transformer engineering design and practice 1_phần 8 doc

single-It is shown in [2] that for a particular case of YNd connected transformer with adelta connected inner winding, the single-line-to-ground fault is more severe.Except for such spec

Short Circuit Stresses and Strength

The continuous increase in demand of electrical power has resulted in the addition

of more generating capacity and interconnections in power systems Both thesefactors have contributed to an increase in short circuit capacity of networks,making the short circuit duty of transformers more severe Failure of transformersdue to short circuits is a major concern of transformer users The success rateduring actual short circuit tests is far from satisfactory The test data from highpower test laboratories around the world indicates that on an average practicallyone transformer out of four has failed during the short circuit test, and the failurerate is above 40% for transformers above 100 MVA rating [1] There arecontinuous efforts by manufacturers and users to improve the short circuitwithstand performance of transformers A number of suggestions have been made

in the literature for improving technical specifications, verification methods andmanufacturing processes to enhance reliability of transformers under shortcircuits The short circuit strength of a transformer enables it to survive through-fault currents due to external short circuits in a power system network; aninadequate strength may lead to a mechanical collapse of windings, deformation/damage to clamping structures, and may eventually lead to an electrical fault inthe transformer itself The internal faults initiated by the external short circuits aredangerous as they may involve blow-out of bushings, bursting of tank, fire hazard,etc The short circuit design is one of the most important and challenging aspects

of the transformer design; it has been the preferential subject in many CIGREConferences including the recent session (year 2000)

Revision has been done in IEC 60076–5 standard, second edition 2000–07,reducing the limit of change in impedance from 2% to 1% for category III (above

100 MVA rating) transformers This change is in line with the results of many

recent short circuit tests on power transformers greater than 100 MVA, in which

an increase of short circuit inductance beyond 1% has caused significantdeformation in windings This revision has far reaching implications fortransformer manufacturers A much stricter control on the variations in materialsand manufacturing processes will have to be exercised to avoid looseness andwinding movements

This chapter first introduces the basic theory of short circuits as applicable totransformers The thermal capability of transformer windings under short circuitforces is also discussed There are basically two types of forces in windings: axialand radial electromagnetic forces produced by radial and axial leakage fieldsrespectively Analytical and numerical methods for calculation of these forces arediscussed Various failure mechanisms due to these forces are then described It isvery important to understand the dynamic response of a winding to axialelectromagnetic forces Practical difficulties encountered in the dynamic analysisand recent thinking on the whole issue of demonstration of short circuit withstandcapability are enumerated Design parameters and manufacturing processes havepronounced effect on natural frequencies of a winding Design aspects of windingand clamping structures are elucidated Precautions to be taken during design andmanufacturing of transformers for improving short circuit withstand capabilityare given

6.1 Short Circuit Currents

There are different types of faults which result into high over currents, viz line-to-ground fault, line-to-line fault with or without simultaneous ground faultand three-phase fault with or without simultaneous ground fault When the ratio ofzero-sequence impedance to positive-sequence impedance is less than one, asingle-line-to-ground fault results in higher fault current than a three-phase fault

single-It is shown in [2] that for a particular case of YNd connected transformer with adelta connected inner winding, the single-line-to-ground fault is more severe.Except for such specific cases, usually the three-phase fault (which is asymmetrical fault) is the most severe one Hence, it is usual practice to design atransformer to withstand a three-phase short circuit at its terminals, the otherwindings being assumed to be connected to infinite systems/sources (of constantvoltage) The symmetrical short circuit current for a three-phase two-windingtransformer is given by

(6.1)

where V is rated line-to-line voltage in kV, Z T is short circuit impedance of the

transformer, and Z S is short circuit impedance of the system given by

where S F is short circuit apparent power of the system in MVA and S is three-phase

rating of the transformer in MVA Usually, the system impedance is quite small ascompared to the transformer impedance and can be neglected, giving an extrasafety margin In per-unit quantities using sequence notations we get

(6.3)

where Z1 is positive-sequence impedance of the transformer (which is leakageimpedance to positive-sequence currents calculated as per the procedure given inSection 3.1 of Chapter 3) and V pF is pre-fault voltage If the pre-fault voltages areassumed to be 1.0 per-unit (p.u.) then for a three-phase solid fault (with a zerovalue of fault impedance) we get

(6.4)The sequence components of currents and voltages are [3]

Since a three-phase short circuit is usually the most severe fault, it is sufficient ifthe withstand capability against three-phase short circuit forces is ensured.However, if there is an unloaded tertiary winding in a three-winding transformer,its design must be done by taking into account the short circuit forces during asingle-line-to-ground fault on either LV or HV winding Hence, most of thediscussions hereafter are for the three-phase and single-line-to-ground faultconditions Based on the equations written earlier for the sequence voltages andcurrents for these two types of faults, we can interconnect the positive-sequence,negative-sequence and zero-sequence networks as shown in figure 6.1 Thesolution of the resulting network yields the symmetrical components of currentsand voltages in windings under fault conditions [4].

