orf, R.C., Wn,. T- Equivlent Networks The Eletril Engineering Hndook Ed. Rihrd C. orf Bo Rton: CRC Press LLC, 000
9 T P Equivlent Networks hen Wn University of Cliforni, vis Rihrd C. orf University of Cliforni, vis 9. Introdution 9. Three-Phse Connetions 9. Wye Û elt Trnsformtions 9. Introdution Two very importnt two-ports re the T nd P networks shown in Fig. 9.. Beuse we enounter these two geometril forms often in two-port nlyses, it is useful to determine the onditions under whih these two networks re equivlent. In order to determine the equivlene reltionship, we will exmine -prmeter equtions for the T network nd the -prmeter equtions for the P network. For the T network the equtions re V ( + )I + I nd for the P network the equtions re V I + ( + )I I ( + )V V I V + ( + )V Solving the equtions for the T network in terms of I nd I, we otin I I æ ç è + ö V ø V V æ + ö + ç V è ø where + +. Compring these equtions with those for the P network, we find tht 000 y CRC Press LLC
000 y CRC Press LLC or in terms of the impednes of the P network If we reverse this proedure nd solve the equtions for the P network in terms of V nd V nd then ompre the resultnt equtions with those for the T network, we find tht (9.) FIGURE 9. T nd P two-port networks.
where + +. Eqution (9.) n lso e written in the form + + + + + + The T is wye-onneted network nd the P is delt-onneted network, s we disuss in the next setion. 9. Three-Phse Connetions By fr the most importnt polyphse voltge soure is the lned three-phse soure. This soure, s illustrted y Fig. 9., hs the following properties. The phse voltges, tht is, the voltge from eh line,, nd to the neutrl n, re given y V n V p Ð0 V n V p Ð 0 (9.) V n V p Ð+0 Blned three-phse power soure FIGURE 9. phse V n phse Blned three-phse voltge soure. + + V n phse + V n n An importnt property of the lned voltge set is tht V n + V n + V n 0 (9.) From the stndpoint of the user who onnets lod to the lned three-phse voltge soure, it is not importnt how the voltges re generted. It is importnt to note, however, tht if the lod urrents generted y onneting lod to the power soure shown in Fig. 9. re lso lned, there re two possile equivlent onfigurtions for the lod. The equivlent lod n e onsidered s eing onneted in either wye () or delt () onfigurtion. The lned wye onfigurtion is shown in Fig. 9.. The delt onfigurtion is shown in Fig. 9.4. Note tht in the se of the delt onnetion, there is no neutrl line. The tul funtion of the n FIGURE 9. Wye ()-onneted lods. FIGURE 9.4 elt ()-onneted lods. 000 y CRC Press LLC
neutrl line in the wye onnetion will e exmined nd it will e shown tht in lned system the neutrl line rries no urrent nd therefore my e omitted. 9. Wye Û elt Trnsformtions For lned system, the equivlent lod onfigurtion TABLE 9. Current-Voltge Reltionships for the Wye my e either wye or delt. If oth of these onfigurtions nd elt Lod Configurtions re onneted t only three terminls, it would e very Prmeter Wye Configurtion elt Configurtion dvntgeous if n equivlene ould e estlished Voltge V line to line V V line to line V etween them. It is, in ft, possile hrteristis re Current I line I I line I the sme. Consider, for exmple, the two networks shown in Fig. 9.5. For these two networks to e equivlent t eh orresponding pir of terminls it is neessry tht the input impednes t the orresponding terminls e equl, for exmple, if t terminls nd, with open-iruited, the impedne is the sme for oth onfigurtions. Equting the impednes t eh port yields + + + Solving this set of equtions for,, nd yields ( + ) + + ( + ) + + ( + ) + + (9.4) + + + + + + (9.5) FIGURE 9.5 Generl wye- nd delt-onneted lods. 000 y CRC Press LLC
Similry, if we solve Eq. (9.4) for,, nd, we otin + + + + + + (9.6) Equtions (9.5) nd (9.6) re generl reltionships nd pply to ny set of impednes onneted in wye or delt onfigurtion. For the lned se where nd, the equtions ove redue to (9.7) nd (9.8) efining Terms Blned voltges of the three-phse onnetion: The three voltges stisfy V n + V n + V n 0 where V n V p Ð0 V n V p Ð 0 V n V p Ð+0 T network: The equtions of the T network re V ( + )I + I V I + ( + )I P network: The equtions of P network re I ( + )V V I V + ( + )V T nd P n e trnsferred to eh other. Relted Topi.5 Three-Phse Ciruits 000 y CRC Press LLC
Referenes J.. Irwin, Bsi Engineering Ciruit Anlysis, 4th ed., New ork: MMilln, 995. R.C. orf, Introdution to Eletri Ciruits, rd ed., New ork: John Wiley nd Sons, 996. Further Informtion IEEE Trnstions on Power Systems IEEE Trnstions on Ciruits nd Systems, Prt II: Anlog nd igitl Signl Proessing 000 y CRC Press LLC