Electric Power Generation, Transmission, and Distribution

The voltage difference between phase conductor ‘‘A’’ and

its image conductor is generated by all charges (Qa,Qb,Qc,

andQa,Qb,Qc) in the system. Using the voltage dif-

ference equations, we obtained the voltage difference be-

tween conductor A and its image:

Va,A¼

QA

2 p«o

ln

Da,A

rcond



þ

QA

2 p«o

ln

rcond

Da,A



þ

QB

2 p«o

ln

Da,B

da,b



þ

QB

2 p«o

ln

da,b

Da,B



þ

QC

2 p«o

ln

Da,C

da,c



þ

QC

2 p«o

ln

da,c

Da,C



This equation can be simplified by combining theþQand

Qterms. The result is

Va,A¼ 2 Va_ ln¼

2 QA

2 p«o

ln

Da,A

rcond



þ

2 QB

2 p«o

ln

Da,B

da,b



þ

þ

2 QC

2 p«o

ln

Da,C

da,c



Further simplification is the division of both sides of the

equation by 2, which results in an equation for the line to

neutral voltage. Similar equations can be derived for phases B

and C. The results are

Va_ ln¼

QA

2 p«o

ln

Da,A

rcond



þ

QB

2 p«o

ln

Da,B

da,b



þ

QC

2 p«o

ln

Da,C

da,c



Vb_ ln¼

QA

2 p«o

ln

Db,A

da,b



þ

QB

2 p«o

ln

Db,B

rcond



þ

QC

2 p«o

ln

Db,C

db,c



Vc_ ln¼

QA

2 p«o

ln

Dc,A

dc,b



þ

QB

2 p«o

ln

Dc,B

db,c



þ

QC

2 p«o

ln

Dc,C

rcond



In these equations, the line to neutral voltages and dimensions are given. The equations can be solved for
the charges (Qa,Qb,Qc).

19.4.2 Electric Field Calculation

The horizontal and vertical components of the electric field generated by the six charges (Qa,Qb,Qc,
andQa,Qb,Qc) are calculated. The sum of the horizontal components and vertical components
gives theXandYcomponents of the total electric field. The vector sum of theXandYcomponents gives
the magnitude of the total field.
Figure 19.10 shows aQcharge generated electric field. The field lines are radial to the charge.
The absolute value of electric field generated by a chargeQis described by the Gauss equation. The
observation point coordinates areXandY. The conductor coordinates arexiandyi.
The electric field magnitude is

Ei¼

Qi

2 pr

¼

Qi

2 p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

(xiX)^2 þ(yiY)^2

q

TheFangle between theEvector and its vertical components is

F¼atn

xiX

yiY



−Qa −Qb −Qc

Da,A Da,B Da,C

Qa Qb Qc

r = da,a

da,b

da,c

FIGURE 19.9 Representation of three-
phase line generated electric field by image
conductors.