Figure 6.1 Sequence networks

and for a double-line-to-ground fault,

(6.10)

The calculation of three-phase fault current is straight-forward, whereas thecalculation of single-line-to-ground fault current requires the estimation of zero-sequence reactances and interconnection of the three sequence networks at thecorrect points The calculation of fault current for two transformers under thesingle-line-to-ground fault condition is described now.

Consider a case of delta/star (HV winding in delta and LV winding in star withgrounded neutral) distribution transformer with a single-line-to-ground fault on

LV side The equivalent network under the fault condition is shown in figure 6.2(a), where the three sequence networks are connected at the points of fault

(corresponding LV terminals) The impedances denoted with subscript S are the system impedances; for example Z 1HS is the positive-sequence system impedance

on HV side The impedances Z 1HL , Z 2HL and Z 0HL are the positive-sequence,negative-sequence and zero-sequence impedances respectively between HV and

LV windings The zero-sequence network shows open circuit on HV system sidebecause the zero-sequence impedance is infinitely large as viewed/measured from

a delta side as explained in Chapter 3 (Section 3.7) When there is no in-feed from

LV side (no source on LV side), system impedances are effectively infinite and thenetwork simplifies to that given in figure 6.2 (b) Further, if the systemimpedances on HV side are very small as compared to the inter-windingimpedances, they can be neglected giving the sequence components and fault

current as (fault assumed on a phase)

Figure 6.2 Single-line-to-ground fault on star side of delta/star transformer

(6.12)

Now, let us consider a three-winding transformer with an unloaded tertiarywinding (HV and LV windings are star connected with their neutrals grounded,and tertiary winding is delta connected) The interconnection of sequencenetworks is shown in figure 6.3 (a) A single-line-to-ground fault is considered on

phase a of LV winding Since it is a three-winding transformer, the corresponding

star equivalent circuits are inserted at appropriate places in the network In thepositive-sequence and negative-sequence networks, the tertiary is shown open-circuited because it is unloaded; only in the zero-sequence network the tertiary is

in the circuit since the zero-sequence currents can flow in a closed delta If the fault currents are neglected, both the sources in positive-sequence network areequal to 1 per-unit voltage The network gets simplified to that shown in figure 6.3(b) The positive-sequence impedance is

where Z 1HS and Z 1LS are positive-sequence system impedances, and Z 1H and Z 1L arepositive-sequence impedances of HV and LV windings respectively in the starequivalent circuit

Figure 6.3 Single-line-to-ground fault in three-winding transformer

Similarly, the negative-sequence and zero-sequence impedances are given by

Z0=([(Z 0HS +Z 0H )//Z 0T ]+Z 0L )//Z 0LS (6.15)

The impedances Z1 and Z2 are equal because the corresponding positive-sequenceand negative-sequence impedances in their expressions are equal The total faultcurrent is then calculated as

zero-sequence current flowing through the impedance Z 0T

An unloaded tertiary winding is used for the stabilizing purpose as discussed in

Chapter 3 Since its terminals are not usually brought out, an external short circuit

