THE INFLUENCE OF TRANSFORMER CONNECTIONS ON THIRDHARMONIC VOLTAGES AND
CURRENTS
It is the purpose of this section, firstly, to state the fundamental principles
of third harmonic voltages and currents in symmetrical threephase systems;
secondly, to indicate their origin in respect of transformers; thirdly, to
marshal the facts and present them in tabular form; and finally, to indicate
their undesirable features.
No new theories are introduced, but facts, often understood in a more or less
vague sort of way, are hopefully crystallized and presented in a clear manner.
The treatment is confined to threephase transformers with double windings,
as the principles, once clearly understood, are easily applicable to polyphase
autotransformers.
Principles of third harmonics in symmetrical threephase systems The two forms
of connections of threephase systems behave differently as regards thirdharmonic
voltages and currents and so need to be considered separately.
(a) Star In any starconnected system of conductors it is a basic law that
the instantaneous sum of the currents flowing to and from the common junction
or star point is zero.
In a symmetrical threephase, threewire starconnected system, the currents
and voltages of each phase at fundamental frequency are spaced 120 apart. At
any instant the instantaneous current in the most heavily loaded phase is equal
and opposite in direction to the sum of the currents in the other two phases,
and at fundamental frequency this balance is maintained throughout the cycle.
At thirdharmonic frequency, however, currents flowing in each phase would
be 3 x 120 = 360º apart, that is in phase with one another and flowing in the
same relative direction in the phases at the same instant. The sum of the currents
in the star connection would therefore not be zero, and consequently in a symmetrical
threephase, threewire starconnected system thirdharmonic currents cannot
exist.
* With ungrounded neutral, or from each line to neutral with grounded neutral.
FIG. 140 Phenomenon of 'oscillating neutral' in a symmetrical threephase,
threewire starconnected system with ungrounded neutral.
If, however, a connection is taken from the neutral point in such a manner
that it completes the circuit of each phase independently (though through a
common connection), a current at 3 times the fundamental frequency can circulate
through each phase winding and through the lines and the fourth wire from the
neutral point. The fourth wire acting as a drain for thirdharmonic currents
preserves the current balance of the system; it has, of course, no effect on
the currents at fundamental frequency, as these are already balanced.
Thirdharmonic voltages, on the other hand, can exist in each phase of a symmetrical
threephase, threewire starconnected system, that is from each line to ground,*
but they cannot appear in the voltages between lines. The third harmonic voltages
in each phase are in phase with one another, so that there is one thirdharmonic
phasor only, and the neutral point of the star is located at the end of this
phasor. The potential of the neutral point is consequently not zero, but oscillates
round the zero point at triplefrequency and thirdharmonic voltage. FIG. 140 illustrates this and also shows how the thirdharmonic voltages to ground
cancel out, so far as the voltages between lines are concerned, leaving the
line terminal voltages free from their influence.
When a connection is taken from the neutral point in such a manner as to allow
thirdharmonic currents to flow, the thirdharmonic voltages to neutral are
expended in forcing the currents round the circuits. It will be seen subsequently
that according to the characteristics of the circuit in which these currents
flow, the thirdharmonic voltages may be suppressed totally or only partially.
(b) Delta
In any deltaconnected system of conductors the resultant fundamental voltage
round the delta is zero. That is, the addition of the voltage phasors at fundamental
frequency which are spaced 360º/m (where m = number of phases) apart forms
a regular closed polygon.
In a symmetrical threephase deltaconnected system thirdharmonic volt ages
tending to occur in each phase would be spaced 360 degr. apart, and so would
be in phase with each other and act in the closed delta circuit as a singlephase
voltage of thirdharmonic frequency. Such a voltage could not actually exist
in a closed delta system, so that thirdharmonic currents circulate round the
delta without appearing in the lines and the thirdharmonic voltages are suppressed.
In discussing the thirdharmonic aspect of various combinations of star and
delta connections for threephase transformer operation, we therefore have
the following bases to work upon:
(1) With a threewire star connection, thirdharmonic voltages may exist between
lines and neutral or ground, but not between lines.
(2) With a threewire connection, thirdharmonic currents cannot exist.
(3) With a fourwire star connection, thirdharmonic voltages from lines to
neutral or ground are suppressed partially or completely according to the impedance
of the thirdharmonic circuit.
