Earth Return Imped v1

1

1

Prof. György VarjuHead of the Power Systems and Environment Group

Budapest University of Technology & Economics

e-mail: [email protected]

Earth return,

phenomena and impedance

2

2

Impedance with earth return

Presentation items:

Introductory notes

Basic references

Earth return phenomenon

Symbols and parameters used

Proposed methods and expressions

Impedances for lines of finite length

3

3


Introductory notes

methods described in detail:

Carson - Clem method

Complex image method.

expressions used in both methods are derived from the Carson/Pollaczek

theory with reasonable simplifications

circumstances affecting the accuracy:

uncertainty in the specific resistivity of the earth, which comes both

from the resistance value itself and the non-homogeneous

(stratified) nature of the soil

inducing line with finite length

Note: If we assume an uncertainty of factor two in the earth resistivity, e.g. value

ranging from 50 to 100 m, this results in 2.6% to 18.4% change in the self

impedance (radius 0.001 m) and mutual impedance (relevant to separation

distance 100 m), respectively.

4

4


Basic references

[1] J.R. CARSON: Wave propagation on overhead wires with ground return, Bell Syst. techn., p.539-554., Vol. 5, 1926,

[2] F. POLLACZEK: Über das Feld einer unendlich langen wechselstrometischflossenen Entfachleitung. Elektrische Nachrichtentechnik, Vol. 3, p. 339, 1926.

[3] CCITT Directives concerning the protection of telecommunication lines against harmful effects from electricity lines, UIT, 1963 (New Delhi, 1960).

[4] J.R. WAIT, K.P. Spies, On the image representation of the quasi-static fields of a line current source above the groung, Can. J. Phys., Vol. 47, pp. 2731-2733, 1969.

[5] G. Tevan-A. Deri: Mathematical verification of Dubanton’s simplified calculation of overhead transmission line parameters and its physical interpretation, Archiv für Elektrotechnik, No. 64., 1981

[6] G. Tevan-A. Deri: Some remarks about the accurate evaluation of the Carson integral for mutual impedance of lines with earth return, Archiv für Elektrotechnik, No. 67., 1984

[7] G. Tevan-A. Deri-G. Varju: Calculation of the inductive coupling between transmission lines, Rail International, 1985

[8] ITU-T, Calculating induced voltages and currents in practical cases, CCITT Directives concerning the protection of telecommunication lines against harmful effects from electric power and electrified railway lines: Volume II., ITU, 1989 Geneva

[9] F. Rachidi-C. A. Nucci-M. Ianoz: Transient Analysis of Multiconductor Lines Above a Lossy Ground, IEEE Transactions on Power Delivery, Vol. 14, No1, January 1999

5

5


Phenomenon of current return via earth

Classification of the methods proposed for induction calculations:

The four ITU (CCITT) methods {2}

1) Expression in form of series form

This is a complete representation of the Carson’s formula with accuracy determined by the number of terms considered in the calculation.

2) Complex image formula

The geometric term (conductor height) is considered and the earth correction term is only approximated, but the accuracy is good enough for the most practical engineering purposes.

3) Expression in polynomial form

The geometric term (conductor height) is neglected, the representation of the correction term can consider any separation distance and frequency.

4) Carson-Clem formula

The geometric term (conductor height) is neglected, the representation of the correction term can consider only limited separation distance and frequency.

6

6

Current return via the earthReturn paths for dc and ac

Return of direct current:

follows the shortest way to minimize R

Return of alternative current:

follows the line route to minimize Z

7

7

Current return via the earthCurrent density distribution

8

8

Interpretation of the earth return impedanceby the elementary Ohm’s law

Elementary Ohm’s law:

Self impedance (blue) and

mutual (red) impedace with

earth return by definition:

I

zy

I

zyEzyZ

,,,

km

zyzyE ,, zyzyE ,,

),(),( zyzyE

9

9

Impedance with earth returnBasic terms used

o= 1/(36·)·10-9

permittivity of free space [As/Vm]

o=4·10-7

permeability of free space [Vs/Am]

r relative permeability of conductor material [ - ]

re 4…10…40 relative permittivity of the earth (default value = 10)

r conductor radius [m]

the specific resistivity of the earth, [m]

=1/ the specific conductivity of the earth, [S/m]

f frequency, [Hz]

=2 f angular frequency, [rad/s]

10

10

Impedance with earth returnParameters used

The parameter and complex :

0 0

j j j0 0

The complex :

j j j jre re0 0 0 0

1

The equivalent depth (radius) of the hypothetical return path (tube) of the earth current:

Dfe

1852659

.

