Dorf, R.C., Wan, Z. “T-∏ Equivalent Networks” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 9 T–P Equivalent Networks 9.1 Introduction 9.2 Three-Phase Connections 9.3 Wye ? Delta Transformations 9.1 Introduction Two very important two-ports are the T and P networks shown in Fig. 9.1. Because we encounter these two geometrical forms often in two-port analyses, it is useful to determine the conditions under which these two networks are equivalent. In order to determine the equivalence relationship, we will examine Z-parameter equations for the T network and the Y-parameter equations for the P network. For the T network the equations are V 1 = (Z 1 + Z 3 )I 1 + Z 3 I 2 V 2 = Z 3 I 1 + (Z 2 + Z 3 )I 2 and for the P network the equations are I 1 = (Y a + Y b )V 1 – Y b V 2 I 2 = –Y b V 1 + (Y b + Y c )V 2 Solving the equations for the T network in terms of I 1 and I 2 , we obtain where D 1 = Z 1 Z 2 + Z 2 Z 3 + Z 1 Z 3 . Comparing these equations with those for the P network, we find that I ZZ D V ZV D I ZV D ZZ D V 1 23 1 1 32 1 2 31 1 13 1 2 = + ? è ? ? ? ÷ =+ + ? è ? ? ? ÷ – – Zhen Wan University of California, Davis Richard C. Dorf University of California, Davis ? 2000 by CRC Press LLC or in terms of the impedances of the P network If we reverse this procedure and solve the equations for the P network in terms of V 1 and V 2 and then compare the resultant equations with those for the T network, we find that (9.1) FIGURE 9.1 T and P two-port networks. Y Z D Y Z D Y Z D a b c = = = 2 1 3 1 1 1 Z D Z Z D Z Z D Z a b c = = = 1 2 1 3 1 1 Z Y D Z Y D Z Y D 1 2 2 2 3 2 = = = c a b ? 2000 by CRC Press LLC where D 2 = Y a Y b + Y b Y c + Y a Y c . Equation (9.1) can also be written in the form The T is a wye-connected network and the P is a delta-connected network, as we discuss in the next section. 9.2 Three-Phase Connections By far the most important polyphase voltage source is the bal- anced three-phase source. This source, as illustrated by Fig. 9.2, has the following properties. The phase voltages, that is, the voltage from each line a, b, and c to the neutral n, are given by V an = V p D0° V bn = V p D–120° (9.2) V cn = V p D+120° An important property of the balanced voltage set is that V an + V bn + V cn = 0 (9.3) From the standpoint of the user who connects a load to the balanced three-phase voltage source, it is not important how the voltages are generated. It is important to note, however, that if the load currents generated by connecting a load to the power source shown in Fig. 9.2 are also balanced, there are two possible equivalent configurations for the load. The equivalent load can be considered as being connected in either a wye (Y) or a delta (D) configuration. The balanced wye configuration is shown in Fig. 9.3. The delta configuration is shown in Fig. 9.4. Note that in the case of the delta connection, there is no neutral line. The actual function of the FIGURE 9.3 Wye (Y)-connected loads. FIGURE 9.4 Delta (D)-connected loads. Z ZZ ZZZ Z ZZ ZZZ Z ZZ ZZZ 1 2 3 = ++ = ++ = ++ ab abc bc abc ac abc FIGURE 9.2Balanced three-phase voltage source. V an + – Balanced three-phase power source V bn V cn a b c n phase a phase b phase c + + a b c n Z Y Z Y Z Y a b c Z D Z D Z D ? 2000 by CRC Press LLC neutral line in the wye connection will be examined and it will be shown that in a balanced system the neutral line carries no current and therefore may be omitted. 9.3 Wye ? Delta Transformations For a balanced system, the equivalent load configuration may be either wye or delta. If both of these configurations are connected at only three terminals, it would be very advantageous if an equivalence could be established between them. It is, in fact, possible characteristics are the same. Consider, for example, the two networks shown in Fig. 9.5. For these two networks to be equiva- lent at each corresponding pair of terminals it is necessary that the input impedances at the corresponding terminals be equal, for example, if at terminals a and b, with c open-circuited, the impedance is the same for both configurations. Equating the impedances at each port yields (9.4) Solving this set of equations for Z a , Z b , and Z c yields (9.5) FIGURE 9.5General wye- and delta-connected loads. TABLE 9.1Current-Voltage Relationships for the Wye and Delta Load Configurations Parameter Wye Configuration Delta Configuration Voltage V line to line = V Y V line to line = V D Current I line = I Y I line = I D 3 3 ZZZ ZZ Z ZZZ ZZZ ZZZ ZZZ ZZZ ZZZ ZZZ ab a b bc b c ca c a =+= + ++ =+= + ++ =+= + ++ 12 3 123 31 2 123 21 3 123 () () () Z ZZ ZZZ Z ZZ ZZZ Z ZZ ZZZ a b c = ++ = ++ = ++ 12 123 13 123 23 123 ? 2000 by CRC Press LLC Similary, if we solve Eq. (9.4) for Z 1 , Z 2 , and Z 3 , we obtain (9.6) Equations (9.5) and (9.6) are general relationships and apply to any set of impedances connected in a wye or delta configuration. For the balanced case where Z a = Z b = Z c and Z 1 = Z 2 = Z 3 , the equations above reduce to (9.7) and Z D = 3Z Y (9.8) Defining Terms Balanced voltages of the three-phase connection:The three voltages satisfy V an + V bn + V cn = 0 where V an = V p D0° V bn = V p D–120° V cn = V p D+120° T network: The equations of the T network are V 1 = (Z 1 + Z 3 )I 1 + Z 3 I 2 V 2 = Z 3 I 1 + (Z 2 + Z 3 )I 2 P network: The equations of P network are I 1 = (Y a + Y b )V 1 – Y b V 2 I 2 = –Y b V 1 + (Y b + Y c )V 2 T and P can be transferred to each other. Related Topic 3.5 Three-Phase Circuits Z ZZ ZZ ZZ Z Z ZZ ZZ ZZ Z Z ZZ ZZ ZZ Z 1 2 3 = ++ = ++ = ++ ab bc ca c ab bc ca b ab bc ca a ZZ Y = 1 3 ? 2000 by CRC Press LLC References J.D. Irwin, Basic Engineering Circuit Analysis, 4th ed., New York: MacMillan, 1995. R.C. Dorf, Introduction to Electric Circuits, 3rd ed., New York: John Wiley and Sons, 1996. Further Information IEEE Transactions on Power Systems IEEE Transactions on Circuits and Systems, Part II: Analog and Digital Signal Processing ? 2000 by CRC Press LLC