is not possible and it may not be necessary to design it for withstanding a shortcircuit at its own terminals However, the above analysis of single-line-to-groundfault in a three-winding transformer has shown that the tertiary winding must beable to withstand the forces produced in it by asymmetrical fault on LV or HVwinding Consider a case of star/star connected transformer with a delta connectedtertiary winding, in which a single-line-to-ground fault occurs on the LV sidewhose neutral is grounded If there is no in-feed from the LV side (no source onthe LV side), with reference to figure 6.3, the impedances Z 1LS , Z 2LS and Z 0LS will beinfinite There will be open circuit on the HV side in the zero-sequence networksince HV neutral is not grounded in the case being considered If thesemodifications are done in figure 6.3, it can be seen that the faulted LV windingcarries all the three sequence currents, whereas the tertiary winding carries onlythe zero-sequence current Since all the three sequence currents are equal for asingle-line-to-ground fault condition (equation 6.7), the tertiary winding carriesone-third of ampere-turns of the faulted LV winding As explained in Chapter 3, anunloaded tertiary winding is used to stabilize the neutral voltage under asymmetricalloading conditions The load on each phase of the tertiary winding is equal to one-third of a single-phase/unbalanced load applied on one of the main windings.Hence, the rating of the unloaded tertiary winding is commonly taken as one-third ofthe rating of the main windings In single-line-to-ground fault conditions, theconductor of the tertiary winding chosen according to this rule should also help thetertiary winding in withstanding forces under a single-line-to-ground fault

condition in most of the cases This is particularly true for the case discussedpreviously in which the neutral terminal of one of the main windings is grounded(in this case the tertiary winding carries one-third of ampere-turns of the faultedwinding) For the other connections of windings and neutral groundingconditions, the value of zero-sequence current flowing in the tertiary windingdepends on the relative values of impedances of windings and system impedances

in the zero-sequence network For example, in the above case if the HV neutral isalso grounded, the zero-sequence current has another path available, and themagnitude of zero-sequence current carried by LV, HV and tertiary windings

depends on the relative impedances of the parallel paths (Z 0T in parallel with

(Z 0HS +Z 0H) in figure 6.3) Hence, with the HV neutral also grounded, the forces onthe tertiary winding are reduced

As seen in Chapter 3, the stabilizing unloaded tertiary windings are provided toreduce the third harmonic component of flux and voltage by providing a path forthird harmonic magnetizing currents and to stabilize the neutral by virtue ofreduction in the zero-sequence impedance For three-phase three-limbtransformers of smaller rating with star/star connected windings having groundedneutrals, the tertiary stabilizing winding may not be provided This is because thereluctance offered to the zero-sequence flux is high, which makes the zero-sequence impedance low and an appreciable unbalanced load can be taken bythree-phase three-limb transformers with star/star connected windings Also, asshown in Appendix A, for such transformers the omission of stabilizing windingdoes not reduce the fault current drastically, and it should get detected by theprotection circuitry The increase in zero-sequence impedance due to its omission

is not significant; the only major difference is the increase in HV neutral current,which should be taken into account while designing the protection system Theremoval of tertiary winding in three-phase three-limb transformers with both HVand LV neutrals grounded, eliminates the weakest link from the short circuitdesign considerations and reduces the ground fault current to some extent Thisresults in reduction of the short circuit stresses experienced by the transformersand associated equipment Hence, as explained in Section 3.8, the provision ofstabilizing winding in three-phase three-limb transformers should be criticallyreviewed if permitted by the considerations of harmonic characteristics andprotection requirements

The generator step-up transformers are generally subjected to short circuitstresses lower than the interconnecting autotransformers The higher generatorimpedance in series with the transformer impedance reduces the fault currentmagnitude for faults on the HV side of the generator transformer There is a lowprobability of faults on its LV side since the bus-bars of each phase are usuallyenclosed in a metal enclosure (bus-duct) But, since generator transformers are themost critical transformers in the whole network, it is desirable to have a highersafety factor for them Also, the out-of-phase synchronization in generatortransformers can result into currents comparable to three-phase short circuit

currents It causes saturation of the core due to which an additional magnetizingtransient current gets superimposed on the fault current [5] Considerable axialshort circuit forces are generated under these conditions [6].

The nature of short circuit currents can be highly asymmetrical like inrushcurrents A short circuit current has the maximum value when the short circuit isperformed at zero voltage instant The asymmetrical short circuit current has twocomponents: a unidirectional component decreasing exponentially with time and

an alternating steady-state symmetrical component at fundamental frequency The

rate of decay of the exponential component is decided by X/R ratio of the

transformer The IEC 60076–5 (second edition: 2000–07) for power transformersspecifies an asymmetry factor corresponding to switching at the zero voltage

instant (the worst condition of switching) For the condition X/R>14, an

asymmetrical factor of 1.8 is specified for transformers upto 100 MVA rating,whereas it is 1.9 for transformers above 100 MVA rating Hence, the peak value ofasymmetrical short circuit current can be taken as

where I sym is the r.m.s value of the symmetrical three-phase short circuit current.The IEEE Standard C57.12.00–2000 also specifies the asymmetrical factors for

various X/R ratios, the maximum being 2 for the X/R ratio of 1000.