(4) With a fourwire star connection, thirdharmonic currents may flow through
the phases and through the line wires and fourth wire from the neutral.
(5) With a threewire delta connection, thirdharmonic voltages in the phases
and hence between the lines are suppressed.
(6) With a threewire delta connection, thirdharmonic current may flow round
the closed delta, but not in the lines.
Origin of thirdharmonic voltages and currents in transformers
It should be understood that this discussion is quite distinct and apart from
higher harmonic functions of the source of supply, and it is limited to those
which are inherent in the magnetic and electric circuits of the transformer.
The two circuits being closely interlinked, it is a natural sequel that the
higher harmonic phenomena occurring in both should be interdependent.
There are two characteristics in the behavior of sheetsteel transformer laminations
when under the influence of an alternating electromagnetic field, which produce
an appreciable distortion in the waveform (from the standard sine wave) of
certain alternating functions. These functions are noload cur rent, flux and
induced voltages, any distortion of which is due to the varying permeability
of the core steel plates and to cyclic magnetic hysteresis. For the purpose
of this section, the range of the phenomena involved is more briefly and cogently
explained by means of diagrams with short explanations than by lengthy dissertation
and tedious mathematical equations. FIGs. 141147 inclusive, together
with the following remarks, aim at attaining this end. FIG. 141 shows a
typical B/H curve with hysteresis loop for coldrolled steel; the hysteresis
loop illustrates the general shapes that would occur in practice.
FIG. 142 shows the waveform relation between the noload current, flux and
induced e.m.f. when the e.m.f. is a sine wave and when hysteresis is absent.
From a study of these curves it will be seen that the current is a true magnetizing
current, being in phase with the flux, its peaked form showing the presence
of a prominent third harmonic. It will also be noted that this wave is symmetrical
about the horizontal axis, and each halfwave about a vertical axis. The flux
must, of course, be sinusoidal on account of the assumption of a sine waveinduced
e.m.f.
FIG. 141 Typical B = H curve and hysteresis loop for coldrolled steel
FIG. 142 Noload current, flux and induced voltage waves, with a sine
wave of applied voltage: i0 = 100 sin θ  54.7 sin θ + 31.5 sin 5 θ
+…
FIG. 143 Noload current, flux and induced voltage waves, with a sine
wave of applied voltage; hysteresis effects included
FIG. 143 is similar to Fig. 142 with the exception that hysteresis is
taken into account. In this case the current is not a true magnetizing current
on account of the hysteresis component which is introduced, which makes the
noload current lead the flux by a certain angle θ, the hysteretic angle of
advance. This figure also shows that for the same maximum flux the maxi mum
values of the true magnetizing and noload current are the same, but that when
taking hysteresis into account the noload current becomes unsymmetrical about
a vertical axis passing through its peak. It will, however, be seen by comparing
Figs 142 and 143 that the thirdharmonic component is contained almost
entirely in the true magnetizing current, and very little, if any, in the current
component due to hysteresis, thus indicating that thirdharmonic currents are
produced as a result of the varying permeability of the core steel, and only
in a very minor degree by magnetic hysteresis.
FIG. 144 shows the waveform relation between the noload current, flux and induced
e.m.f. when the current is a sine wave and when hysteresis is absent. As in the
case of FIG. 142, the current is a true magnetizing cur rent and in phase with
the flux. The flux wave is flat topped, which indicates the presence of a third
harmonic in phase with the fundamental, the harmonic having a negative maximum
coincident with the positive maximum of the fundamental, and so producing a flattopped
resultant wave. It will be noticed that the flux wave is symmetrical about the
horizontal axis, and each halfwave about a vertical axis. The induced e.m.f.
is, of course, affected by the departure of the shape of the flux wave from the
sine, a flattopped flux wave producing a highly peaked wave of induced e.m.f.
(as shown in the figure), in which also appears a prominent third harmonic. In
the case of the voltage wave the third harmonic is in opposition to the fundamental,
the positive maximum of fundamental and harmonic waves occurring at the same
instant, so that the resultant voltage wave becomes peaked.
