for investigating forces, and for large units where calculation is difficult it may be more economical to construct a model and measure the forces than to carry out elaborate calculations.
The voltage per turn method has proved very useful in detecting small accidental axial displacements of two windings from the normal position.
Calculation of axial electromagnetic forces
The problem of calculating the magnitude of the radial leakage field and hence the axial forces of transformer windings has received considerable attention and precise solutions have been determined by various authors. These methods are complex and a computer is necessary if results are to be obtained quickly and economically. The residual ampere-turn method gives reliable results, and attempts to produce closer approximations add greatly to the complexity with out a corresponding gain in accuracy. This method does not give the force on individual coils, but a number of simple formulae of reasonable accuracy are available for this purpose.
Residual ampere-turn method
The axial forces are calculated by assuming the winding is divided into two groups, each having balanced ampere-turns. Radial ampere-turns are assumed to produce a radial flux which causes the axial forces between windings.
The radial ampere-turns at any point in the winding are calculated by taking the algebraic sum of the ampere-turns of the primary and secondary windings between that point and either end of the windings. A curve plotted for all points is a residual or unbalanced ampere-turn diagram from which the method derives its name. It is clear that for untapped windings of equal length and without displacement there are no residual ampere-turns or forces between windings. Nevertheless, although there is no axial thrust between windings, internal compressive forces and forces on the end coils are present. A simple expedient enables the compressive forces present when the ampere-turns are balanced to be taken into account with sufficient accuracy for most design purposes.
The method of determining the distribution of radial ampere-turns is illustrated in FIG. 88 for the simple case of a concentric winding having a fraction a of the total length tapped out at the end of the outer winding. The two components I and II of FIG. 88(b) are both balanced ampere-turn groups which, when superimposed, produce the given ampere-turn arrangement. The diagram showing the radial ampere-turns plotted against distance along the winding is a triangle, as shown in FIG. 88(c), having a maximum value of a (NImax), where (NImax) represents the ampere-turns of either the primary or secondary winding.
To determine the axial forces, it is necessary to find the radial flux produced by the radial ampere-turns or, in other words, to know the effective length of path for the radial flux for all points along the winding. The assumption is made that this length is constant and does not vary with axial position in the winding. Tests show that this approximation is reasonably accurate, and that the flux does, in fact, follow a triangular distribution curve of a shape similar to the residual ampere-turn curve.
The calculation of the axial thrust in the case shown in FIG. 88 can now be made as follows. If leff is the effective length of path for the radial flux, and since the mean value of the radial ampere-turns is a(NImax), the mean radial flux density at the mean diameter of the transformer is:
...and the axial force on either winding of NImax ampere-turns is:
The second factor of this expression, πDm/leff, is the permeance per unit axial length of the limb for the radial flux, referred to the mean diameter of the transformer. It is independent of the physical size of the transformer and depends only upon the configuration of the core and windings. Forces are greatest in the middle limb of a three-phase transformer, and therefore the middle limb only need to be considered. A review of the various factors involved indicates that the forces are similar in a single-phase transformer wound on two limbs.
Thus if Eq. (8) is written as:
where ?= pi Dl meff/ and is the permeance per unit axial length of limb, it gives the force for all transformers having the same proportions whatever their physical size. Since the ampere-turns can be determined without difficulty, in order to cover all cases it is necessary to study only how the constant fi varies with the proportions of the core, arrangement of tappings, dimensions of the winding duct and proximity of tank.
Reducing the duct width increases the axial forces slightly, and this effect is greater with tapping arrangements which give low values of residual ampere turns. However, for the range of duct widths used in practice the effect is small.
Where the equivalent duct width is abnormally low, say less than 8 percent of the mean diameter, forces calculated using the values given in Table 3 should be increased by approximately 20 percent for tappings at two points equidistant from the middle and ends, and 10 percent for tappings at the middle. The axial forces are also influenced by the clearance between the inner winding and core.
The closer the core is to the windings, the greater is the force.
The effect of tank proximity is to increase ? in all cases, and for the outer limbs of a three-phase transformer by an appreciable amount; however, the middle limb remains practically unaffected unless the tank sides are very close to it. As would be expected, the presence of the tank has the greatest effect for tappings at one end of the winding, and the least with tappings at two points equidistant from the middle and ends of the winding. As far as limited tests can show, the presence of the tank never increases the forces in the outer limbs to values greater than those in the middle limb, and has no appreciable effect upon the middle limb with practical tapping arrangements.
The only case in which the tank would have appreciable effect is in that of a single-phase transformer wound on one limb, and in this case the value of ? would again not exceed that for the middle phase of a three-phase transformer.
The location of the tappings is the predominating influence on the axial forces since it controls the residual ampere-turn diagram. Forces due to arrangement E in Table 3 are only about one thirty-second of those due to arrangement A. The value of ? is only slightly affected by the arrangement of tappings so that practically the whole of the reduction to be expected from a better arrangement of tappings can be realized. It varies slightly with the ratio of limb length to core circle diameter, and also if the limbs are more widely spaced.
In Table 3 values of flare given for the various tapping arrangements and for two values of the ratio, window height/core circle diameter. The formula for calculating the axial force on the portion of either winding under each triangle of the residual ampere-turn diagram is given in each case. The values of ? apply to the middle limb with three-phase excitation, and for the tapping sections in the outer winding.