6.2 Thermal Capability at Short Circuit

A large current flowing in transformer windings at the time of a short circuitresults in temperature rise in them Because of the fact that the duration of shortcircuit is usually very short, the temperature rise is not appreciable to cause anydamage to the transformer The IEC publication gives the following formulae forthe highest average temperature attained by the winding after a short circuit,

(6.17)

(6.18)

whereθ0is initial temperature in °C

J is current density in A/mm2 during the short circuit based on the r.m.s.value of symmetrical short circuit current

t is duration of the short circuit in seconds

While arriving at these expressions, an assumption is made that the entire heatdeveloped during the short circuit is retained in the winding itself raising itstemperature This assumption is justified because the thermal time constant of awinding in oil-immersed transformers is very high as compared to the duration ofthe short circuit, which allows us to neglect the heat flow from windings to thesurrounding oil The maximum allowed temperature for oil-immersedtransformers with the insulation system temperature of 105°C (thermal class A) is250°C for a copper conductor whereas the same is 200°C for an aluminumconductor Let us calculate the temperature attained by a winding with the ratedcurrent density of 3.5 A/mm2 If the transformer short circuit impedance is 10%,the current density under short circuit will be 35 A/mm2 (corresponding to thesymmetrical short circuit current) Assuming the initial winding temperature as105°C (worst case condition), the highest temperature attained by the windingmade of copper conductor at the end of the short circuit lasting for 2 seconds(worst case duration) is about 121°C, which is much below the limit of 250°C.Hence, the thermal withstand capability of a transformer under the short circuitconditions is usually not a serious design issue.

6.3 Short Circuit Forces

The basic equation for the calculation of electromagnetic forces is

where B is leakage flux density vector, I is current vector and L is winding length.

If the analysis of forces is done in two dimensions with the current density in the z

direction, the leakage flux density at any point can be resolved into two

components, viz one in the radial direction (Bx) and other in the axial direction

(By) Therefore, there is radial force in the x direction due to the axial leakage flux

density and axial force in the y direction due to the radial leakage flux density, as

shown in figure 6.4

Figure 6.4 Radial and axial forces

The directions of forces are readily apparent from the Fleming’s left hand rulealso, which says that when the middle finger is in the direction of current and thesecond finger in the direction of field, the thumb points in the direction of force(all these three fingers being perpendicular to each other).

We have seen in Chapter 3 that the leakage field can be expressed in terms ofthe winding current Hence, forces experienced by a winding are proportional tothe square of the short circuit current, and are unidirectional and pulsating innature With the short circuit current having a steady state alternating component

at fundamental frequency and an exponentially decaying component, the forcehas four components: two alternating components (one at fundamental frequencydecreasing with time and other at double the fundamental frequency with aconstant but smaller value) and two unidirectional components (one constantcomponent and other decreasing with time) The typical waveforms of the shortcircuit current and force are shown in figure 6.5 Thus, with a fully offset currentthe fundamental frequency component of the force is dominant during the initialcycles as seen from the figure

Figure 6.5 Typical waveforms of short circuit current and force

As described earlier, the short circuit forces are resolved into the radial andaxial components simplifying the calculations The approach of resolving theminto the two components is valid since the radial and axial forces lead to thedifferent kinds of stresses and modes of failures There are number of methodsreported in the literature for the calculation of forces in transformers Once theleakage field is accurately calculated, the forces can be easily determined usingequation 6.19 Over the years, the short circuit forces have been studied from astatic consideration, that is to say that the forces are produced by a steady current.The methods for the calculation of static forces are well documented in 1979 by aCIGRE working group [7], The static forces can be calculated by any one of thefollowing established methods, viz Roth’s method, Rabin’s method, the method

of images and finite element method Some of the analytical and numericalmethods for the leakage field calculations are described in Chapter 3 Thewithstand is checked for the first peak of the short circuit current (with appropriateasymmetry factor as explained in Section 6.1)

A transformer is a highly asymmetrical 3-D electromagnetic device Under athree-phase short circuit, there is heavy concentration of field in the core windowand most of the failures of core-type transformers occur in the window region Inthree-phase transformers, the leakage fields of adjacent limbs affect each other.The windings on the central limb are usually subjected to higher forces There is aconsiderable variation of force along the winding circumference Although,within the window the two-dimensional formulations are sufficiently accurate, thethree-dimensional numerical methods may have to be used for accurate estimation

of forces in the regions outside the core window [8]

