Transmission Line Electromagnetic Transients with …catedraendesa.us.es/documentos/sem_Nucci/Course Sevilla...Transmission Line Electromagnetic Transients with special Reference to

Transmission Line Electromagnetic Transients with

special Reference to the Lightning Performance of

Transmission and Distribution Lines

Short course

by

Carlo Alberto Nucci

University of Bologna – Faculty of Engineering – Department of Electrical Engineering

University of Sevilla

June 27 and 28, 2011

The material contained in this lectures is based on the results obtained within the framework of a joint research collaboration among the

University of Bologna, the Swiss Federal Institute of Technolgy – Lausanne, the University of Rome ‘La Sapienza’, and the University of Florida.

© Carlo Alberto Nucci

Course Outline 1/3 1. Transient Perturbations in Power Networks: a General Overview

2. ‘Generalized’ Transmission Line Equations (with Illumination of

an External Electromagnetic Field)

2.1. Preface and Main Assumptions

2.2. Single Conductor Line above a Perfectly Conducting Ground

- Agrawal, Price, and Gurbaxani model

- Taylor, Satterwhite, and Harrison model

- Rachidi model

- Contribution of the different components of the electromagnetic field in the

coupling mechanism

- Other models

- Classical Transmission Line Equations (no External Field Illumination)

2.3. Single Conductor Line above a Lossy Ground

- Agrawal, Price, and Gurbaxani model extended to a lossy ground

- Classical Transmission Line equations (no External Field Illumination)


Course Outline 2/3 2.4. Coupling Equations for a Multi-Conductor Line

2.5. Line Parameters for Multi Conductor Line above a Perfectly

Conducting Ground

2.6. Line Parameters for Multi Conductor Line above a Lossy Ground

3. Solving the ‘Generalized’ Transmission Line equations 3.1. Analytical or numerical solution? Frequency or time domain

solution?

3.2. An interesting Case of Analytical Solution: Single Conductor

Line above a Perfectly Conducting Ground, no External Field

Illumination.

3.3. Propagation of surges along an ideal non-illuminated

transmission line: reflection coefficient and transmission

coefficient

3.4. Time-Domain Solution: Finite Difference Method

3.5. Frequency Domain Solution

3.6. An example of Time Domain Code: LIOV


Course Outline 3/3 4. Lightning Transient Overvoltages 4.1 Lightning as Source of Transients

− The Phenomenon of the Lightning Discharge

− The Lightning Current and Relevant Statistics

− Lightning Return Stroke Current Models

− Lightning Electromagnetic Fields and Relevant characteristics

− Lightning Location Systems

4.2 Lightning as a Source of Voltage Dips (Sags)

4.3 Lightning-Induced Overvoltages − The IEEE St 1410 for Evaluating the Lightning Performance of a

Distribution Line and the CIGRE proposed methodology, based on the

LIOV code and LIOV-EMTP

4.4 Lightning Overvoltages due to Direct Strokes − The IEEE St 1243 for Evaluating the Lightning Performance of a

Transmission Line and the CIGRE proposed methodology

4.5 Mitigation of Lightning-Originated Overvoltages − Direct Lightning: use of the FLASH code for optimal line design

− Indirect Lightning: use of the LIOV and LIOV-EMTP codes to optimize

the use of shielding wires and of surge arresters.


Transient Perturbations in Power

Networks: a General Overview

Part 1


4. Thermodynamic phenomena

(from boyler control action in steam power plants) 10 ÷ 10 4 s

2. Electromagnetic phenomena (machine windings following a disturbance, operation of the protection system

or interaction between electrical machines and network) 10-4 ÷ 1 s

3. Electromechanical phenomena (oscillations of rotating masses of the generators and motors occurring after a

disturbance, operation of the protection system or voltage and prime movers

control) 10-1 ÷ 10 s

1. ‘Wave’ phenomena

(e.g. surges caused by lightning and switching operation) 10-7 ÷ 10-3

s

Time frame of the basic power system

dynamic phenomena

(adapted from Machowski et al., 2005)


Different types of transients

Transient recovery voltage across a circuit breaker

following interruption of the fault current (type 2)

Modification of surges due to direct

lightning stroke with distance

traveled (type 1)

Voltage variation during bank-to-bank

capacitor switching (type 2)

Adapted from [A. Greenwood, “Electrical Transients in Power Systems”, 2nd Ed., J. Wiley and Sons, NY, 1991]


Line energization (NON-matched) © Carlo Alberto Nucci

Zc SA

SA SA SA SA Zc

SA

200 m

Stroke

location 5

0 m

370 m © Carlo Alberto Nucci

‘Generalized’ Transmission Line

Equations (with Illumination of an

External Electromagnetic Field)

Part 2


2.1. Preface and main assumptions

Voltages and currents propagating along transmission lines are in general

evaluated using the Transmission Line (TL) approximation.

And this for both steady state and transient conditions.

Both steady state equations and transient equations are used by power

engineers.

Clearly, steady state equations are a particular case of transient ones.

