Ecuaciones de Carson 216-220

8/13/2019 Ecuaciones de Carson 216-220

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C Transient propagation in HV OH lines

195

C.4 Carsons formula

C.4.1 Definition

Carsons formula is based on the following assumptions (more details can be found in

[19],[20], [34]):

a. The conductors are perfectly horizontal above ground, and long enough sothat three-dimensional end effects can be neglected (this makes the field

problem two-dimensional). The sag is taken into account indirectly by using

the average height above ground.

b. The aerial space is homogeneous without loss, with permeability 0 andpermittivity 0 .

c. The earth is homogeneous with uniform resistivity , permeability 0 andpermittivity 0 , and it is bounded by a flat plane with infinite extent, to

which the conductors are parallel. The earth behaves as a conductor.

d. The spacing between conductors is at least an order of magnitude larger thanthe radius of the conductors.

The elements of the series impedance matrix can be calculated from the geometry of the

tower configuration, Fig. 117, and from the characteristics of the conductors.

For self impedances:

gextii ZZZZ ++= int (78)

And for mutual impedances

gmijij ZZZ += ' (79)

where:

intZ is the internal impedance of the wire, [/m],

extZ is the external impedance of the wire, [/m],

gZ is Carsons correction term for earth self impedance, [/km],

'ijZ is the mutual impedance of the two conductors i and j, [/km],

gmZ is Carsons correction term for earth mutual impedance, [/km],

The symbols used in Eqs. (78) and (79) are explained in the following, Appendix C.4.2.


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C.4.2 Terms and definitions

C.4.2.1 Terms in Eqs. (78) and (79)

Fig. 120 Allocation of conductors i and j and their images.

The mutual impedance of conductors iandj, Fig. 120

)ln

2

(' 0

ij

ij

ij

d

DjZ

= (80)

Internal and external impedance of a conductor

Internal impedance of conductor i

)4

1(

2

0intintintint

jRLjRZ +=+= (81)

External impedance of conductor i, Fig. 120

)2

(ln2

0

i

iiiext

r

hjLjZ

== (82)

Carsons correction terms for earth return effects

The ground self impedance

ggg jXRZ +=

(83)

ih2

i

'i

jh2

j

ijd

ijD

ijX

Images

ir2


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The ground mutual impedance

gmgmgm jXRZ += (84)

The symbols used in Eqs. (80) to (84) are explained in the following:

C.4.2.2 Terms in Eqs. (80) to (84)

Carsons correction terms for earth return effects

For self resistance

++=

...])ln[(81044

4

3

3

2

22

1

1

7

kdkbkkCbkbRg

(85)

For self reactance

+++=

...])ln[()ln6159315.0(2

1104

4

44

3

3

2

2

1

1

7kkCbkbkdkbkXg (86)

For mutual resistance

+

++=

...4cos3cos

]2sin2cos)ln[(cos8104

4

4

3

3

22

22

1

17

mm

mmmm

gm

kdkb

kkkCbkbR (87)

For mutual reactance

++

++=

...]4cos4cos)ln[(

3cos2coscos)ln6159315.0(2

1

10444

44

3

3

2

2

1

17

mmm

mmmm

gm

kkkCb

kbkdkbkX (88)

The symbols used in Eqs. (85) to(88) are explained in the following:

C.4.2.3 Terms of Eq. (85) to(88)

The coefficients ib , iC and id are constants which can be calculated as follows:


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Coefficients b

6

21 =b (89)

161

2 =b (90)

)2(2

+=

ii

signbb ii , ...,5,4,3=i

(91)

Note sign=1 and changes after each successive term (sign=1 for i=1,2,3,4 and sign=-1

for i=5,6,7,8 etc.)

Coefficients C

3659315.12 =C (92)

and

2

112

+++=

iiCC ii , ...,8,6,4=i (93)

Coefficients d

ii bd4

= , ...,5,4,3=i (94)

Variables kand mk are frequency related and are given by:

/)2(10544 fhk

i

= (95)

and

/)(10544 fDk

ijm

= (96)

The angle is indicated in Fig. 120

)/sin( ijij Dxa= (97)


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Other symbols

ih is the average height above ground of conductor i, [m],

ijD is the distance between conductor iand the image of conductorj, [m],

ijd is the distance between conductors iandj, [m],

ir is the radius of conductor i, [m],

is the angular frequency, [rad],

intR is the ac resistance of the conductor, [/km],

intL is the internal inductance of the conductor, [H/km],

gR Carsons correction term for self resistance due to the earth, [/km],

gmR Carsons correction term for mutual resistance due to the earth, [/km],

gX Carsons correction term for self reactance due to the earth, [/km],

gmX Carsons correction term for mutual reactance due to the earth, [/km].

Ecuaciones de Carson 216-220

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