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Home The Transformer 3Phase Transformer Connections Third Harmonics in 3Phase Transformers  
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Third Harmonics in 3Phase Transformer Operation
It was shown in Section 53 that the sinusoidal flux in iron cores requires a thirdharmonic component in the exciting current, which, although small in relation to the rated current, may produce undesirable effects in 3phase transformer operation. Consider three identical unloaded, singlephase transformers connected wyewye to a 3phase generator with their primary neutral connected to the generator neutral as shown in Fig. 632. The sum of the instantaneous currents flowing in the primary must equal zero, i.e.
The fundamental components, as well as harmonics  not including the third and multiples thereof, are 120° apart, and, being of equal amplitudes, their sum is
where the subscript h stands for the order of the harmonics 1, 5, 7, 11, but not for 3, 9, 15, etc. It should be remembered that harmonics in the exciting current of an ironcore transformer are odd for sinusoidal flux when there is no dc component of flux. It follows from Eqs. 694 and 695 that the neutral current in the unloaded transformers, or in such as deliver balanced sinusoidal 3phase currents, is comprised of thirdharmonic current, which is the sum of the third harmonics in the three phases, thus
or
Figures 633(a), 633(b), and 633(c) show balanced 3phase waves that are comprised of a fundamental and a third harmonic. The sum of the three
balanced waves is shown in Fig. 633(d) and is a pure third harmonic having an amplitude equal to three times that of the third harmonic in any one phase. If the neutral connection between the transformer primaries and the generator is broken, then the path for the thirdharmonic currents is interrupted and the third harmonics in the exciting current will be suppressed. As a result, the flux cannot be sinusoidal, as it will contain a third harmonic, which in turn produces a third harmonic in the transformer voltages. These third harmonics show up only in the linetoneutral voltage if the transformers are identical, and will not appear in the linetoline voltages because the linetoline voltages are the phasor difference between the linetoneutral voltages, i.e.
The third harmonics in the linetoneutral voltages of all three phases are equal and in phase with each other and, therefore, cancel in the linetoline
voltages. This becomes evident when the difference is taken between any two of three waves a, b, and c of Fig. 633. Since
and
When the primaries are connected in delta, the thirdharmonic components in the current are free to flow, but will not show up in the line currents because the line currents are the differences between the currents flowing in the delta as shown in Fig. 621. The delta connection on the secondary side of a wyedelta arrangement also provides a path for the thirdharmonic components in the exciting mmf. Figure 634 shows the primaries of a wyedelta arrangement connected in wye with the neutral isolated. One corner of the delta is shown open. Since the neutral is isolated, there is no return path on the primary side for the third harmonics in the exciting current, causing third harmonics to appear in the voltage across each primary winding. There will be corresponding third harmonics in the voltages across each secondary winding if one or more corners of the delta are open. The voltage appearing across the open corner of the delta in Fig. 634 is the sum of the voltages in the three secondary windings, and, if the exciting characteristics of the three phases are identical, the sum of the fundamentals, as well as that of all harmonics  except the third and its multiples  will be zero since these are all equal and 120° apart. The multiples of the third harmonics are usually negligible. The third harmonics are equal and in phase with each other. And the voltage across the open corner of the delta is three times the thirdharmonic voltage in one phase of the secondary. Thus, if V_{3} is the third harmonic voltage per phase in the delta, then 3V_{3} is the voltage across the open corner of the delta. Closing the open corner of the delta in Fig. 634, for normal operation, short circuits the thirdharmonic emf 3 V_{3}, causing a thirdharmonic current to circulate in the delta, thus producing a substantially sinusoidal flux. If, in addition, the primary neutral is closed, the thirdharmonic components of the mmf required by the sinusoidal flux divide between the primary and secondary, depending upon their relative thirdharmonic leakage impedances. Since the delta connection provides a path for the thirdharmonic current, and because it assures balanced voltages, most 3phase transformations include a delta winding, which makes the wyedelta or deltawye arrangement very common. Where wyewye transformation is required, it is quite common to incorporate a third winding, known as a tertiary, connected in delta. Generally, the rating of the deltaconnected tertiary is considerably lower than that of the primary and secondary wyeconnected windings.


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