Transient phenomena have probably provided transformer design engineers with
their most interesting and stimulating challenge. For many years the very elusiveness
of the subject coupled with the difficulties often met with in reproducing
in the laboratory or test room the identical conditions to those which occur
in practice undoubtedly provided the most significant aspect of that challenge.
Until the advent of computers quantitative calculations were often very difficult
since, under extremely abnormal conditions (for instance, when dealing with
voltages at lightning frequencies and with supersaturation of magnetic circuits),
the qualities of resistance, inductance and capacitance undergo very material
temporary apparent changes compared with their values under normal conditions.
A considerable amount of connected investigation has been carried out on transient
phenomena of different kinds, by many brilliant investigators, and it is largely
to these that we owe our present knowledge of transients.
A number of individual papers have been presented before technical engineering
institutions in the UK, the USA and Europe, and these have formed valuable
additions to the literature of the subject. We cannot hope, in a volume of
this nature, to cover anything approaching the whole field of the subject,
but we have here endeavored to present a brief survey of the chief disturbances
to which transformers are particularly liable.
The transients to which transformers are mainly subjected are:
• Impact of highvoltage and highfrequency waves arising from various causes,
including switching in.
• System switching transients with slower wavefronts than the above.
• Switching in current rushes.
• Shortcircuit currents.
It is not intended to discuss specifically the results of faulty operations,
such as paralleling out of phase or the opening on load of a system isolator
link, as the resulting transients would be of the nature of one or more of
those mentioned above.
Impact of highvoltage and highfrequency waves
Transformer windings may be subject to the sudden impact of highfrequency
waves arising from switching operations, atmospheric lightning discharges,
load rejections, insulator flashovers and short circuits, and, in fact, from
almost any change in the electrostatic and electromagnetic conditions of the
circuits involved. An appreciable number of transformer failures occurred in
the past, particularly in the earlier days of transformer design, due to the
failure of inter turn insulation, principally of those end coils connected
to the line terminals, though similar insulation failures have also occurred
at other places within the windings, notably at points at which there is a
change in the winding characteristics. The failures which have occurred on
the lineend coils have been due chiefly to the concentration of voltage arising
on those coils as a result of the relative values and distribution of the inductance
and capacitance between the turns of the coils.
In the early stages when these breakdowns occurred, considerable discussion
took place on the relative merits of external protection in the form of choke
coils and reinforced insulation of the end coils, but actual experience with
external choke coils showed that in many cases their provision did not eliminate
the necessity for reinforcement of the end coils, while, on the other hand,
added reinforcement of the end coils was itself occasionally still subject
to failure, and more frequent breakdown of the interturn insulation occurred
beyond the reinforcement. For many years there was in use in the UK a British
Standard, BS 422, which provided recommendations for the extent of reinforcement
of end turns of highervoltage windings. Now external protection is provided
by means of coordinating gaps or surge diverters coupled with the use of insulation
coordination and winding design has developed to a stage
at which more effective measures are available than reinforcement of end turns.
This development followed on from a fuller under standing of the response of
windings to highfrequency transients and a recognition of the part played
by capacitances at these high frequencies.
Lightning impulses
The following description of the effect of lightning impulses on transformer
windings is based on material contained in a book 'Power Transformers for High
Voltage Transmission with Special Reference to their Design' by Duncan McDonald
(now the late Sir Duncan McDonald), formerly Chief Designer, Transformer Department,
Bruce Peebles, published by Bruce Peebles and Company Limited.
FIG. 52 Equivalent circuit of transformer for simplified uniform winding.
