SYMMETRICAL COMPONENTS CALCULATION FOR TRANSFORMERS
As a passive device, the positive- and negative-sequence impedance magnitudes for transformers are identical and are equal to the nameplate leakage reactance provided by the manufacturer. However, in modeling transformers in symmetrical components, recognizing that an inherent phase shift is associated with delta-connected windings is sometimes necessary.
Wye-delta and delta-wye transformers built under ANSI standards are designed so that high-voltage quantities always lead the corresponding low-voltage quantities by 30°. The complete positive-sequence model for a delta-wye or wye-delta transformer, therefore, should include a 30° phase shift.
Negative-sequence quantities, however, are shifted in the opposite direction, and so the negative-sequence representation should include a phase shift opposite to the shift considered in positive sequence. These relationships are illustrated in Figure 2-7a.
Figure 2-7a—Positive- and negative-sequence equivalent circuits for delta-wye or wye-delta transformer
1—The phase shift in positive sequence is in the same direction as in the physical transformer: high voltage leads low voltage by 30° for ANSI standard transformers.
2—The phase shift in the negative-sequence circuit is opposite in direction.
Inclusion of these phase shifts is important only if a rigorous calculation is needed to determine exact phase currents and voltages on both sides of the transformer, including phase angles. Analysts often take the shortcut of neglecting phase shifts if the calculations are restricted to determining information on only one side of the transformer.
No inherent phase shift occurs in wye-wye transformers; therefore, the positive- and negativesequence equivalent circuits for these transformers also do not require phase shifts.
ZERO SEQUENCE IMPEDANCE OF POWER TRANSFORMER
The zero-sequence impedance of a transformer is controlled by a number of factors. The best way to determine a magnitude of this impedance is by an actual test, but the following comments, supplemented by information in some of the references, may be used to predict a value that is close enough for many applications.
First, the zero-sequence impedance seen looking into a transformer depends upon the configuration of the winding. The zero-sequence impedance of a delta winding is infinite (an open circuit), whereas the zero-sequence impedance of a wye-connected winding is a series composite of the zero-sequence impedance of the transformer and the impedance of any neutral grounding devices that might be present.
Thus, an ungrounded wye winding would present an infinite zero-sequence impedance because the absence of a neutral grounding connection appears as an open circuit in series with the zero-sequence impedance of the transformer winding itself (see Figure 2-7b).
Figure 2-7b—Zero-sequence equivalent circuit for delta-wye-grounded transformer
NOTE—The circuit is open on the side corresponding to the delta winding on the physical transformer
The impedance of the transformer itself depends upon several factors in the construction of the transformer. Three-phase transformers, which are constructed so that a closed, low-impedance path exists for the flow of zero-sequence flux within the transformer, have a lower zerosequence impedance than transformers without such a path.
One such path is the transformer core. Transformers with core-form construction have lower zero-sequence impedances than units with shell-form cores.
Three-phase transformers with delta windings have the lowest zero-sequence impedance, and in the absence of actual test data, it is often assumed that the zero-sequence impedance of core-form transformers with delta windings is about 0.85 times the positive-sequence leakage reactance of such transformers.
The zero-sequence impedance of shell-form transformers has about the same magnitude as the positive sequence leakage reactance of such transformers. Conversely, a three-phase transformer bank consisting of three, single-phase transformers connected wye-wye has a very high zero-sequence impedance.