Industrial Power Transformers -- Operation and maintenance [9]

THE INFLUENCE OF TRANSFORMER CONNECTIONS ON THIRD-HARMONIC VOLTAGES AND CURRENTS

It is the purpose of this section, firstly, to state the fundamental principles of third harmonic voltages and currents in symmetrical three-phase 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 three-phase transformers with double windings, as the principles, once clearly understood, are easily applicable to polyphase autotransformers.

Principles of third harmonics in symmetrical three-phase systems The two forms of connections of three-phase systems behave differently as regards third-harmonic voltages and currents and so need to be considered separately.

(a) Star In any star-connected 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 three-phase, three-wire star-connected 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 third-harmonic 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 three-phase, three-wire star-connected system third-harmonic currents cannot exist.

* With ungrounded neutral, or from each line to neutral with grounded neutral.

FIG. 140 Phenomenon of 'oscillating neutral' in a symmetrical three-phase, three-wire star-connected 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 third-harmonic 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.

Third-harmonic voltages, on the other hand, can exist in each phase of a symmetrical three-phase, three-wire star-connected 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 third-harmonic 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 triple-frequency and third-harmonic voltage. FIG. 140 illustrates this and also shows how the third-harmonic 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 third-harmonic currents to flow, the third-harmonic 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 third-harmonic voltages may be suppressed totally or only partially.

(b) Delta

In any delta-connected 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 three-phase delta-connected system third-harmonic 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 single-phase voltage of third-harmonic frequency. Such a voltage could not actually exist in a closed delta system, so that third-harmonic currents circulate round the delta without appearing in the lines and the third-harmonic voltages are suppressed.

In discussing the third-harmonic aspect of various combinations of star and delta connections for three-phase transformer operation, we therefore have the following bases to work upon:

(1) With a three-wire star connection, third-harmonic voltages may exist between lines and neutral or ground, but not between lines.

(2) With a three-wire connection, third-harmonic currents cannot exist.

(3) With a four-wire star connection, third-harmonic voltages from lines to neutral or ground are suppressed partially or completely according to the impedance of the third-harmonic circuit.

(4) With a four-wire star connection, third-harmonic currents may flow through the phases and through the line wires and fourth wire from the neutral.

(5) With a three-wire delta connection, third-harmonic voltages in the phases and hence between the lines are suppressed.

(6) With a three-wire delta connection, third-harmonic current may flow round the closed delta, but not in the lines.

Origin of third-harmonic 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 sheet-steel 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 no-load 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. 141-147 inclusive, together with the following remarks, aim at attaining this end. FIG. 141 shows a typical B/H curve with hysteresis loop for cold-rolled steel; the hysteresis loop illustrates the general shapes that would occur in practice.

FIG. 142 shows the waveform relation between the no-load 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 half-wave about a vertical axis. The flux must, of course, be sinusoidal on account of the assumption of a sine wave-induced e.m.f.

FIG. 141 Typical B = H curve and hysteresis loop for cold-rolled steel

FIG. 142 No-load 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 No-load 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 no-load 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 no-load current are the same, but that when taking hysteresis into account the no-load current becomes unsymmetrical about a vertical axis passing through its peak. It will, however, be seen by comparing Figs 142 and 143 that the third-harmonic 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 third-harmonic 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 no-load 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 flat-topped resultant wave. It will be noticed that the flux wave is symmetrical about the horizontal axis, and each half-wave 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 flat-topped 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 No-load current, flux and induced voltage waves, with a sine wave of no-load 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 no-load 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 half-wave 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 No-load current, flux and induced voltage waves with a sine wave of no-load 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 no-load 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 third-harmonic 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 no-load 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 no-load current the flux and consequently the induced voltages contain prominent third harmonics which produce a flat-topped 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 Third-harmonic distribution of inductance, resistance and capacitance in an ungrounded neutral three-phase circuit consisting of the secondaries of a three-phase group of single phase transformers supplying an open-ended transmission line

Undesirable features of third harmonics

These are summarized under two headings as follows:

Due to third-harmonic 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 third-harmonic voltages (d) Increased transformer insulation stresses.

(e) Electrostatic charging of adjacent lines and telephone cables.

(f) Possible resonance at third-harmonic 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 third-harmonic 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 third-harmonic component of the no-load 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 third-harmonic 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 third-harmonic 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 delta-connected transformer winding or omitting the fourth wire and grounding at one point of the circuit only.(c) In the case of a three-phase bank of single-phase 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 third-harmonic 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 three-phase shell-type transformers.

Under certain conditions, the third-harmonic component of the phase voltage of star/star-connected three-phase shell-type transformers or banks of single-phase transformers may be amplified by the line capacitances.

This occurs when the HV neutral is grounded, so that third-harmonic 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 third-harmonic 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 third-harmonic component of the fluxes therefore increases, which in turn produces an increase in the third-harmonic voltages, and a further increase of the third-harmonic 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 three-phase core-type transformers on account of the absence of third harmonics.