The circuit parameters, uniformly distributed, are:
L _ Inductance Cs _ Series (turntoturn) capacitance Cg _ Shunt (turntoground)
capacitance rL _ Loss component of inductance (winding resistance) rs _ Loss
component of series capacitance r _ Loss component of shunt capacitance
FIG. 53 Transient voltage response of transformer winding: (a) initial and final
distribution of impulse voltage and (b) transitional timespace distribution
of impulse voltage
In assessing its surge behavior the transformer may be represented by an equivalent
network possessing capacitance, inductance and resistance elements as shown
in FIG. 52. The voltage response of the transformer, the space distribution
of potential through its windings at any instant of time, is a function of
the magnitude and disposition of these circuit elements and of the nature of
the incident voltage. In practice impulse voltages are characterized by a rapid
rise to their crest value followed by a relatively slow decline to zero 
by a front of high and a tail of low equivalent frequency. The steeper the
front and the flatter the tail of the wave, the more severe its effect on the
windings and for this reason, coupled with the analytical convenience and clearer
understanding of the principles involved, it is convenient to regard the incident
impulse voltage as a unit function wave having a front of infinite and a tail
of zero equivalent frequency.
The orthodox explanation of the transient behavior of the windings is based
on the time response of the circuit elements to these equivalent frequencies.
At the instant of incidence of the impulse the capacitance elements alone
react to the front of the wave establishing an initial distribution of potential
which is usually nonuniform (FIG. 53(a)). At the end of the phenomenon,
during the tail of the wave, the resistance elements govern the response establishing
a final distribution which is usually uniform (FIG. 53(a)). The transitional
behavior between the initial and final extremes takes the form of damped transference
of electrostatic and electromagnetic energy during which complex oscillations
are usually developed (FIG. 53(b)). It can be seen that all parts of the
winding may be severely stressed at different instants in time; initially,
concentrations of volt age may appear at the line end of the winding; during
the transitional period, concentrations may appear at the neutral end whilst
voltages to ground considerably in excess of the incident impulse may develop
in the main body of the winding.
Generally, under steadystate conditions, equal voltages are induced between
turns and consequently, ideally, equal amounts of insulation are required between
turns. To utilize this uniformly disposed insulation to best advantage, the
volt ages appearing between turns throughout the winding under impulse conditions
should also be equal. To approach this ideal, in which oscillation voltages
are completely eliminated, the initial distribution, like the final, must also
be uniform. Unfortunately, for many years, the basic theories and practice
always showed that uniform windings, with a uniform final distribution, inherently
exhibited a grossly nonuniform initial distribution. Faced with this paradox,
the designer has concentrated in determining by what artifice he might improve
the initial distribution whilst striving to maintain winding uniformity.
The initial voltage distribution It will be recalled that the initial distribution
is determined wholly by the equivalent capacitance network. Consequently two
circuit elements are avail able for controlling and improving the initial response
 the shunt capacitance Cg and the series capacitance Cs. When a unit function
wave is applied to the line terminal of a winding whose equivalent network
is shown in FIG. 54, the initial distribution of impulse voltage is determined
from the differential equation of the capacitance network (FIG. 54(a)). This
equation may (for a uniform winding of length L, of uniform interturn capacitance
Cs, of uniform turntoground capacitance Cg) be expressed in terms of the
instantaneous volt age to ground ex at any point x (measured from the neutral
terminal) as:
FIG. 54 Initial distribution of impulse voltage in a uniform winding with
grounded neutral: (a) equivalent capacitance circuit of winding and (b) curves
of initial distribution of impulse voltage corresponding to various values
of factor a
Solution of Eq. (eqn. 49) may be found in the form:
(eqn. 50)
where the constants of integration A and B are defined by substituting the
boundary conditions. In particular, if the winding neutral is solidly grounded
ex _ 0 when x _ 0 and from Eq. (eqn. 50)
(eqn. 51)
In addition, ex _ E, the incident surge, when x _ L and, from Eq. (eqn. 50) E
_ A(ea _ e_a)
Substituting this value of A in equation (eqn. 51) it is seen that
(eqn. 52)
FIG. 54(b), which is prepared from Eq. (eqn. 52), illustrates the variation
of the initial distribution with a. It will be seen that when a _ 0 (when the
shunt capacitance is zero, or the series capacitance infinite) the initial
distribution is uniform and coincident with the final distribution; it will
also be seen that as a increases the nonuniformity is aggravated. Clearly
the distribution may be improved by decreasing the shunt capacitance (or nullifying
it partially or wholly by electrostatic shields) and/or by increasing the series
capacitance.