FIG. 144 Noload current, flux and induced voltage waves, with a sine
wave of noload current; hysteresis effects excluded:
M = 100 sin θ + 22.9 sin 3 θ + 5.65 sin 5 θ +…
E = 100 cos θ + 69.0 cos 3 θ +
28.4 cos 5θ +…
FIG. 145 is similar to FIG. 144 with the exception that hysteresis is
taken into account. In this case the noload current leads the flux, thereby
producing the hysteretic angle of advance flas in the case of FIG. 143.
The flux wave is somewhat more flat topped, and while still symmetrical about
the horizontal axis, each halfwave is unsymmetrical about a vertical axis
passing through its peak.
The induced voltage waves of Figs 144 and 145 do not take into account
harmonics above the fifth, and this accounts for the ripples on the zero axis.
Hysteresis does not alter the maximum value of the flux wave, though it increases
its dissymmetry; the wider the hysteresis loop the greater the dissymmetry
of the flux wave.
FIG. 145 Noload current, flux and induced voltage waves with a sine wave
of noload current, hysteresis effects included:
Fm =100 sinθ + 22.9 sin 3θ + 5.65 sin 5θ +…
E= 100 cosθ + 69.0 cos 3θ
+ 28.4 cos 5θ + …
FIG. 146 Harmonic analysis of peaked noload current wave of FIG. 142
i0 = 100 sinθ +31.5 sin 5 θ + …
FIGs. 146 and 147 show the analysis up to the fifth harmonic of the
magnetizing current wave, i0, FIG. 142, and the induced voltage wave E,
FIG. 144; in each case waves are given showing the sum of the fundamental
and third harmonic, and indicating the degree of the error involved in ignoring
harmonics beyond the third. In order to obtain some idea at a glance of the
approximate phase of the thirdharmonic relative to the fundamental in a composite
wave FIG. 148 shows the shape of the resultant waves obtained when combining
the fundamental and third harmonic alone with different positions of the harmonic.
From the foregoing discussion on the origin of third harmonics the following
conclusions are to be drawn:
(1) With a sine wave of flux, and consequently induced voltages, the noload
current contains a prominent third harmonic which produces a peakiness in the
wave. The third harmonic is introduced mainly into the true magnetizing current
component through the variation in the permeability of the sheet steel and
only in a very small degree into the hysteresis component of the current by
the cyclic hysteresis.
(2) With a sine wave of noload current the flux and consequently the induced
voltages contain prominent third harmonics which produce a flattopped flux
wave and peaked induced voltage waves.
FIG. 147 Harmonic analysis of peaked induced voltage wave of FIG. 144
E = 100 cosθ + 69 cosθ = 28.4 cos 5θ +…
FIG. 148 Thirdharmonic distribution of inductance, resistance and capacitance
in an ungrounded neutral threephase circuit consisting of the secondaries
of a threephase group of single phase transformers supplying an openended
transmission line
Undesirable features of third harmonics
These are summarized under two headings as follows:
Due to thirdharmonic currents:
(a) Overheating of transformer windings and
of load.
(b) Telephone and discriminative protective gear magnetic disturbances.
(c) Increased iron loss in transformers.
Due to thirdharmonic voltages (d) Increased transformer insulation stresses.
(e) Electrostatic charging of adjacent lines and telephone cables.
(f) Possible resonance at thirdharmonic frequency of transformer windings
and line capacitance.
These disadvantages may briefly be referred to as follows:
(a) In practice, overheating of the transformer windings and load due to the
circulation of thirdharmonic current rarely occurs, as care is taken to design
the transformer so that the flux density in the core is not so high as to increase
the thirdharmonic component of the noload current unduly.
Apart from the question of design, a transformer might, of course, have a
higher voltage impressed upon it than that for which it was originally designed,
but in this case the increased heating from the iron loss due to the resulting
higher flux density would be much more serious than the increased heating in
the transformer windings due to larger values of the thirdharmonic circulating
current. These remarks hold good, irrespective of whether the transformer windings
are delta connected or star connected with a fourth wire system.
The only case where the circulation of the thirdharmonic currents is likely
to become really serious in practice is where the transformer primary windings
are connected in interconnected star, the generator and transformer neutrals
being joined together.
