Dorf,RC, Wan, Z " T- Equivalent Networks The electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Dorf, R.C., Wan, Z. “T-∏ Equivalent Networks” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
T-II Equivalent networks Zhen Wan University of California, davis 9.1 Introduction Richard C. dorf 9.2 Three-Phase Connections 9.3 Wye s Delta Transformations 9.1 Introduction Two very important two-ports are the T and Il 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 Il network For the T network the equations are V1=(Z1+Z3)I1+Z3L2 V2=Z3I1+(Z and for the Il network the equations are ,=(Y+Y),-Yv ,=-Y,V,+(Y,+y Solving the equations for the T network in terms of I, and I,, we obtain +zyl_Z,v2 where D,=Z,,+ Z2Z,+ ZZ, Comparing these equations with those for the II network, we find that c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 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 V1 = (Z1 + Z 3)I1 + Z 3 I2 V2 = Z3 I1 + (Z 2 + Z 3)I2 and for the P network the equations are I1 = (Ya + Yb)V1 – YbV2 I2 = –YbV1 + (Yb + Yc)V2 Solving the equations for the T network in terms of I1 and I2, we obtain where D1 = Z1Z2 + Z2Z3 + Z1Z3. Comparing these equations with those for the P network, we find that I Z Z D V Z V D I Z V D Z Z D V 1 2 3 1 1 3 2 1 2 3 1 1 1 3 1 2 = Ê + Ë Á ˆ ¯ ˜ = + Ê + Ë Á ˆ ¯ ˜ – – Zhen Wan University of California, Davis Richard C. Dorf University of California, Davis
FIGURE 9.1 T and nI two-port networks. D Z Y or in terms of the impedances of the ni network D If we reverse this procedure and solve the equations for the Il network in terms of V, and V2 and then compare the resultant equations with those for the T network, we find that (91) D, e 2000 by CRC Press LLC
© 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 V1 and V2 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
where D2=Y,Y,+Y.+ Y,Y Equation(9.1)can also be written in the for 2.+2+2 Za z+z +Z1+Z The T is a wye-connected network and the nl 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=V,∠-120° =V.∠+120° FIGURE 9.2 Balanced three-phase voltage source An important property of the balanced voltage set is that Van+Vl+Ⅴa=0 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(yor a delta(A)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 lo FIGURE9.4 Delta(A)-connected loads e 2000 by CRC Press LLC
© 2000 by CRC Press LLC where D2 = YaYb + YbYc + YaYc . 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 balanced 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 cto the neutral n, are given by Van = Vp –0° Vbn = Vp ––120° (9.2) Vcn = Vp –+120° An important property of the balanced voltage set is that Van + Vbn + Vcn = 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 Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z 1 2 3 = + + = + + = + + a b a b c b c a b c a c a b c FIGURE 9.2 Balanced three-phase voltage source. Van + – Balanced three-phase power source Vbn Vcn a b c n phase a phase b phase c + + a b c n ZY ZY ZY a b c ZD ZD ZD
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 TABLE 9. 1 Current-Voltage Relationships for the Wye ay be either wye or delta. If both of these configurations and Delta Load Configurations are connected at only three terminals, it would be very Parameter Wye Configuration Delta Configuration advantageous if an equivalence could be established Voltage Viane to line=3v, v in=v the same. Consider, for example the two networks Curre shown in Fig. 95. For these two networks to be equiva lent at each corresponding pair of terminals it is necessary that the input impedances at the corresponding rminals 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 Z1(Z2+Z3) z1+Z2+Z3 (94) Z Solving this set of equations for Z, Zi and Z yields (95) z1+Z2+Z3 FIGURE 9.5 General wye- and delta-connected loads. e 2000 by CRC Press LLC
© 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 equivalent 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 Za, Zb, and Zc yields (9.5) FIGURE 9.5 General wye- and delta-connected loads. TABLE 9.1 Current-Voltage Relationships for the Wye and Delta Load Configurations Parameter Wye Configuration Delta Configuration Voltage Vline to line = VY Vline to line = VD Current Iline = IY Iline = ID 3 3 Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z ab a b bc b c ca c a = + = + + + = + = + + + = + = + + + 1 2 3 1 2 3 3 1 2 1 2 3 2 1 3 1 2 3 ( ) ( ) ( ) Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z a b c = + + = + + = + + 1 2 1 2 3 1 3 1 2 3 2 3 1 2 3
Z+zz. +ZZ Z Zb +zz +z (96) Zaz+ziz + zza Z Equations(9.)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,=Zb=Z, and Z,=Z2=Z3, the equations above reduce to Z=-Z Z、=3Z (98) Defining Ter Balanced voltages of the three-Phase connection: The three voltages satisfy 0 where T network: The equations of the T network are V1=(Z1+Z3)I1+Z3l v2=ZI1+(Z2+Z3)I2 I,=(Y, YbV-Y,v2 =-Y VI+(Yb+Yv T and li can be transferred to each other Related T 3.5 Three-Phase Circuits e 2000 by CRC Press LLC
© 2000 by CRC Press LLC Similary, if we solve Eq. (9.4) for Z1, Z2, and Z3, 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 Za = Zb = Zc and Z1 = Z2 = Z3, the equations above reduce to (9.7) and ZD = 3ZY (9.8) Defining Terms Balanced voltages of the three-phase connection: The three voltages satisfy Van + Vbn + Vcn = 0 where Van = Vp –0° Vbn = Vp ––120° Vcn = Vp –+120° T network: The equations of the T network are V1 = (Z1 + Z3)I1 + Z3I2 V2 = Z3I1 + (Z2 + Z3)I2 P network: The equations of P network are I1 = (Ya + Yb )V1 – YbV2 I2 = –YbV1 + (Yb + Yc)V2 T and P can be transferred to each other. Related Topic 3.5 Three-Phase Circuits Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z 1 2 3 = + + = + + = + + a b b c c a c a b b c c a b a b b c c a a Z Z Y = 1 3
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 System IEEE Transactions on Circuits and Systems, Part II: Analog and Digital Signal Processing e 2000 by CRC Press LLC
© 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