The former is not very practicable and it is therefore the latter approach
which has formed the main strategy for improving the response of transformer
windings to lightning surges. Section 4.4 describes the methods which have
been developed for increasing the series capacitance for practical windings.
The use of interleaving, which is now one of the most common methods of increasing
series capacitance in fact enables near uniform initial distributions to be
obtained thus achieving the ideal of utilizing the same uniform interturn insulation
structure for both impulse and steadystate withstand. It should be stressed,
however, that winding design is a matter of economics and not necessarily one
of achieving ideals. Interleaving is an expensive method of winding and where
acceptable stress distributions can be obtained without recourse to this method,
say, by the use of shields between end sections, designers will always prefer
to do this. As unit ratings get larger there will be a tendency for Cg to get
smaller relative to Cs anyway due to increase in physical size and increased
clearances. In addition, the volts per turn will be greater in a larger unit
so that the total number of turns, and hence number of turns per section, will
be reduced. The next most critical area after the lineend interturn stress
is usually the stress between the first two sections. A reduction in the number
of turns per section will help to reduce this. These factors usually mean that
impulse stress in a large highvoltage transformer is less than that in one
of lesser rating but having the same rated voltage. In the smaller rated unit
inter leaving might be essential, whereas for the larger unit it will probably
be possible to avoid this.
The final voltage distribution
The form of the final voltage distribution can be calculated in a similar
manner to that for the initial distribution. For an incident wave with an infinite
tail the capacitance and inductance elements of FIG. 52 appear respectively
as open and short circuits and the resulting final distribution is governed
wholly by the resistive elements. It will be seen that these resistive elements
form a network identical to that of the capacitance network (FIG. 54(a))
if Cs is replaced by
[...]
The differential equation for this network may therefore be written:
(eqn. 53)
The solution of this equation, which is of the same form as Eq. (eqn. 49) is
clearly given by:
(eqn. 54)
(eqn. 55)
which is a uniform distribution of potential from line to ground.
Partwinding resonances
As in any network consisting of inductances and capacitances, transformer
windings are capable of oscillatory response to certain incident disturbances.
When the disturbance has the appropriate properties severe dielectric stresses
and, on occasions, failure can result.
In the discussion above relating to lightning impulses the incident disturbance
is a onceonly occurrence. The oscillatory circuits receive a single burst
of energy and return by free oscillations at their natural frequencies to a
steady state. Since in most cases the maximum voltage developed in the transformer
windings occurs during the first one or two oscillations, the natural frequency
and damping of the oscillatory circuits are of only secondary importance.
In contrast, however, certain switching transients may consist of an initial
peak voltage followed by an oscillatory component. If the frequency of this
oscillation coincides with a natural frequency of the windings a resonance
can develop which can take several cycles to reach its maximum amplitude.
The value of this maximum amplitude is dependent on the damping of both the
incident transient and of the windings themselves but it can on occasions be
greater than the voltage resulting from a lightning impulse. It should be recognized
that, unlike the case of designing in resistance to lightning impulses, the
solution to resonance problems cannot be achieved by transformer manufacturers
acting in isolation. Resonance always requires a passive structure, namely
the transformer windings, and an active component represented by the various
sources of oscillating voltages in the electrical system.
Resonances became recognized as a cause of dielectric failures in the early
1970s and a number of technical papers dealing with the subject were published
over the next decade. The majority of these described specific incidents which
had led to the failure of EHV transformers and although the mechanism of failure
was ascribed to resonance phenomena the papers generally pro vided little information
concerning the source and the nature of the initiating disturbance. In 1979
CIGRE set up a Working Group to deal with resonance problems and to report
on the state of the art, including the provision of a description of the response
of transformers to oscillating voltages and making a survey of the possible
sources of oscillating voltages in electrical systems. The Working Group's
findings were presented at the August/September, 1984, session [6.6] and the
following notes represent a summary of the salient points from their report.