(b) It is well known that harmonic currents circulating in lines paralleling
telephone wires or through the ground where a telephone ground return is adopted
produce disturbances in the telephone circuit. This is only of practical importance
in transmission or distribution lines of some length (as distinct from short
connections to load), and then as a rule it only occurs with the star connection
using a fourth wire, which may be one of the cable cores or the ground.
Similar interference may take place in the pilot cores of discriminative protective
gear systems, and unless special precautions are taken relays may operate incorrectly.
The remedy consists either of using a deltaconnected transformer winding
or omitting the fourth wire and grounding at one point of the circuit only.(c)
In the case of a threephase bank of singlephase transformers using a star/
star connection, it has been proved experimentally that a fourth wire connection
on the primary side between the transformer bank and generator neutrals (which
allows the circulation of thirdharmonic currents) results in increasing the
iron loss of the transformers to 120 percent of that obtained with the neutrals
disconnected. This figure varies according to the design of the transformers
and the impedance of the primary circuit. The conditions are similar for threephase
shelltype transformers.
Under certain conditions, the thirdharmonic component of the phase voltage
of star/starconnected threephase shelltype transformers or banks of singlephase
transformers may be amplified by the line capacitances.
This occurs when the HV neutral is grounded, so that thirdharmonic cur rents
may flow through the transformer windings, returning through the ground and
the capacitances of the line wires to ground. The amplification occurs only
when the capacitance of the circuit is small as compared to its inductance,
in which case the thirdharmonic currents will lead the third harmonic voltages
almost by 90º, and they will be in phase with the third harmonic component
of the magnetic fluxes in the transformer cores. The thirdharmonic component
of the fluxes therefore increases, which in turn produces an increase in the
thirdharmonic voltages, and a further increase of the thirdharmonic capacitance
currents. This process continues until the transformer cores become saturated,
at which stage it will be found the induced voltages are considerably higher
and more peaked than the normal voltages, and the iron loss of the transformer
is correspondingly greater. In practice, the iron loss has been found to reach
3 times the normal iron loss of the transformer, and apparatus has failed in
consequence.
This phenomenon does not occur with threephase coretype transformers on
account of the absence of third harmonics.
(d) It has been pointed out previously that with the threewire star connection
and isolated neutral a voltage occurs from the neutral point to ground having
a frequency of 3 times the fundamental, so that while measurements between
the lines and from lines to neutral indicate no abnormality, a measurement
from neutral to ground with a sufficiently low reading voltmeter would indicate
the magnitude of the third harmonic. In practice, with singlephase transformers
the thirdharmonic voltages may reach a magnitude of 60 percent of the fundamental,
and this is a measure of the additional stress on the transformer windings
to ground. While due to the larger margin of safety it may not be of great
importance in the case of distribution transformers, it will have considerable
influence on the reliability of transformers at higher voltages.
(e) Due also to the conditions outlined in (d), starconnected banks of single
phase transformers connected to an overhead line or underground cable, and
operated with a grounded or ungrounded neutral, may result in an electrostatic
charging at thirdharmonic frequency of adjacent power and telephone cables.
This produces abnormalinduced voltages to ground if the adjacent circuits
are not grounded, the whole of the circuit being raised to an indefinite potential
above ground even though the voltages between lines remain normal. The insulation
to ground, therefore, becomes unduly stressed, and the life of the apparatus
probably reduced.
(f ) A further danger due to the conditions outlined under (e) is the possible
resonance which may occur at thirdharmonic frequency of the transformer windings
with the line capacitance. This can happen if the transformer neutral is grounded
or ungrounded, and the phenomenon occurs perhaps more frequently than is usually
appreciated, but due to the presentday complicated networks and the resulting
large damping constants, the magnitude of the quantities is such as often to
render the disturbances innocuous.
Further notes on third harmonics with the star/star connection It is generally
appreciated that threephase shelltype transformers and three phase groups
of singlephase transformers should not have their windings connected star/star
on account of the thirdharmonic voltages which may be generated in the transformers
at the normal flux densities usually employed. It is, however, not so equally
well known that under certain operating conditions the star/star connection
of the type of transformers referred to above may pro duce serious overheating
in the iron circuit in addition to augmented stresses in the dielectrics. The
conditions referred to are when the secondary neutral of the transformer or
group is grounded, the connecting lines having certain relative values of electrostatic
capacitance.