11

11

Impedance with earth returnRelations between the different parameters

The complex image depth (see in the next figure):

p j ej j

j45 ( )12

2

1 1 1

0 0

The amended complex image depth:

p

j jj jre

re

1 1 1

10 0

0 0

Note: on the relations between the penetration depth (d) and the equivalent earth return

complex image depth.

2 2

0 0

21 1 1

2;

p

1

20 701 .

31.12

852.1852.1eD

( . )D pe 1852

12

12


Geometrical (real and complex) data

ri radius of the conductor i

r’i Geometric Mean Radius (GMR) of conductor i

hi, hj height above the ground-level of conductors i and j

aij separation (horizontal distance) between conductors i and j

dij physical distance between conductors i and j:

d a h hij ij i j 22

Dij distance between conductor i and image of conductor j or vice versa:

D a h hij ij i j 22

qij angle shown in next figure

cos ; sin ij

i j

ijij

ij

ij

h h

D

a

D

D ij

distance between conductor i and complex image of j or vice versa:

D a h h pij ij i j 22

2

13

13

Impedance with earth returnConductor distances for the complex image method

d ij

Complex image

plane

hj

aij

hi

hj

hj

Dij

Dij

p

p

Geometric layout of conductors, and their images

p e j 1

45

14

14

Impedance with earth returnComposition of the impedance with earth return

Mutual impedance

The mutual impedance with earth return, per unit length, composed of the following two terms:

Z Z Zij Gij E ij

where:

ZGij is the geometric term (reactance involved in the magnetic field in the air):

Z ja h h

a h h

jD

dGij

ij i j

ij i j

ij

ij

0

22

22

0

2 2ln ln

Its value is zero when both conductors are on (or under) the surface of the earth.

ZEij the earth contribution (ground impedance {3}) term:

Z R j XEij ij ij 2

15

15

Impedance with earth returnTerms constituting the mutual impedance with earth return

Earth correction term accordingly to the original Carson formula

0 0

2cos

22

ddj

ejXjRZ ij

hh

ijijEij

ji

0 0

2cos

2ln

2

ddj

ej

d

DjZ ij

hh

ij

ijoij

ji

Geometric term Earth correction term

16

16

Impedance with earth returnSelf impedance

The self impedance with earth return, per unit length, is, in general, composed of the following tree terms:

Z Z Z ZS i G ii E iiii

External part of the self impedance (the 2nd and 3rd terms):

Z Z Zii G ii E ii

ZGii , the geometric term:

Z jh

rGijii

ii

0

2

2ln

This term can be explained as a contribution due to the inductance / reactance involved in the magnetic field in the air.

ZEii, the earth contribution (ground impedance {3}) to the external part of the self impedance:

Z R j XEii ii ii 2

17

17


Internal impedance

The internal impedance of the conductor i can, in general, be expressed as:

Z R jXi i i

where:

Ri is the a.c. internal resistance (calculated or measured)

Xi is the internal reactance relevant to the internal inductance, Li.

In general, the internal impedance can be expressed as (point 4.1.5.2 of {2}):

r

r

r

rL o

i ln102.0ln102

33

[H/km]

The relation between the GMR r’ and the radius r is given:

r = m.r

where the factor m is given for different conditions in the following table:

18

18


Factor m for calculating GMR

Case no.