Only power transformers above 110 kV were considered and furnace and other
special transformers were excluded. The Working Group also noted that their
findings were in line with those of an American IEEE working group dealing
with the same subject.
FIG. 55 Equivalent network and response to a periodic and oscillating
voltages
The report described studies carried out on a 405/115/21 kV, 300 MVA sub station
transformer having tappings of _13 _ 4.675 kV per step on the HV winding. The
arrangement is shown in FIG. 55. The application of a step voltage on an
arbitrarily chosen terminal of the transformer will cause the 'net work' to
oscillate. In principle, since the step function contains all frequencies,
each natural frequency inherent in the network will be excited. The values
of the frequencies and their related amplitudes depend on the parameters of
the network and the boundary conditions. The total number of natural frequencies
is given by the number n of free nodes. The values of the amplitudes are also
a function of the location, that is of the ordering number j of a node. Because
of the presence of resistances the oscillations are more or less damped, so
finally the response at an arbitrary chosen node j to a step voltage US is
(disregarding slight phase shifts) given by:
(eqn. 56)
Aj00 describes the final steadystate voltage distribution a is the damping
constant.
The response to a standard 1.2/50 full wave (FIG. 55(a)) is similar to the
response (eqn. 56) to a step voltage. FIG. 55(b) shows the voltage generated
at the free oscillating end of the tapped winding (node m) under the given
boundary conditions. From the oscillogram (FIG. 55(b)) it can be seen that
there is a dominant natural frequency f i
*
of about 40 kHz, the related amplitude Ai*,m has a value of about 0.2 per
unit. Application of a steadystate sine wave (FIG. 55(c)) with an amplitude
UR (1 per unit, that is 420 _ _2/_3) and a frequency fi
* causes resonance and, according to oscillogram (FIG. 55(d)), a volt age
with a maximum peak of ARi*,m of 3.85 _ UR is generated at node m. The amplitude
is limited to this value due to the inner damping of the transformer d.
The report gives two methods of determining the degree of this internal damping.
The first is to vary the frequency of the applied voltage and to make a second
measurement, for instance at 0.9 _ f i*. FIG. 56 shows a plot of maximum
peak amplitude against frequency. From a measured second peak of 1.04 per unit
a ratio of 1.04/3.85 _ 0.27 is derived and plotting this on the curve of FIG. 56 gives a value for d of 0.85, which is stated by the report to accord with
the value which can be estimated from the impulse response.
FIG. 56 Influence of frequency and inner damping on the resonance amplitude
(? _ 1)
The second method makes use of another mathematical relationship which can
be used to determine the ratio:
(eqn. 57)
FIG. 57 V and t as a function of damping
Values of vmax are shown in FIG. 57 plotted for varying values of inner
damping, d, and external damping, that is the damping of the applied oscillatory
voltage, ?. For a value of vmax of 3.85/0.2 _ 19.25 and ? _ 1 (i.e. no external
damping) it can again be seen that this gives a value of d _ 0.85. FIG. 57
also permits the time, expressed as number of cycles t at which the maximum
amplitude occurs to be determined.
Formula (eqn. 57) also enables a comparison to be made of the voltage stresses
in the case of resonance with those generated during impulse testing, especially
if a natural frequency is dominating the impulse voltage response. Comparing
oscillograms (FIG. 57(b)) and (FIG. 57(d)) reveals that the voltage generated
during undamped resonance conditions is significantly higher than that under
impulse test conditions. In order that this stress should not be exceeded by
a resonance excitation of the same amplitude (i.e. 1 per unit) external damping
? must be less than 0.9. Hence, from the curve of FIG. 57 vmax _ 8.7. In
this case the resonance voltage would be:
[...]
which is in good agreement with oscillogram (FIG. 57(f )).
Another way of describing the resonance response is the comparison with the
voltage under rated conditions. This relationship is quantified by the socalled
qfactor.
(eqn. 58)
where rj
is the actual turns ratio of the node j.
For the example for which rm _ 0.2, a qfactor of qi*,m _ 19.25 can be calculated.