Axial forces for various tapping arrangements
Additional axial forces due to tappings can be avoided by arranging the tap pings in a separate coil so that each tapping section occupies the full winding height. Under these conditions there are no ampere-turns acting radially and the forces are the same as for untapped windings of equal length. Another method is to arrange the untapped winding in a number of parallel sections in such a way that there is a redistribution of ampere-turns when the tapping position is changed and complete balance of ampere-turns is retained.
(i) Transformer with tappings at the middle of the outer winding
To calculate the radial field the windings are divided into two components as shown in FIG. 89. Winding group II produces a radial field diagram as shown in FIG. 89(c). The two halves of the outer winding are subjected to forces in opposite directions towards the yokes while there is an axial compression of similar magnitude at the middle of the inner winding.
Measured curves are given in FIG. 90 for the case of 131- 3 percent tapped out of the middle of the outer winding. The maximum compression in the outer winding is only slightly greater than the end thrust, and it occurs at four to five coils from the ends. The maximum compression in the inner winding is at the middle.
Axial end thrust The axial end thrust is given by:
If Pc is the sum of both compressions as given by Eq. (9) and it is assumed that two-thirds of this is the inner winding, then the maximum compression in the inner winding is given by:
The maximum compression in the outer winding is slightly less than this.
FIG. 91 shows curves of maximum compression in the inner and outer windings, and of end thrust plotted against the fraction of winding tapped out for the same transformer. Equation (10) represents the line through the origin.
Most highly stressed turn or coil
The largest electromagnetic force is exerted upon the coils immediately adjacent to the tapped out portion of a winding and it is in these coils that the maximum bending stresses occur when sector spacers are used. The force upon a coil or turn in the outer winding immediately adjacent to the gap is given theoretically by:
...where Pr is the total radial bursting force of transformer, kN
q is fraction of total ampere-turns in a coil or winding
w is the axial length of coil including insulation, mm
a' is axial length of winding tapped out
There is reasonable agreement between calculated and measured forces; the calculated values are 10-20 percent high, no doubt owing to the assumption that the windings have zero radial thickness.
The coils in the inner winding exactly opposite to the most highly stressed coils in the outer winding have forces acting upon them of a similar, but rather lower, magnitude.
(ii) Tappings at the middle of the outer winding but with thinning of the inner winding
The forces in the previous arrangement may be halved by thinning down the ampere-turns per unit length to half the normal value in the portion of the untapped winding opposite the tappings. Alternatively, a gap may be left in the untapped winding of half the length of the maximum gap in the tapped winding. With these arrangements there is an axial end thrust from the untapped winding when all the tapped winding is in circuit, and an end thrust of similar magnitude in the tapped winding when all the tappings are out of circuit.
In the mid-position there are no appreciable additional forces compared with untapped windings.
(a) Axial end thrust
When all tappings are in circuit the end thrust of the untapped winding may be calculated by means of Eq. (10), substituting for a the fractional length of the gap in the untapped winding. When all tappings are out of circuit the end thrust is given by:
where a, the fraction of the axial length tapped out, is partially compensated by a length [1/2 x a ] omitted from the untapped winding. The constant Λ has the same value as in Eq. (10). The forces are similar when the ampere-turns are thinned down instead of a definite gap being used.
(b) Maximum compression
In either of the two preceding cases the maximum compression exceeds the end thrust by an amount rather less than the force given by Eq. (7).
(c) Most highly stressed coil or turn
When all tappings are in circuit, the force upon the coil or turn adjacent to the compensating gap in the untapped winding may be calculated by applying Eq. (12); in such a case a would be the length of the gap expressed as a fraction. It should be noted, however, that since thinning or provision of a compensating gap is usually carried out on the inner winding, the presence of the core increases the force slightly. Hence this equation is likely to give results a few percent low in this case. On the other hand, when thinning is used, the force upon the coil adjacent to the thinned out portion of winding is rather less than given by Eq. (12).
(iii) Two tapping points midway between the middle and ends of the outer winding
(a) Without thinning of the untapped winding A typical example of the compression in the inner and outer windings is given in FIG. 92 for the case of approximately 13 percent tapped out of the outer winding, half being at each of two points midway between the middle and ends of the winding.
There are three points of maximum compression in the outer winding, the middle one being the largest. In the inner winding there are two equal maxima opposite the gaps in the outer winding.
The axial force upon each quarter of either winding due to the tappings is given by:
...where a is the total fraction of axial length tapped out, and the constant Λ has the value given in Table 3.
This force acts towards the yokes in the two end sections of the outer winding, so that Eq. (10) gives the axial end thrust for the larger values of a. The curve of end thrust plotted against the fraction tapped out can be estimated with out difficulty since it deviates only slightly from the straight line of Eq. (14).
The forces with this arrangement of tappings are only about one-sixteenth of the forces due to tappings at one end of the winding, and they are of the same order as the forces in the untapped winding.
The most highly stressed coils are those adjacent to the tapping points, and the forces may be calculated from Eq. (12) by substituting 1/2 a for a.
(b) With thinning of the untapped winding This practice represents the optimum method of reducing forces when a section is tapped out of a winding, and the dashed curves in FIG. 92 show the forces obtained when the inner winding is thinned opposite each of the two gaps in the outer winding to an extent of 50 percent of the total tapping range. The force upon each quarter of either winding is
...when all tappings are out of circuit. In these equations Λ has the value given in Table 3, and a represents the total fraction tapped out.
The forces upon the coils immediately adjacent to the gaps may be calculated as described in Eq. (12), since these forces are determined by the lengths of the gaps and not by their positions in the winding.