6.3.1 Radial forces

The radial forces produced by the axial leakage field act outwards on the outerwinding tending to stretch the winding conductor, producing a tensile stress (alsocalled as hoop stress); whereas the inner winding experiences radial forces actinginwards tending to collapse or crush it, producing a compressive stress Theleakage field pattern of figure 6.4 indicates the fringing of the leakage field at theends of the windings due to which the axial component of the field reducesresulting into smaller radial forces in these regions For deriving a simple formulafor the radial force in a winding, the fringing of the field is neglected; theapproximation is justified because the maximum value of the radial force isimportant which occurs in the major middle portion of the winding

Let us consider an outer winding, which is subjected to hoop stresses Thevalue of the leakage field increases from zero at the outside diameter to amaximum at the inside diameter (at the gap between the two windings) The peakvalue of flux density in the gap is

where NI is the r.m.s value of winding ampere-turns and H w is winding height inmeters The whole winding is in the average value of flux density of half the gap

value The total radial force acting on the winding having a mean diameter of D m

(in meters) can be calculated by equation 6.19 as

(6.21)

For the outer winding, the conductors close to gap (at the inside diameter)experience higher forces as compared to those near the outside diameter (forcereduces linearly from a maximum value at the gap to zero at the outside diameter).The force can be considered to be transferred from conductors with high load(force) to those with low load if the conductors are wound tightly [9] Hence,averaging of the force value over the radial depth of the winding as done in theabove equation is justified since the winding conductors share the load almostuniformly If the curvature is taken into account by the process of integrationacross the winding radial depth as done in Section 3.1.1 of Chapter 3, the meandiameter of the winding in the above equation should be replaced by its insidediameter plus two-thirds of the radial depth

The average hoop stress for the outer winding is calculated as for a cylindrical

boiler shell shown in figure 6.6 The transverse force F acting on two halves of the

winding is equivalent to pressure on the diameter [10]; hence it will be given byequation 6.21 with πDm replaced by D m If the cross-sectional area of turn is A t (in

m2), the average hoop stress in the winding is

Figure 6.6 Hoop stress calculation

Let I r be the rated r.m.s current and Z pu be the per-unit impedance of a transformer.Under the short circuit condition, the r.m.s value of current in the winding is equal

to (I r /Z pu) To take into account the asymmetry, this current value is multiplied by

the asymmetry factor k If we denote copper loss per phase by P R, the expression

for σavg under the short circuit condition is

(6.23)

Substituting the values of µ0(=4π×10-7) and ρ (resistivity of copper at 75°

=0.0211×10-6) we finally get

(6.24)or

(6.25)

where P R is in watts and H w in meters It is to be noted that the term P R is only the

DC I 2 R loss (without having any component of stray loss) of the winding per

phase at 75°C Hence, with very little and basic information of the design, theaverage value of hoop stress can be easily calculated If an aluminum conductor isused, the numerical constant in the above equation will reduce according to theratio of the resistivity of copper to aluminum giving,

(6.26)

As mentioned earlier, the above value of average stress can be assumed to beapplicable for an entire tightly wound disk winding without much error This isbecause of the fact that although the stress is higher for the inner conductors of theouter winding, these conductors cannot elongate without stressing the outerconductors This results in a near uniform hoop stress distribution over the entirewinding In layer/helical windings having two or more layers, the layers do notfirmly support each other and there is no transfer of load between them Hence,

the hoop stress is highest for the innermost layer and it decreases towards the outerlayers For a double-layer winding, the average stress in the layer near the gap is1.5 times higher than the average stress for the two layers considered together.