In this case either vmax or the qfactor permit the estimation of the maximum
stresses developed under resonance condition at node m. However, as the dominating
natural frequency of the regulating winding also influences the other windings
it might be inferred that a resonance identified on one arbitrary node on the
tapping winding might be indicative of other high stresses in other windings.
FIG. 58 shows the calculated spatial distribution of the amplitudes Ai*,
j throughout the HV winding. Although the qfactors for volt ages to ground
are rather moderate or even zero, high qfactors result between certain parts
of the winding. From the gradients of the spatial amplitude distribution qfactors
of up to 22 can be calculated and such values have indeed been reported.
FIG. 58 Distribution of Aj *,j along the HV winding.
In summary, the Working Group came to the following conclusions as regards
resonances within transformer windings:
• There is a close interdependence between impulse voltage response and resonance
response.
• Amplitudes of harmonics can therefore to a certain degree be influenced
by controlling the initial voltage distribution of a standard impulse wave
for individual windings.
• There must be an awareness of the fact that transferred oscillations from
other windings cannot be suppressed and may cause severe stresses.
• Internal damping is a decisive factor on the resonance response.
• qfactors may be misleading and should not be used in assessing transformer
behavior.
Determination of resonance response
The Working Group considered that a very
detailed analysis was necessary to get precise information about the resonance
of a particular transformer. Three different approaches are possible  calculation,
measurement or a combination of the two. The calculable number of harmonics
depends on the degree of subdivision of the equivalent network. To get sufficient
information about the spatial amplitude distribution demands a large number
of elements. Elaborate computer programs have to be used but the accuracy
of the results still depends on the validity of the parameters inserted. From
FIG. 56 it can be seen that a deviation from the resonance frequency of only
a few percent, considerably less than the margin of error in many instances,
can affect the apparent amplitude by a large amount. In addition, at the
present time there are no exact methods available for determining damping factors,
the computation has to be based on empirical values and is therefore of limited
accuracy. On the other hand to obtain a full assessment of the resonant response
from measurements is very laborious, costly and even risky.
There is the problem of making tappings on inner windings, and measurements
taken out of the tank are inaccurate due to the difference in the permittivities
of air and oil. Hence, the compromise solution of performing a calculation
and checking this by means of measurements taken at easily accessible points
may prove to be the best option. Even this approach will be costly and should
be adopted only if it is considered that a problem may exist.
Sources of oscillating voltages in networks
The Working Group also reported their conclusions concerning the sources of
oscillations in networks. They found that their existence stems from one of
three possible sources:
(1) Lightning
(2) Faults
(3) Switching
Oscillations created by lightning need only be considered if this causes a
switching operation or triggers a fault. Faults comprise single phase to ground
faults and two or threephase short circuits with or without ground fault
involvement. Switching may be initiated by the operator or automatically by
the sys tem protection.
The Working Group investigated 21 categories of incidents including remote
and closeup faults, clearance of faults, reclosing onto faults, energization
of a transformerterminated line, deenergization of an unloaded and loaded
transformer, with and without prestrikes or reignitions as appropriate. Their
analysis revealed that in only three of these categories was there a likelihood
of oscillations which might coincide with a natural frequency of the transformer.
These were:
• Polyphase closeup faults on a single line.
• Energization of a short transformerterminated line from a strong bus.
• Repetitive reignitions during the deenergization of a transformer loaded
with a reactive load.
Closeup faults
These are defined as occurring at a distance of less than 15 km from the transformer,
while the line itself is considerably longer. The transformer is likely to
be struck by a dangerous oscillatory component only in those cases where one
line is connected to the transformer (FIG. 59) and a two or threephase
fault occurs at the critical distance l, given by:
(eqn. 59)
where c is the velocity of the travelling wave, which is about 300 km/ms for
overhead lines and 150 km/ms for cables.
FIG. 59 Close fault
Energization of a transformerterminated line
Switching in a short line through a circuitbreaker fed from a strong busbar
(FIG. 60) creates standing waves which can be within the critical frequency
range. Their frequency can be calculated from Eq. (eqn. 59), where l corresponds
to the length of the line.