Consider a threephase stepup group of singlephase transformers having their
windings star/star connected, each transformer of such a group having a flux
density in the core of approximately 1.65 Tesla.
With isolated neutrals on both sides, no thirdharmonic currents can flow,
and consequently the magnetic fluxes and induced voltages would contain large
thirdharmonic components, the flux waves being flat topped and the induced
voltage waves peaked. The magnetizing current waves would be sinusoidal. At
the flux density stated, the flux waves would have a thirdharmonic component
approximately equal in amplitude to 20 percent of that of the fundamental,
and the resulting induced voltage waves would have third harmonic components
of amplitudes of approximately 60 percent of that of their fundamentals. With
isolated neutrals the thirdharmonic components of the voltage waves would
be measurable from each neutral to ground by an electro static voltmeter. Their
effects would be manifested when measuring the volt ages from each line terminal
to the neutral point by an ordinary moving iron or similar voltmeter. There
would be no trace of them when measuring between line terminals on account
of their opposition in the two windings which are in series between any two
line terminals so far as third harmonics are concerned.
With isolated neutrals the only drawback to the thirdharmonic voltage components
is the increased dielectric stress in the transformer insulation.
It should be borne in mind that so far as third harmonics of either voltage
or current are concerned the transformer windings of each phase are really
in parallel and the harmonics in each winding have the same time phase position.
When such transformers are connected to transmission or distribution lines
on open circuit, the parts which are effective so far as third harmonics are
concerned can be represented as shown in FIG. 149(a) where we have three
circuits in parallel, each consisting of one limb of the transformer with the
capacitance to ground of the corresponding line, this parallel circuit being
in series with the capacitance between ground and the neutral point of the
transformer. By replacing the three parallel circuits by a simple equivalent
circuit consisting of a resistance, inductance and capacitance, FIG. 149(a)
can be simplified to that shown in FIG. 149(b). The inductance is that of
the three phases of the transformer in parallel, and the voltage across these
is the third harmonic voltage generated in each secondary phase of the transformer.
As the thirdharmonic voltages are generated in the transformer windings on
account of the varying permeability of the magnetic cores, the inductance shown
in FIG. 149(b) can be looked upon as being a triplefrequency generator supplying
a voltage equal to the thirdharmonic voltage of each phase across the two
capacitors in series. The capacitor 3CL is equal to 3 times the capacitance
to ground of each line while the capacitor CN represents the capacitance from
the neutral point to ground. By comparison the latter capacitor is infinitely
small, so that as a voltage applied across series capacitors divides up in
inverse proportion to their capacitances, practically the whole of the thirdharmonic
voltage appears across the capacitor formed between the transformer neutral
point and ground. This explains why, in star/starconnected banks having isolated
neutrals, the thirdharmonic voltage can be measured from the neutral point
to ground by means of an electrostatic voltmeter.
Now consider the conditions when the secondary neutral point is grounded,
the secondary windings being connected to a transmission or distribution line
on open circuit. This line, whether overhead or underground, will have certain
values of capacitance from each wire to ground, and so far as third harmonics
are concerned the circuit is as shown diagrammatically in FIG. 150(a).
It will be seen that the only difference between this figure and FIG. 149(a)
is that the capacitor CN between the neutral point and ground has been short
circuited. The effect of doing this may, under certain conditions, produce
undesirable results. The compound circuit shown in FIG. 150(a) may be replaced
by that shown in FIG. 150(b), where resistance, inductance and capacitance
are respectively the single equivalents of the three shown in parallel in FIG.
150(a), and from this diagram it will be seen that all the thirdharmonic voltage
is concentrated from each line to ground. Under this condition the third harmonic
component cannot be measured directly, but its effects are manifested when
measuring from each line terminal to ground by an ordinary moving iron or similar
instrument.
FIG. 149 Thirdharmonic distribution of inductance, resistance and capacitance
in a grounded neutral threephase circuit consisting of the secondaries of
a threephase group of single phase transformers supplying an openended transmission
line.
The chief difference between the conditions illustrated in Figs 149(a) and
150(a) is that where in the first case no appreciable thirdharmonic current
could flow on account of the small capacitance between the neutral point and
ground, in the second case triplefrequency currents can flow through the transformer
windings completing their circuit through the capacitances formed between the
lines and ground. We thus see that the conditions are apparently favorable
for the elimination of the thirdharmonic voltages induced in the transformer
windings on account of the varying permeability of the magnetic cores.