Generalizing, if there are L layers, the average stress in kth layer (from gap) is

[2-((2k-1)/L)] times the average stress of all the layers considered together Thus, the

design of outer multi-layer winding subjected to a hoop stress requires specialconsiderations

For an inner winding subjected to radial forces acting inwards, the averagestress can be calculated by the same formulae as above for the outer winding.However, since the inner winding can either fail by collapsing or due to bendingbetween the supports, the compressive stresses of the inner winding are not thesimple equivalents of the hoop stresses of the outer winding Thus, the innerwinding design considerations are quite different, and these aspects along with thefailure modes are discussed in Section 6.5

6.3.2 Axial forces

For an uniform ampere-turn distribution in windings with equal heights (idealconditions), the axial forces due to the radial leakage field at the winding ends aredirected towards the winding center as shown in figure 6.4 Although, there ishigher local force per unit length at the winding ends, the cumulative compressiveforce is maximum at the center of windings (see figure 6.7) Thus, both the innerand outer windings experience compressive forces with no end thrust on theclamping structures (under ideal conditions) For an asymmetry factor of 1.8, thetotal axial compressive force acting on the inner and outer windings taken together

is given by the following expression [11]:

(6.27)

Figure 6.7 Axial force distribution

where S is rated power per limb in kVA, H w is winding height in meters, Z pu is

per-unit impedance, and f is frequency in Hz The inner winding being closer to the

limb, by virtue of higher radial flux, experiences higher compressive force ascompared to the outer winding In the absence of detailed analysis, it can beassumed that 25 to 33% of force is taken by the outer winding, and the remaining

75 to 67% is taken by the inner winding

Calculation of axial forces in the windings due to the radial field in non-idealconditions is not straightforward Assumptions, if made to simplify thecalculations, can lead to erroneous results for non-uniform windings Thepresence of tap breaks makes the calculations quite difficult The methodsdiscussed in Chapter 3 should be used to calculate the radial field and the resultingaxial forces The forces calculated at various points in the winding are added tofind the maximum compressive force in the winding Once the total axial force foreach winding is calculated, the compressive stress in the supporting radial spacers(blocks) can be calculated by dividing the compressive force by the total area ofthe radial spacers The stress should be less than a certain limit, which depends onthe material of the spacer If the pre-stress (discussed in Section 6.7) applied ismore than the value of force, the pre-stress value should be considered whilecalculating the stress on the radial spacers

The reasons for a higher value of radial field and consequent axial forces are:mismatch of ampere-turn distribution between LV and HV windings, tappings inthe winding, unaccounted shrinkage of insulation during drying and impregnationprocesses, etc When the windings are not placed symmetrically with respect tothe center-line as shown in figure 6.8, the resulting axial forces are in such adirection that the asymmetry and the end thrusts on the clamping structuresincrease further It is well known that even a small axial displacement of windings

or misalignment of magnetic centers of windings can eventually cause enormousaxial forces leading to failure of transformers [12,13] Hence, strict sizing/dimension control is required during processing and assembling of windings sothat the windings get symmetrically placed

Figure 6.8 Axial asymmetry

6.4 Dynamic Behavior Under Short Circuits

The transformer windings along with the supporting clamping structure form amechanical system having mass and elasticity The applied electromagnetic forcesare oscillatory in nature and they act on the elastic system comprising of windingconductors, insulation system and clamping structures The forces aredynamically transmitted to various parts of the transformer and they can be quitedifferent from the applied forces depending upon the relationship betweenexcitation frequencies and natural frequencies of the system Thus, the dynamicbehavior of the system has to be analyzed to find out the stresses anddisplacements produced by the short circuit forces The dynamic analysis,although quite complex, is certainly desirable which improves the understanding

of the whole phenomenon and helps designers to enhance the reliability of thetransformers under short circuit conditions The dynamic behavior is associatedwith time-dependence of the instantaneous short circuit current and thecorresponding force, and the displacement of the windings producinginstantaneous modifications of these forces The inertia of conductors, frictionalforces and reactionary forces of the various resilient members of the system play

an important role in deciding the dynamic response

In the radial direction, the elasticity of copper is large and the mass is small,resulting into natural frequency much higher than 50/60 Hz and 100/ 120 Hz (thefundamental frequency and twice the fundamental frequency of the excitationforce) Hence, there exists a very remote possibility of increase in displacements

by resonance effects under the action of radial forces Therefore, these forces may

be considered as applied slowly and producing a maximum stress corresponding

to the first peak of an asymmetrical fault current [10] In other words, the energystored by the displacement of windings subjected to radial forces is almostentirely elastic and the stresses in the windings correspond closely with theinstantaneous values of the generated forces [14]

Contrary to the radial direction, the amount of insulation is quite significantalong the axial direction, which is easily compressible With the axial forcesacting on the system consisting of the conductor and insulation, the naturalfrequencies may come quite close to the excitation frequencies of the short circuitforces Such a resonant condition leads to large displacements and eventual failure

of transformers Hence, the dynamic analysis of mechanical system consisting ofwindings and clamping structures is essential and has been investigated in detail

by many researchers

The transformer windings, made up of large number of conductors separated

by insulating materials, can be represented by an elastic column with distributedmass and spring parameters, restrained by end springs representing the insulationbetween the windings and yokes Since there is heavy insulation at the windingends, these springs are usually assumed as mass-less When a force is applied to anelastic structure, the displacement and stress depend not only on the magnitude of

force and its variation with time, but also on the natural frequencies of thestructure.