FIG. 60 Energization of transformerterminated line
Repetitive reignitions
Breaking of small inductive currents ( 1 kA), in particular the interruption
of magnetizing currents of transformers, may cause oscillations, but these
are in the kHz range and strongly damped, therefore these do not create a risk
of resonance. The interruption of reactive loaded transformer currents can
cause repetitive reignitions at nearly constant time intervals. If the repetition
frequency coincides with one of the lower natural frequencies of the transformer,
resonance may result. A typical configuration for which this can happen is
the case of an unloaded threephase transformer with a reactor connected to
the tertiary winding.
Very fast transients
The majority of switching transients occurring on the system will have slower
wavefronts and lower peak voltages than those resulting from lightning strikes
and will therefore present a less severe threat to the insulation of the transformer
HV windings. The exceptions are certain transients which can arise as a result
of switching operations and fault conditions in gasinsulated substations (GIS).
These are known as very fast transient overvoltages (VFTOs). The geometry and
dielectric of GIS (metallic sheath, coaxial structure and short dielectric
distances) lend themselves well to the generation and propagation of VFTOs.
Studies of the characteristics of VFTOs have indicated that typically these
might have rise times of 20 ns and amplitudes of 1.5 per unit. In the worst
condition a rise time of 10 ns and an amplitude of 2.5 per unit is possible.
The steep fronted section of the wave is often followed by an oscillatory component
in the frequency range 110 MHz, the precise value being dependent on the travelling
wave characteristics of the GIS system.
The VFTOs arriving at a transformer winding are more difficult to predict
since magnitude and fronttime depends on the transformer parameters and the
precise nature and length of the winding connection to the GIS. In the worst
case, the fronttime will be only slightly increased and the amplitude increased
by possibly 30 percent.
When the VFTOs reach the transformer windings two problems can arise.
Their very much higher frequency compared with standard impulse waves results
in high intersection stresses which are usually concentrated in the sections
near to the line end. These stresses cannot be controlled by interleaving in
the same way as can lightning impulse stresses. The second problem is the production
of partwinding resonance resulting from the oscillatory wavetail of the VFTO.
This can create oscillatory voltages within the end sections of the transformer
windings, producing intersection stresses many times greater than those resulting
from lightning impulses.
FIG. 61 Capacitance and reluctance networks FIG. 62 220 kV winding
under test
In attempting to predict the response of transformer windings to VFTOs it
is necessary to represent the winding structure in a similar way to that employed
in performing calculations of impulse voltage distribution, in that capacitances
predominate, however it is no longer sufficient to consider a simple network
having constant values of series and shunt capacitance, Cs and Cg, respectively.
In their paper presented to the 1992 CIGRE Summer Meeting, Cornick and others
used multiconductor transmission line theory to produce a turn by turn
mathematical model of a 40 MVA 220 kV partially interleaved winding.
FIG. 61 shows the type of network considered, in which the capacitance
of each turn is taken into account and, in order to predict the resonant frequencies,
they also take into account the inductance network. Though laborious, the method
lends itself well to computer calculation and, because it is the end sections
which are known to be critical, computing time can be reduced by restricting
the solution to the end, say the first four, sections of the winding.
The authors compared their predicted intersection and interturn voltages with
measurements made on the actual winding following application of the output
from a recurrent surge generator producing front chopped impulses. These had
a prospective fronttime of 1.2 µs chopped at that time. Voltage collapse time
was 230 ns, relatively slow for a VFTO. FIG. 62 shows the arrangement of
the winding end sections and the applied, predicted and measured intersection
voltage between the end two sections. The resonance frequency predicted by
calculation was 2.12 MHz and that obtained by measurement 2.22 MHz, considered
by the authors to represent good agreement.
Apart from noting the high level of the intersection voltages observed, the
above paper makes little general recommendation regarding the need for protection
when connecting transformers directly to GIS, or the form which any protection
might take. The authors do note, however, that the use of inductances, suitably
damped, in series with the transformer windings might be justified in specific
cases.
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