Conductor condition Factor m

for GMR calculation

1 Circular of ferromagnetic material 4

r

e

2 Circular of non ferromagnetic material 7788.04

1

e

3 Tubular conductor Analytical expressions

4 Any with given (measured) internal inductance Li [H/km]

3102,0 iL

e

Notes: 1) It is worth mentioning that the expression for case 4 allows for specifying the r’ on conductors of any shape

and materials, e.g. for railway rails. When the factor m and thus r’ is evaluated from a measured Li then r is

the radial distance till the boundary of the inductance measurements. This boundary is, normally, the outer surface of the conductor for cylindrical conductors. While for conductors having other shapes the boundary is

at a distance where the directional dependence of Li becomes, practically, negligible, e.g. for a railway rail 220 mm.

2) In fact, the radius r’ is that of a fictitious conductor assumed to have no internal flux but with the same

inductance as the actual conductor of radius r.

19

19

Carson-Clem method

20

20

Impedance with earth returnCarson-Clem method

Description

The Carson-Clem formula is derived from the Carson’s formula {2}) by the following assumptions:

(1) the geometrical term is neglected,

(2) the terms considered from the Carson’s series expressions are:

only the first term of R

only the first two term from X.

The earth return path is represented by a pipe having a

unit length resistance of Re and a

radius of De (see next figure)). Applicability limit:

dij < 0.25 or dij(max) < 0.135 De

In this case the resulting errors are usually less than 2.5%. (For limit values see the table.)

21

21


Carson-Clem method, equivalent return tube

De

Re

dij

Ii

Uj

dij(max)

<=0.135 De

Re = 10-3 . 0.987 . f [/km]

The equivalent tube of earth return for the Carson-Clem method

22

22


Values of De and dmax as functions of frequency f and earth resistivity

De [m] dmax [m]

f [m] [m]

[Hz] 10 100 1 000 10 000 10 100 1 000 10 000

16.7 510 1613 5100 16128 37.2 217.7 688.5 2177

50 295 932 2948 9321 21.5 125.8 397.9 1258

60 269 851 2691 8509 19.6 114.9 363.2 1149

800 74 233 737 2330 5.4 31.5 99.5 315

1200 60 190 602 1903 4.4 25.7 81.2 257

2000 47 147 466 1474 3.4 19.9 62.9 199

10000 21 66 208 659 1.5 8.9 28.1 89

100000 7 21 66 208 0.5 2.8 8.9 28

1000000 2 7 21 66 0.2 0.9 2.8 9

23

23

Impedance with earth returnCarson-Clem formula for the mutual impedance

and mutual inductance

The expression for mutual impedance with earth return, per unit length, is:

ij

eij

d

DjfZ ln102.0987.010 33

[/km]

The expression for the complex mutual inductance with earth return, per unit length, is:

MD

dj

D

dj

H

kmije

ij

e

ij

2 10

210 200 157 12 2ln ln .

Note:

1) For calculating the self impedance the mutual conductor distance dij is substituted by the conductor radius r or conductor GMR r’

2) The following relation applies between the impedance and inductance, both with earth return:

Z j Mij ij

24

24

Complex image method

25

25

Impedance with earth returnComplex image method

Description Applicable for considering:

high spacing or separation distance

high frequency

conductor height above the earth.

Relevant complex image:

normally the complex image depth given by (5) and

for high frequency the complex image depth given by (6)

26

26

Impedance with earth returnNumerical values for the complex image depth

Modulus of p complex image depth as function of frequency f and earth

resistivity

p[m]

f [m]

[Hz] 10 100 1 000 10 000

16.7 275 871 2 754 8 709

50 159 503 1 592 5 033

60 145 459 1 453 4 594

800 40 126 398 1 258

1 200 32 103 325 1 027

2 000 25 80 252 796

10 000 11 36 113 356

100 000 4 11 36 113

1 000 000 1 4 11 36

Note: The highest error occurs at =90, i.e. when both conductors are placed at earth surface, but even this maximum error does nor exceeds 16 %.