The methods for calculating dynamic response are quite complex They have totake into account the boundary conditions, viz degree of pre-stress, stiffness ofclamping structure and the proximity of tank/other windings It should also takeinto account the effects of displacement of conductors The method reported in[15] replaces a model of ordinary linear differential equations representing thesystem by an approximate equivalent model of linear difference equations with aconstant time step-length The non-linear insulation characteristics obtained fromthe experimental data are used to solve the difference equations by a digitalcomputer In [16,17], the dynamic load and displacement at any point in thewinding are calculated by using a generalized Fourier series of the normal modes(standing wave approach) The analysis presented can be applied to an arbitraryspace distribution of electromagnetic forces with actual time variation of a fullyasymmetric short circuit current taken into account The dynamic forces arereported to have completely different magnitudes and waveshapes as compared tothe applied electromagnetic forces

A rigorous analytical solution is possible when linear insulation characteristicsare assumed The insulation of a transformer has non-linear insulationcharacteristics The dynamic properties of pressboard are highly non-linear andconsiderably different from the static characteristics The dynamic stiffness anddamping characteristics can be experimentally determined [18,19] The use ofstatic characteristics was reported to be acceptable [19], which leads to pessimisticresults as compared to that obtained by using the dynamic characteristics It wasshown in [20] that the dynamic value of Young’s modulus can be derived from thestatic characteristics However, it is explained in [17] that this approximation maynot be valid for oil-impregnated insulation Oil provides hydrodynamic masseffect to the clamping parts subjected to short circuit forces, and it alsosignificantly influences the insulation stiffness characteristics These complexitiesand the non-linearity of the systems involved can be effectively taken into account

by numerical methods A dynamic analysis is reported in [21] which accounts forthe difference in the electromagnetic forces inside and outside the core window It

is shown that a winding displacement inside the window is distinctly different andhigher than that outside the window A simplified model is proposed in [22]whereby the physical aggregation of conductors and supports is considered as acontinuous elastic solid represented by a single partial differential equation.Thus, a number of numerical methods are available for determining thedynamic response of a transformer under short circuit conditions The methodshave not been yet perfected due to the lack of precise knowledge of dynamiccharacteristics of various materials used in transformers The dynamiccalculations can certainly increase the theoretical knowledge of the wholephenomenon, but it is difficult to ascertain the validity of the results obtained Onthe contrary, it is fairly easy to calculate the natural frequencies of windings andcheck the absence of resonance Hence, a more practical approach can be to check

the withstand for the worst possible peak value of an asymmetrical fault current(static calculation as explained in Section 6.3) In addition, the natural frequencies

of windings should be calculated to check that they are far away from the powerfrequency or twice the power frequency If the natural frequencies are close toeither 50 or 100 Hz (60 or 120 Hz), these can be altered (to avoid resonance) byusing a different pre-stress value or by changing the modes of vibration by asuitable sub-division of windings Hence, the well-established static calculationsalong with the determination of natural frequencies could form a basis of shortcircuit strength calculations [23,24] until the dynamic analysis is perfected andstandardized

In a typical core type power transformer, windings are commonly clampedbetween top and bottom clamping plates (rings) of insulating material Theconstruction of the winding is quite complicated consisting of many differentmaterials like kraft paper, pre-compressed board, copper/aluminum conductor,densified wood, etc The winding consists of many disks and insulation spacers.Thus, the winding is a combination of spacers, conductors and pre-compressedboards Strictly speaking, the winding is having multiple degrees of freedom Thewinding is considered as a distributed mass system in the analysis The windingstiffness is almost entirely governed by the insulation only The top and bottomend insulations are considered as mass-less linear springs The winding can berepresented by an elastic column restrained between the two end springs as shown

length

Figure 6.9 Representation of winding [16]