This, however, is not all the story, for in order that the thirdharmonic
voltages induced in the transformer windings shall be eliminated, the third
harmonic currents must have a certain phase relationship with regard to the
fundamental sine waves of magnetizing currents which flow in the primary windings.
In practice the thirdharmonic currents flowing in such a circuit as shown
in FIG. 150(a) may or may not have the desired phase relationship, for the
following reasons.
The circuit shown in FIG. 150(b) is a simple series circuit of inductance
L resistance R and capacitance C, the impedance of which is given by the equation,
The resistance R is the combined resistance of the three circuits in parallel
shown in FIG. 150(a), namely, the transformer windings which are grounded,
the lines, and the ground. The capacitance C is the combined capacitance of
the three lines to ground in parallel, as the capacitance of the transformer
windings to ground is so small that it can be ignored. The inductance L is
the combined inductance of the three transformer windings in parallel which
are grounded, the inductance between the lines and ground being ignored on
account of their being very small. The inductance of the transformer windings
corresponds to open circuit conditions, as the triplefrequency currents are
confined to the secondary windings only, on account of the connections adopted.
FIG. 150 Thirdharmonic distribution of inductance, resistance and capacitance
in a grounded neutral threephase circuit consisting of the secondaries of
a threephase group of singlephase transformers supplying an openended transmission
line.
For a circuit of this description the power factor is given by the expression,
and the angle of lead or lag of the current with respect to the applied voltage
is:
If the value of 2pfL is greater than that of 1/2pfC the angle f is lagging,
and if smaller the angle f is leading.
There are three extreme conditions to consider:
(1) when C is very large compared with L;
(2) when L is very large compared with C;
(3) when L and C are equal.
If C is large compared with L the impedance of the combined circuit is relatively
low, so that the line capacitances to ground form, more or less, a short circuit
to the thirdharmonic voltage components induced in the transformer secondary
windings. Under this condition the resulting thirdharmonic cur rents will
be lagging with respect to the thirdharmonic voltage components.
The thirdharmonic currents will act with the fundamental waves of primary
magnetizing current to magnetize the core, and the resulting total ampereturns
will more or less eliminate the thirdharmonic components of the flux waves,
bringing the latter nearer to the sine shape. This will correspondingly reduce
the thirdharmonic voltage components, making the induced voltage waves also
more sinusoidal. The reduction in thirdharmonic voltage components will have
a reflex action upon the thirdharmonic currents circulating through the transformer
secondary windings and the line capacitances, and a balance between thirdharmonic
voltages and currents will be reached when the third harmonic voltage components
are reduced to such an extent as to cause no further appreciable flow of secondary
thirdharmonic currents.
In the extreme case where the line capacitances are so large as to make the
capacitive reactance practically zero, almost the full values of lagging thirdharmonic
currents flow in the secondary windings to eliminate practically the whole
of the thirdharmonic voltages, so that from the thirdharmonic point of view
this condition would be equivalent to deltaconnected secondary transformer
windings. FIG. 151 shows the different current, flux and induced voltage
wave phenomena involved, assuming that C L.
FIG. 151 Induced voltage, flux and magnetizing current waves in a threephase
star/starconnected group of singlephase transformers with secondary neutral
solidly grounded and supplying an openended line such that CL LL
The diagrams of FIG. 151 show the phase relationship of all the functions
involved, but they do not show the actual thirdharmonic flux and voltage reduction
phenomena. The composite diagram of FIG. 151 shows clearly that the thirdharmonic
secondary current is in opposition to the thirdharmonic flux component, and
the result is a reduction in amplitude of the latter. As a consequence the
induced voltage waves become more nearly sinusoidal, and ultimately they approach
the true sine wave to an extent depending upon the value of the capacitance
reactance of the secondary circuit.
When, however, the inductance of the transformer windings is high com pared
with the line capacitances to ground, the thirdharmonic components of the
voltage waves become intensified. In this case the inductive reactance is very
high compared with the capacitive reactance, so that the thirdharmonic voltage
components impressed across the line capacitances produce third harmonic secondary
currents which lead the thirdharmonic secondary voltages.