27

27


Complex image formula

a) For general applications:

the mutual impedance with earth return:

Z

j p h h a

h h aij

i j ij

i j ij

0

2 2

2 22

2ln

or with the use of the distances to the other conductor dij and to the image of the

other conductor Dij (see Fig.1):

Z jD

dij

ij

ij

0 2 10 3. ln

the external part of the self impedance:

Zj p h

rii

i

i

0

2

2ln

( )

28

28


Complex image formula

b) For high frequency applications:

the mutual impedance with earth return:

Z

j p h h a

h h aij

i j ij

i j ij

0

2 2

2 22

2ln

(1)

the external part of the self impedance:

Zj p h

rii

i

i

0

2

2ln

( )

Note: The error involved in calculation of the external impedance due to the neglecting the ore term in the calculation of the complex image depth can be demonstrated by the following data:

around 0.5 %, when ore = 5.55 %, for = 10 m and re = 10, at 10 MHz

around 11%, when ore = 555 %, for = 10000 m and re = 10, at 1 MHz.

29

29

Impedance with earth returnRelative importance of the term representing the displacement current in the earth

The normalised % value of ore with the base of

Earth resistivity [m]

Parameter 10 10000

f [Hz] Relativ permittivity of the earth ere Relativ permittivity of the earth ere

4 10 40 4 10 40

10 0.0000022 0.000006 0.00002 0.0022 0.006 0.02

16.7 0.0000037 0.000009 0.00004 0.0037 0.009 0.04

50 0.0000111 0.000028 0.00011 0.0111 0.028 0.11

100 0.0000222 0.000056 0.00022 0.0222 0.056 0.22

800 0.0001778 0.000444 0.00178 0.1778 0.444 1.78

1 000 0.0002222 0.000556 0.00222 0.2222 0.556 2.22

10.106 0.0022222 0.005556 0.02222 2.2222 5.556 22.22

100 106 0.0222222 0.055556 0.22222 22.2222 55.556 222.22

1000 106 0.2222222 0.555556 2.22222 222.2222 555.556 2 222.22

10000 106 2.2222222 5.555556 22.22222 2222.2222 5555.556 22 222.22

30

30

Earth return impedance

for lines of finite length

31

31

Plots showing the mutual inductancefor lines of finite length [1]

[1] Kostenko M.V., Mihajlov K.E.,

Potnov E.L., Pozdnjakov L.G., Sokolov

V.B., Zarehi M., -

Resistive and inductive interference on

communication lines entering large

power plants.

CIGRE, 36.02, 1970 Session.

32

32

Numerical values of the parameters for the ranges of interest

Values of a Values of = l /a

For frequencies of For lengths of

Separation

„a”

m 16 2/3 Hz 50 Hz 40m 160m 500m

5 0.5810-2 110-2 8 32 100

10 1.1510-2 210-2 4 16 50

20 2.29-2 410-2 2 8 25

Note: For earth resistivity of 100 m

33

33

Numerical values for the mutual impedance

R, X and Z, for the ranges of interest

Values of R and X m/kma)

For frequencies of

16 2/3 Hz 50 Hz

For lengths (in m) of

Separation „a”

m

Real, imaginary part and modulus

40 160 500 40 160 500

R 0.43 1.6 4.2 15 2.2 8.8 22 50

X 54 84 104 125 170 251 314 330 5

Z 54 84 104 126 170 251 315 334

R 0.42 1.6 4.2 15 2.2 8.8 22 50

X 42 70 94 104 127 204 267 286 10

Z 42 70 94 105 127 204 268 290

R 0.43 1.6 4.2 15 2.2 8.8 22 50

X 31 54 77 94 91 158 220 242 20

Z 31 54 77 95 91 158 221 247

Note: a) For earth sensitivity of p=100m

34

34

Numerical values for the modulus of the mutual impedance, Z

Normalized values in per cent, base the value for infinite length

0

50

100

%

a=5 m 57 67 83 100 51 75 94 100

a=20 m 33 57 81 100 37 64 90 100

40 m 160 m 500 m infinite 40 m 160 m 500 m infinite

16 2/3 Hz 50 Hz

Note: For earth sensitivity of p=100m

35

35

Thank you for your attention

?György Varju

e-mail: [email protected]

Earth Return Imped v1

Documents

Transcript of Earth Return Imped v1