The expression for natural frequency, ωn (in rad/sec) can be derived fromequation 6.28 with the boundary conditions that the displacement and velocity at

any position x are zero at t=0, and the net force acting at positions x=0 (winding bottom) and x=L (winding top) is zero The expression is

where A is area of insulation, E eq is equivalent Young’s modulus of winding, and

Leq is equivalent length of winding Thus, the natural frequency of a winding is afunction of its mass, equivalent height, cross sectional area and modulus ofelasticity The conductor material (copper) is too stiff to get compressedappreciably by the axial force Hence, all the winding compression is due to thosefractions of its height occupied by the paper and press-board insulation Theequivalent Young’s modulus can therefore be calculated from [20]

(6.31)

where E eq is modulus of elasticity of the combined paper and pressboard

insulation system, E p is modulus of elasticity of paper, and E b is modulus of

elasticity of pressboard The terms L p, L b and L eq represent thickness of paper,thickness of pressboard and total equivalent thickness of paper and pressboardrespectively

The eigen values (λ) are calculated [16] from the equation

(6.32)

where K1 and K2 are the stiffness values of bottom and top end insulationrespectively In equation 6.32, the only unknown is λ which can be found by aniterative method Subsequently, the values of natural frequencies can be calculatedfrom equation 6.29

The natural frequencies can be more accurately calculated by numericalmethods such as FEM analysis [24], If any of the calculated natural frequencies isclose to the exciting frequencies, they can be altered by making suitable changes

in the winding configuration and/or pre-stress value

6.5 Failure Modes Due to Radial Forces

The failure modes of windings are quite different for inward and outward radialforces Winding conductors subjected to outward forces experience the tensile(hoop) stresses The compressive stresses are developed in conductors of awinding subjected to the inward forces In concentric windings, the strength ofouter windings subjected to the outward forces depends on the tensile strength ofthe conductor; on the contrary the strength of inner windings subjected to theinward forces depends on the support structure provided The radial collapse ofthe inner windings is common, whereas the outward bursting of the outerwindings usually does not take place

6.5.1 Winding subjected to tensile stresses

If a winding is tightly wound, the conductors in the radial direction in a diskwinding or in any layer of a multi-layer winding can be assumed to have a uniformtensile stress Since most of the space in the radial direction is occupied withcopper (except for the small paper covering on the conductors), the ratio ofstiffness to mass is high As mentioned earlier, the natural frequency is muchhigher than the exciting frequencies, and hence chances of resonance are remote.Under a stretched condition, if the stress exceeds the yield strength of theconductor, a failure occurs The conductor insulation may get damaged or therecould be local bulging of the winding The conductor may even break due toimproper joints The chances of failure of windings subjected to the tensile hoopstresses are unlikely if a conductor with a certain minimum 0.2% proof strength isused The 0.2% proof stress can be defined as that stress value which produces apermanent strain of 0.2% (2 mm in 1000 mm) as shown in figure 6.10 One of thecommon ways to increase the strength is the use of work-hardened conductor; thehardness should not be very high since there could be difficulty in windingoperation with such a hard conductor A lower value of current density is also used

to improve the withstand characteristics

6.5.2 Windings subjected to compressive stresses

Conductors of inner windings, which are subjected to the radial compressiveload, may fail due to bending between supports or buckling The former case isapplicable when the inner winding is firmly supported by the axially placedsupporting spacers (strips), and the supporting structure as a whole has higherstiffness than conductors (e.g., if the spacers are supported by the corestructure)

In that case, the conductors can bend between the supports all along thecircumference as shown in figure 6.11 (a) if the stress exceeds the elastic limit of

the conductor material This form of buckling is termed as forced buckling [25,

discussion of 26], which also occurs when the winding cylinder has a significantstiffness as compared to the winding conductors (i.e., when thick cylinders of astiff material are used)

The latter case of buckling, termed as free buckling, is essentially an

unsupported buckling mode, in which the span of the conductor buckle bears norelation to the span of axial supporting spacers as shown in figure 6.11 (b) Thiskind of failure occurs mostly with thin winding cylinders, where conductor hashigher stiffness as compared to that of inner cylinders and/or the cylinders (andthe axial spacers) are not firmly supported from inside The conductors bulgeinwards as well as outwards at one or more locations along the circumference.There are many factors which may lead to the buckling phenomenon, viz windinglooseness, inferior material characteristics, eccentricities in windings, lowerstiffness of supporting structures as compared to the conductor, etc

Figure 6.10 0.2% Proof stress

Figure 6.11 Buckling phenomena