vvv Ib Ig=Ia+Ib+Ic+In FIGURE 61.1 A three-phase transmission line with one FIGURE 61.2 Geometric diagr onductors a and b Though the symmetrical component method has been used to simplify many of the problems in power system analysis, the following Paragraphs, which describe the formulas in the calculation of the line parameters, are much more general and are not limited to the application of symmetrical components. The sequence impedances and admittances used in the symmetrical components method can be easily calculated by a matrix transformation[Chen and Dillon, 1974. A detailed discussion of symmetrical components can be found in Clarke(1943 Series Impedance The network equation of a three-phase transmission line with one neutral wire(as given in Fig. 61. 1)in which only series impedances are considered is given as follows: an-g V Zbc-g Zm-gIbLvb (61.1) vcIZa-g Zob-g Za-g Zmn-gIve 8 where Zi-g= self-impedance of phase i conductor and Zi-g= mutual impedance between phase i conductor and phase j conductor. The subscript g indicates a ground return. Formulas for calculating Zi-g and Zi-g were developed by J.R. Carson based on an earth of uniform conductivity and semi-infinite in extent [Carson, 1926]. For two con- ductors a and b with earth return, as shown in Fig. 61.2, the self-and mutual impedances in ohms per mile are jo In b+oo(p+jq) (61.3) where the"prime"is used to indicate distributed parameters in per-unit length; z,=r+ jx,= conductor a internal impedance, Q2/mi; h,=height of conductor a, ft; r,=radius of conductor a, ft; da,=distance between conductors a and b, ft; So,= distance from one conductor to image of other, ft; @=2Tff= frequency, cycles/s, Ho= the magnetic permeability of free space, Ho= 4T X 10-X 1609.34 H/mi; and P, g are the correction terms for earth return effect and are given later The conductor internal impedance consists of the effective resistance and the internal reactance. The effective resistance is affected by three factors: temperature, frequency, and current density In coping with the temper ature effect on the resistance, a correction can be applied. c 2000 by CRC Press LLC© 2000 by CRC Press LLC Though the symmetrical component method has been used to simplify many of the problems in power system analysis, the following paragraphs, which describe the formulas in the calculation of the line parameters, are much more general and are not limited to the application of symmetrical components. The sequence impedances and admittances used in the symmetrical components method can be easily calculated by a matrix transformation [Chen and Dillon, 1974]. A detailed discussion of symmetrical components can be found in Clarke [1943]. Series Impedance The network equation of a three-phase transmission line with one neutral wire (as given in Fig. 61.1) in which only series impedances are considered is given as follows: (61.1) where Zii–g = self-impedance of phase i conductor and Zij–g = mutual impedance between phase i conductor and phase j conductor. The subscript g indicates a ground return. Formulas for calculating Zii–g and Zij–g were developed by J. R. Carson based on an earth of uniform conductivity and semi-infinite in extent [Carson, 1926]. For two con￾ductors a and b with earth return, as shown in Fig. 61.2, the self- and mutual impedances in ohms per mile are (61.2) (61.3) where the “prime” is used to indicate distributed parameters in per-unit length; za = rc + jxi = conductor a internal impedance, W/mi; ha = height of conductor a, ft; ra = radius of conductor a, ft; dab = distance between conductors a and b, ft; Sab = distance from one conductor to image of other, ft; w = 2pf; f = frequency, cycles/s; m0 = the magnetic permeability of free space, m0= 4p ¥ 10–7 ¥ 1609.34 H/mi; and p, q are the correction terms for earth return effect and are given later. The conductor internal impedance consists of the effective resistance and the internal reactance. The effective resistance is affected by three factors: temperature, frequency, and current density. In coping with the temper￾ature effect on the resistance, a correction can be applied. FIGURE 61.1 A three-phase transmission line with one neutral wire. FIGURE 61.2 Geometric diagram of conductors a and b. V V V V ZZZZ ZZZZ ZZZZ ZZZZ I I I I A B C N aa g ab g ac g an g ba g bb g bc g bn g ca g cb g cc g cn g na g nb g nc g nn g a b c n È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ = È Î Í Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ ˙ È Î Í Í Í Í ˘ ˚ ˙ – – – – – – – – – – – – – – – – ˙ ˙ ˙ + È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ V V V V a b c n Z ¢ = + z j + + h r p jq aa g a a a – w ln ( ) m p w m p 0 0 2 2 Z j ¢ = + + S d p jq ab g ab ab – w ln ( ) m p w m p 0 0 2