Electric Power Generation, Transmission, and Distribution

Self-impedance:

^zzii¼riþ 4 ˆPiiGþj Xiþ 2 ˆGiiln

Sii

Ri

þ 4 ˆQiiG



V=mile (21:1)

Mutual impedance:

^zzij¼ 4 ˆPijGþj2ˆG ln

Sij

Dij

þ 4 ˆQijG



V=mile (21:2)

where^zzii ¼self-impedance of conductor i inV=mile
^zzij ¼mutual impedance between conductors i and j inV=mile
ri ¼resistance of conductor i inV=mile
v ¼system angular frequency in radians per second
G ¼0.1609347 10 ^7 Vcm=abohm-mile
Ri ¼radius of conductor i in feet
GMRi¼geometric mean radius of conductor i in feet
f ¼system frequency in Hertz
r ¼resistivity of earth inVm
Dij ¼distance between conductors i and j in feet
Sij ¼distance between conductor i and image j in feet
qij ¼angle between a pair of lines drawn from conductor i to its own image and to the image
of conductor j

Xi¼ 2 vGln

Ri

GMRi

V=mile (21:3)

Pij¼

p

8



1

3

ffiffiffi

2

p kijcosuij



þ

kij^2

16

cos 2uij



0 : 6728 þln

2

kij



(21:4)

Qij¼ 0 : 0386 þ

1

2

ln

2

kij

þ

1

3

ffiffiffi

2

p kijcosuij



(21:5)

kij¼ 8 : 565  10 ^4 Sij

ffiffiffi

f

r

s

(21:6)

As indicated above, Carson made use of conductor images; that is, every conductor at a given distance
above ground has an image conductor the same distance below ground. This is illustrated in Fig. 21.2.

21.1.1.2 Modified Carson’s Equations

Only two approximations are made in deriving the ‘‘modified Carson equations.’’ These approximations
involve the terms associated withPijandQij. The approximations are shown below:

Pij¼

p

8

(21:7)

Qij¼ 0 : 03860 þ

1

2

ln

2

kij

(21:8)

It is also assumed
f¼frequency¼60 Hertz
r¼resistivity¼ 100 Vm