0195136047.pdf

490 MAGNETIC CIRCUITS AND TRANSFORMERS

(b) During 24 hours,

energy output=( 8 × 0. 9 × 50 × 0. 8 )+( 12 × 0. 5 × 50 × 0. 8 )=528 kWh

core loss= 24 × 0. 173 = 4 .15 kWh

copper loss=( 8 × 0. 92 × 0. 65 )+( 12 × 0. 52 × 0. 65 )= 6 .16 kWh

The all-day (or energy) efficiency of the transformer is given by

ηAD=

528

528 + 4. 15 + 6. 16

= 0. 9808 , or 98.08%

11.5 Three-Phase Transformers

As we have seen in Chapter 10, three-phase transformers are used quite extensively in power

systems between generators and transmission systems, between transmission and subtransmission

systems, and between subtransmission and distribution systems. Most commercial and industrial

loads require three-phase transformers to transform the three-phase distribution voltage to the

ultimate utilization level.

Transformation in three-phase systems can be accomplished in either of two ways: (1)

connecting three identical single-phase transformers to form athree-phase bank(each one will

carry one-third of the total three-phase load under balanced conditions); or (2) a three-phase

transformer manufactured for a given rating. A three-phase transformer, compared to a bank of

three single-phase transformers, for a given rating will weigh less, cost less, require less floor

space, and have somewhat higher efficiency.

The windings of either core-type or shell-type three-phase transformers may be connected

in either wye or delta. Four possible combinations of connections for the three-phase, two-

winding transformers are Y– , –Y, – , and Y–Y. These are shown in Figure 11.5.1 with

the transformers assumed to be ideal. The windings on the left are the primaries, those on the

right are the secondaries, and a primary winding of the transformer is linked magnetically with

the secondary winding drawn parallel to it. With the per-phase primary-to-secondary turns ratio

(N 1 /N 2 =a), the resultant voltages and currents for balanced applied line-to-line voltagesVand

line currentsIare marked in the figure.

As in the case of three-phase circuits under balanced conditions, only one phase needs to

be considered for circuit computations, because the conditions in the other two phases are the

same except for the phase displacements associated with a three-phase system. It is usually

convenient to carry out the analysis on a per-phase-of-Y (i.e., line-to-neutral) basis, and in such

a case the transformer series impedance can then be added in series with the transmission-line

series impedance. In dealing with Y– and –Y connections, all quantities can be referred to

the Y-connected side. For – connections, it is convenient to replace the -connected series

impedances of the transformers with equivalent Y-connected impedances by using the relation

Zper phase of Y=

1

3

Zper phase of (11.5.1)

It can be shown that to transfer the ohmic value of impedance from the voltage level on one side

of a three-phase transformer to the voltage level on the other side, the multiplying factor is the

square of the ratio of line-to-line voltages, regardless of whether the transformer connection is

Y– Y, –Y, or –. In some cases, where Y–Y transformation is utilized in particular, it is quite

common to incorporate a third winding, known astertiary winding,connected in delta. Such

multiwinding transformers with three or more windings are not considered in this text.