(d) It has been pointed out previously that with the three-wire 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 single-phase transformers the third-harmonic 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), star-connected 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 third-harmonic frequency of adjacent power and telephone cables. This produces abnormal-induced 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 third-harmonic 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 present-day 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 three-phase shell-type transformers and three phase groups of single-phase transformers should not have their windings connected star/star on account of the third-harmonic 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 three-phase step-up group of single-phase 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 third-harmonic currents can flow, and consequently the magnetic fluxes and induced voltages would contain large third-harmonic 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 third-harmonic 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 third-harmonic 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 third-harmonic 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 third-harmonic 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 triple-frequency generator supplying a voltage equal to the third-harmonic 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 third-harmonic voltage appears across the capacitor formed between the transformer neutral point and ground. This explains why, in star/star-connected banks having isolated neutrals, the third-harmonic 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 third-harmonic 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 Third-harmonic distribution of inductance, resistance and capacitance in a grounded neutral three-phase circuit consisting of the secondaries of a three-phase group of single phase transformers supplying an open-ended 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 third-harmonic current could flow on account of the small capacitance between the neutral point and ground, in the second case triple-frequency 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 third-harmonic 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 third-harmonic 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 third-harmonic 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 triple-frequency currents are confined to the secondary windings only, on account of the connections adopted.

FIG. 150 Third-harmonic distribution of inductance, resistance and capacitance in a grounded neutral three-phase circuit consisting of the secondaries of a three-phase group of single-phase transformers supplying an open-ended 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 third-harmonic voltage components induced in the transformer secondary windings. Under this condition the resulting third-harmonic cur rents will be lagging with respect to the third-harmonic voltage components.

The third-harmonic currents will act with the fundamental waves of primary magnetizing current to magnetize the core, and the resulting total ampere-turns will more or less eliminate the third-harmonic components of the flux waves, bringing the latter nearer to the sine shape. This will correspondingly reduce the third-harmonic voltage components, making the induced voltage waves also more sinusoidal. The reduction in third-harmonic voltage components will have a reflex action upon the third-harmonic currents circulating through the transformer secondary windings and the line capacitances, and a balance between third-harmonic 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 third-harmonic 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 third-harmonic currents flow in the secondary windings to eliminate practically the whole of the third-harmonic voltages, so that from the third-harmonic point of view this condition would be equivalent to delta-connected 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 three-phase star/star-connected group of single-phase transformers with secondary neutral solidly grounded and supplying an open-ended 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 third-harmonic flux and voltage reduction phenomena. The composite diagram of FIG. 151 shows clearly that the third-harmonic secondary current is in opposition to the third-harmonic 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 third-harmonic components of the voltage waves become intensified. In this case the inductive reactance is very high compared with the capacitive reactance, so that the third-harmonic voltage components impressed across the line capacitances produce third harmonic secondary currents which lead the third-harmonic secondary voltages.

The angle of lead is given by Eq. (64) and in the extreme case where the capacitance is very small the third-harmonic current will lead the third harmonic voltage almost by 90º. The resulting third-harmonic ampere-turns 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 third-harmonic 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 flat-topped flux wave. In other words, the third-harmonic components of the flux waves are intensified, and on this account the third harmonic voltage components of the induced voltage waves are also intensified.

Higher-third-harmonic voltage waves react upon the secondary circuit to produce larger third-harmonic currents, which in turn increase the third harmonic flux waves, and again the third-harmonic voltage waves. This process of intensification continues until a further increase of magnetizing current produces no appreciable increase of third-harmonic flux, so that the ultimate induced voltages become limited only by the saturation characteristics of the magnetic circuit. It should be noted that the third-harmonic 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 third-harmonic currents lead the third-harmonic voltage components by 90º. The diagrams of this figure do not show the actual amplification phenomena. The composite diagram shows very clearly that the third-harmonic secondary current is in phase with the third-harmonic 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 three-phase star/star-connected group of single-phase transformers with secondary neutral solidly grounded and supplying an open-ended line such that LL CL

FIG. 153 Induced voltage, flux and magnetizing current waves in a three-phase star/star-connected group of single-phase transformers with secondary neutral solidly grounded and supplying an open-ended line such that CC LL

Cases have occurred of transformer failures due to this third-harmonic effect, and one case is known where, on no-load, 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 third-harmonic current is limited only by the resistance of the secondary circuit. The third-harmonic currents would be in phase with the third-harmonic 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/ star-connected transformers with grounded secondary neutral, and unless some provision can be made for allowing the circulation of third-harmonic currents under permissible conditions three-phase shell-type transformers or three phase groups of single-phase transformers should not so be connected.

With three-phase core-type transformers there is still theoretically the same disadvantage, but as in such transformers the third-harmonic 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 third-harmonic voltage component may be serious in star/star-connected three phase core-type 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 three-phase transformers or groups having interconnected star/star windings if it is desired to ground the neutral point on the star-connected side. With this connection the third-harmonic volt ages are eliminated by opposition on the interconnected star side only, but they are present on the star-connected side in just the same way as if the windings were star/star connected, their average magnitudes being of the order 5 percent for three-phase core-type transformers and 50 60 percent in three-phase shell-type and three-phase groups of single-phase transformers.

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