CAPACITANCETRANSMISSION LINE PARAMETERS: A REVIEW

Definition

Transmission line conductors exhibit capacitance with respect to each other due to the potential difference between them.

The amount of capacitance between conductors is a function of the conductor size, spacing and height above ground.

Capacitance is the ratio of charge Q to the potential difference V.

Farad V

QC

Basic Principles

The electric field intensity at a point P due to an infinite line charge is:

The potential difference between two points due to a line charge is:

m

V

2 0

aE L

V ln2 1

21

2

12

o

LdLEV

Capacitance of 1-phase, 2-wire Transmission Lines

Assuming conductor 1 alone to have a charge of ρL1, the voltage between conductors 1 and 2 is:

Dr1 r2

V ln2

112 1 r

DV

o

LL

Capacitance of 1-phase, 2-wire Transmission Lines

Assuming conductor 2 having a charge of ρL2, the voltage between conductors 1 and 2 is:

Dr1 r2

V ln2

212 2 D

rV

o

LL

Capacitance of 1-phase, 2-wire Transmission Lines

Dr1 r2

12

21 0

LL

LL

Capacitance of 1-phase, 2-wire Transmission Lines

Dr1 r2

V ln2

V ln2

1

112 2

r

D

D

rV

o

L

o

LL

Capacitance of 1-phase, 2-wire Transmission Lines

By the Principle of Superposition,

r

DV

VVV

L

LL

ln0

112

121212 21

Capacitance of 1-phase, 2-wire Transmission Lines

The capacitance between conductors is:

Dr1 r2

m

F

ln

0

12

112

rDV

C L

Capacitance of 1-phase, 2-wire Transmission Lines

The capacitance between conductor to neutral is:

m

F

ln

2 0

rD

Cn

Potential Difference in a Multiconductor Configuration

By Superposition,

For i = m or k = m,

km

imM

mLmki

kM

iMLM

k

iL

k

iLki

D

DV

D

D

D

D

D

DV

ln2

1

ln2

...

...ln2

ln2

10

0

2

2

0

2

1

1

0

1

kkk

iii

rD

rD

Capacitance of Transposed 3-Phase Lines

1223

D31

1

2

3

2

3

1

1

12

23

3

I II III

Capacitance of Transposed 3-Phase Lines

Section I:

2

3

1

1

12

23

3

I II III

13

233

122

211

012

ln

lnln

2

1

D

D

D

r

r

D

V

L

LL

I

Capacitance of Transposed 3-Phase Lines

Section II:

2

3

1

1

12

23

3

I II III

12

133

232

231

012

ln

lnln

2

1

D

D

D

r

r

D

V

L

LL

II

Capacitance of Transposed 3-Phase Lines

Section III:

2

3

1

1

12

23

3

I II III

23

123

132

131

012

ln

lnln

2

1

D

D

D

r

r

D

V

L

LL

III

Capacitance of Transposed 3-Phase Lines

The average value of V12 is:

GMD

r

r

GMDV

DDD

r

r

DDDV

DDD

DDD

DDD

r

r

DDD

V

LL

LL

L

LL

lnln2

1

lnln2

1

ln

lnln

23

1

210

12

3

1

132312

2

3

1

1323121

012

132312

1323123

132312

3

23132312

1

012

Capacitance of Transposed 3-Phase Lines

Similarly, the average value of V13 is:

GMD

r

r

GMDV LL lnln

2

131

013

Capacitance of Transposed 3-Phase Lines

Adding V12 and V13

But,

GMD

r

r

GMDVV LLL lnln2

2

1321

01312

132

321 0

LLL

LLL

Capacitance of Transposed 3-Phase Lines

r

GMDV

r

GMDV

r

GMDVV

GMD

r

r

GMDVV

Ln

Ln

L

LL

ln2

ln2

33

ln2

3

lnln22

1

0

11

0

11

0

11312

110

1312

Capacitance of Transposed 3-Phase Lines

The capacitance to neutral, Cn is:

m

F

ln

2 0

rGMD

Cn

Effect of Bundling on Capacitance

The effect of bundling is to introduce an equivalent radius rb.

The equivalent rb is similar to the GMR calculated for inductance with the exception that radius r of each subconductor is used instead of r’ or Ds.

The capacitance per phase (or line-to-neutral) is found to be

F/m ln

2 0

b

n

r

GMDC

Capacitance of Three-phase, Double-Circuit Lines

The capacitance per phase is found to be

Important: The expression for GMD is the same as was found for inductance

calculation.

The GMRCeq of each phase is similar to the GMRLeq with the exception that rb is used instead of Dbs.

F/m ln

2 0

Ceq

eqn

GMR

GMDC

Examples

1. A completely transposed 60-Hz three-phase line has flat horizontal phase spacing with 10 m between adjacent conductors. The conductors are Dove ACSR. Line length is 200 km. Determine the capacitance per phase and susceptance.

Solution to Example 1

D D

1 2 3

mho 10 6013.0F 595.1602fC2

phaseper F 595.1200000

01177.0599.12

ln

2

m 01177.0cm 100

m 1

1"

cm 54.2

2

"927.0

m 599.12201010

3-

0

3

C

r

GMD

Examples

2. A three-phase double circuit line is composed of 300,000 cmil, 26/7 Ostrich conductors arranged as shown in the figure below. Find the 60-Hz capacitive susceptance in ohms per mile per phase.

Solution to Ex. 2

ft 9072.261820

ft 9146.21

5.1910

ft 112.10

5.110

22''

22''''

22''''

ccaa

cbbcbaab

cbbcbaab

DD

DDDD

DDDD

Solution to Ex. 2

ft 14.16

ft 9737.18

ft 886.14

ft 886.14

3

4''''

4''''

4''''

CABCABeq

acaccacaCA

cbcbbcbcBC

babaababAB

GMDGMDGMDGMD

DDDDGMD

DDDDGMD

DDDDGMD

Solution to Ex. 2

ft 8165.0

ft 8731.0

ft 7141.0

ft 8731.0

ft 02833.012

1

2

"680.0

3

4 2'

2

4 2'

2

4 2'

2

cCcBcAceq

ccostrcC

bbostrcB

aaostrcA

ostr

GMRGMRGMRGMR

DrGMR

DrGMR

DrGMR

r

Solution to Ex. 2

Siemenor mho 10 01131.0F 03.0602fC2

phaseper F 03.01609

8165.014.16

ln

2

3-

0

nC

Effect of Earth on the Capacitance

The effect of earth is to increase the capacitance.

However, the height of the conductor is large compared to the distance between conductors and the earth effect is negligible.

Therefore, for all line models used for balanced steady-state analysis, the effect of earth on the capacitance can be neglected. But, for unbalanced analysis such as unbalanced faults, the earth’s effect as well as the shield wires should be considered.

Effect of Earth on the Capacitance

F/m

lnln

2

3321

3312312

0

HHH

HHH

r

GMDCn