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Off-nominal Transformer Tap Settings |
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A transformer with turns ratio a interconnecting two nodes i, k can be represented by an
ideal transformer in series with the nominal transformer leakage admittance as shown in Fig. 2.2 (a).
If the transformer is on nominal tap (a=1), the nodal equations for the network branch in the per unit
system are |
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In this case Iik=-Iki*
For an off-nominal tap setting and letting the voltage on the k side of the ideal transformer be Vt
we
can write |
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Eliminating Vt
between equations (2.2.4) and (2.2.5) we obtain |
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A simple equivalent - circuit can be deduced from equations (2.2.7) and (2.2.8) the elements of which
can be incorporated into the admittance matrix. This circuit is illustrated in Fig. 2.2(b).
The equivalent circuit of Fig 2.2(b) has to be used with care in banks containing delta-connected
windings. In a star-delta bank of single-phase transformer units, for example, with nominal turns ratio,
a value of 1.0 per unit voltage on each leg of the star winding produces under balanced conditions
1.732 per unit voltage on each leg of the delta winding (rated line to neutral voltage as base). The
structure of the bank requires in the per unit representation an effective tapping at √3 nominal turns
ratio on the delta side, i.e. a= 1.732. |
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Figure 2.2
Transformer with off-nominal tap setting |
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For a delta-delta or star-delta transformer with taps on the star winding, the equivalent circuit of Fig.
2.2(b) would have to be modified to allow for effective taps to be represented on each side. The
equivalent-circuit model of the single-phase unit can be derived by considering a delta-delta
transformer as comprising a delta-star transformer connected in series (back to back) via a zeroimpedance link to a star-delta transformer, i.e. star windings in series. Both neutrals are solidly
earthed. The leakage impedance of each transformer would be half the impedance of the equivalent
delta-delta transformer. An equivalent per unit representation of this coupling is shown in Fig. 2.3.
Solving this circuit for terminal currents |
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or in matrix form |
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These admittance parameters form the primitive network for the coupling between a primary and
secondary coil. |
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Phase-shifting Transformers |
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Figure 2.3
Basic equivalent circuit in p.u. for coupling between primary and secondary coils with both primary and
secondary off-nominal tap ratios of α and β |
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To cope with phase shifting, the transformer of Fig. 2.3 has to be provided with a complex
turns ratio. Moreover, the invariance of the product VI* across the ideal transformer requires a
distinction to be made between the turns ratios for current and voltage, i.e. |
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or |
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Thus the circuit of Fig. 2.3 has two different turns ratios, i.e. |
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and |
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Solving the modified circuit for terminal currents. |
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Thus, the general single-phase admittance of a transformer including phase shifting is |
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Note that, although an equivalent lattice network similar to that in Fig. 2.3 could be
constructed, it is no longer a bilinear network as can be seen from the asymmetry of y in
equation (2.2.14). The equivalent circuit of a single-phase phase-shifting transformer is
thus of limited value and the transformer is best represented analytically be its admittance
matrix. |
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