CHAP. 11]

255

POLYPHASE CIRCUITS

Y-to- Transformation

-to-Y Transformation

Z1 Z2 þ Z1 Z3 þ Z2 Z3 Z3 Z1 Z2 þ Z1 Z3 þ Z2 Z3 ZB ¼ Z2 Z1 Z2 þ Z1 Z3 þ Z2 Z3 ZC ¼ Z1 ZA ¼

ZA ZB ZA þ ZB þ ZC ZA ZC Z2 ¼ ZA þ ZB þ ZC ZB ZC Z3 ¼ ZA þ ZB þ ZC Z1 ¼

It should be noted that if the three impedances of one connection are equal, so are those of the equivalent connection, with Z =ZY ¼ 3.

11.9

SINGLE-LINE EQUIVALENT CIRCUIT FOR BALANCED THREE-PHASE LOADS

Figure 11-13(a) shows a balanced Y-connected load. In many cases, for instance, in power calculations, only the common magnitude, IL , of the three line currents is needed. This may be obtained from the single-line equivalent, Fig. 11-13(b), which represents one phase of the original system, with the line-to-neutral voltage arbitrarily given a zero phase angle. This makes IL ¼ IL , where  is the impedance angle. If the actual line currents IA , IB , and IC are desired, their phase angles may be found by adding  to the phase angles of VAN , VBN , and VCN as given in Fig. 11-7. Observe that the angle on IL gives the power factor for each phase, pf ¼ cos . The method may be applied to a balanced -connected load if the load is replaced by its Yequivalent, where ZY ¼ 13 Z (Section 11.8).

Fig. 11-13 EXAMPLE 11.4 Rework Example 11.3 by the single-line equivalent method. Referring to Fig. 11-14 (in which the symbol Y indicates the type of connection of the original load), IL ¼

VLN 98:0 08 ¼ ¼ 4:90 308 A Z 20 308

From Fig. 11-7(b), the phase angles of VAN , VBN , and VCN are 908, 308, and 1508. IA ¼ 4:90 608 A

11.10

IB ¼ 4:90 608 A

Hence,

IC ¼ 4:90 1808 A