The angle of lead is given by Eq. (64) and in the extreme case where the
capacitance is very small the thirdharmonic current will lead the third harmonic
voltage almost by 90º. The resulting thirdharmonic ampereturns of the secondary
winding act together with the fundamental exciting ampere turns in the primary,
and as the two currents are in phase with one another their effect is the same
as that produced by a primary exciting current equal to the sum of the fundamental
primary and thirdharmonic secondary currents. The sum of two such currents,
in phase is a dimpled current wave, and compared with the fundamental sine
wave of exciting current the r.m.s. value of the composite current wave is
higher, though more important than this is the fact that such a current wave
produces a very flattopped flux wave. In other words, the thirdharmonic components
of the flux waves are intensified, and on this account the third harmonic voltage
components of the induced voltage waves are also intensified.
Higherthirdharmonic voltage waves react upon the secondary circuit to produce
larger thirdharmonic currents, which in turn increase the third harmonic flux
waves, and again the thirdharmonic voltage waves. This process of intensification
continues until a further increase of magnetizing current produces no appreciable
increase of thirdharmonic flux, so that the ultimate induced voltages become
limited only by the saturation characteristics of the magnetic circuit. It
should be noted that the thirdharmonic currents circulate in the secondary
windings only, as the connections on the primary side do not permit the transfer
of such currents.
FIG. 152 shows the phase relationship of the different current, flux and
induced voltage waves involved, assuming the thirdharmonic currents lead the
thirdharmonic voltage components by 90º. The diagrams of this figure do not
show the actual amplification phenomena. The composite diagram shows very clearly
that the thirdharmonic secondary current is in phase with the thirdharmonic
flux component, and the result is an amplification of the latter.
Therefore, the induced voltage waves become more highly peaked and ultimately
reach exceedingly high values, producing excessive dielectric stresses, high
iron losses, and severe overheating.
FIG. 152 Induced voltage, flux and magnetizing current waves in a threephase
star/starconnected group of singlephase transformers with secondary neutral
solidly grounded and supplying an openended line such that LL CL
FIG. 153 Induced voltage, flux and magnetizing current waves in a threephase
star/starconnected group of singlephase transformers with secondary neutral
solidly grounded and supplying an openended line such that CC LL
Cases have occurred of transformer failures due to this thirdharmonic effect,
and one case is known where, on noload, the transformer oil reached a temperature
rise of 53ºC in 6 hours, the temperature still rising after that time at the
rate of 3ºC/hour.
In the resonant condition where the capacitive and inductive reactances are
equal, the flow of thirdharmonic current is limited only by the resistance
of the secondary circuit. The thirdharmonic currents would be in phase with
the thirdharmonic voltage components, and being of extremely high values they
would produce exceedingly high voltages from each line to ground and across
the transformer windings. The transformer core would reach even a higher degree
of saturation than that indicated in the previous case, and the transformers
would be subjected to excessive dielectric and thermal stresses. FIG. 153
shows the wave phenomena apart from the amplification due to resonance.
The resonant condition fortunately, however, is one that may be seldom met,
but the other two cases are likely to occur on any system employing star/ starconnected
transformers with grounded secondary neutral, and unless some provision can
be made for allowing the circulation of thirdharmonic currents under permissible
conditions threephase shelltype transformers or three phase groups of singlephase
transformers should not so be connected.
With threephase coretype transformers there is still theoretically the same
disadvantage, but as in such transformers the thirdharmonic voltage components
do not exceed about 5 percent of the fundamental, the dangers are proportionately
reduced. However, at high transmission voltages even a 5 percent thirdharmonic
voltage component may be serious in star/starconnected three phase coretype
transformers when the neutral point is grounded, and it is there fore best
to avoid this connection entirely if neutral points have to be grounded.
Precisely the same reasoning applies to threephase transformers or groups
having interconnected star/star windings if it is desired to ground the neutral
point on the starconnected side. With this connection the thirdharmonic volt
ages are eliminated by opposition on the interconnected star side only, but
they are present on the starconnected side in just the same way as if the
windings were star/star connected, their average magnitudes being of the order
5 percent for threephase coretype transformers and 50 60 percent in threephase
shelltype and threephase groups of singlephase transformers.
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