( , ) ( , )' 0

u x t i x tL

x t

( , ) ( , )' 0

i x t u x tC

x t

0' xx ILj

dx

Ud

0' xx UCj

dx

Id

The above equations are valid for a lossless line


Transients, as earlier mentioned, are due to various sources such as faults,

switching operations and direct lightning strikes to the lines, but also to the

action of external electromagnetic fields (e.g. nearby lightning strokes).

We here use the adjective ‘generalized’ to mean that the transmission line

equation describing voltage and current propagation along transmission line

(that we are going to derive) take into account also the presence of an

external electromagnetic field.

These equations are also called transmission line coupling equations: they

describe, in general, the electromagnetic coupling between an external

electromagnetic pulse and a transmission line.

When the external electromagnetic field is equal to zero, they become the

‘classical’ transmission line equations.

2.1. Preface and main assumptions cont’


To solve the problem of interest we can use the antenna theory, the general

approach based on Maxwell's equations.

However, due to the length of distribution lines, the use of such a theory for

practical evaluations implies long computation time, especially when statistical

studies are desired.

The use of quasi-static approximation, according to which propagation is

neglected and coupling between incident electromagnetic fields and line

conductors can be described by means of lumped elements, is not appropriate.

In fact, such an approach requires that the overall dimensions of the circuit be

smaller than about one tenth of the minimum significant wavelength of the

electromagnetic field, an unacceptable assumption for the most common case

of power lines illuminated by external fields (For instance the frequency spectrum

of lightning electromagnetic field extends up to frequencies of about a few MHz and

even beyond, which corresponds to minimum wavelength of about 100 m or less.)



i

z

y

x The line geometry is

reasonably uniform;

Let us make reference to the

geometry of Fig. 1 and make the

following assumptions:

2a i

h

The transverse dimensions (cross sectional dimensions) of the

line are small compared to the minimum wavelength lmin; we

can then consider that propagation occurs mainly along x axis, and, as

we shall see, the line can be represented by a distributed-parameter

structure along its axis.



The line response is quasi-TEM (transverse electromagnetic), i.e. the

electromagnetic field produced by the electric charges and currents

along the line is confined in the transverse plane and perpendicular to

the line axis. (Note that it is in practical impossible that the response of a line

be purely TEM. In fact, a pure TEM mode would occur only for the case of a

perfectly conducting ground and when the exciting electromagnetic field has no

electric field component tangential to the line conductors)

The sum of the line currents at any cross section of the line is zero, i.e. the

ground -the reference conductor- is the return path for the currents in the n

overhead conductors.(This means that we are considering only ‘transmission line

mode’ currents and neglecting the so-called ‘antenna-mode’ currents. If we desire to

compute the load responses of the line, this assumption is adequate, because the

antenna mode current response is small near the ends of the line. Along the line,

however, the presence of antenna-mode currents makes that the sum of the currents

at a cross section is not necessarily equal to zero. However, the quasi-symmetry due

to the presence of the ground plane results in a very small contribution of antenna

mode currents and consequently, the predominant mode on the line will be TL.)



Incident field

.

ZL ZL Total current

I1(x)

I2(x)

Einc

Hinc k ^

=

ZL ZL TL current

It1(x)

It2(x)

ZL ZL Antenna current

Ia(x)

Ia(x)

Decomposition of line current and charge into antenna and

transmission line (TL) components



Magnitudes of the total wire currents I1

and I2 and the transmission line

component It1 as a function of position

along the line of previous slide

(supposed 30 m long).

Magnitudes of the current distribution

along a 30 m long line over a perfect

ground using the transmission line (TL)

theory and the antenna theory



Incident field

Scattered field

TOTAL FIELD

To each field component we associate the corresponding voltage.

2.2. Single Conductor Line above a Perfectly

Conducting Ground

E = Ei + Es

B= Bi + Bs


Agrawal, Price, and Gurbaxani model

h

x y

z C

2a x

We shall adopt a frequency domain formalism, although the same equations

that we shall achieve could be obtained starting from a time domain

formalism.

Applying Stokes theorem to the first Maxwell equation

where is the electric field, is the magnetic flux density and C is the

contour of surface S.

rot E j B

E l j B S

C S d d

E

B

(1)


Conducting Ground


Considering only the scattered electric and magnetic fields,

identified by superscript s, we can write the first and the second

member of (1) respectively as (C clockwise)

(2)

(3)

By omitting for convenience the notation concerning ordinate y within the

parenthesis (equal to zero in (2) and (3)), equation (1) can be rewritten as

follows

( ) ( ) [ ] ( ) [ ] ( ) [ ]

E x x z E x z z E x h x j B x z x z z

s

z

s

h

x

s

x

x x

y

s h

x

x x

, , , , d d d d 0

0 0 (4)

Es´dlCò = - E

z

s x+ Dx,0,z( ) - Ez

s x,0,z( )éë

ùûdz

0

h

ò + Ex

s x,0,h( )éë

ùû

x

x+Dx

ò dx

- jw Bs´dSS

ò = - jw By

s x,0,z( )dxéë

ùûdz

0

h

òx

x+Dx

ò


Conducting Ground


By dividing both members of (4) by x and taking the limit for x0, we

obtain

(5)

(6)

By adding to both members of (5) the horizontal incident electric field at

height h, identified by superscript i, and reminding that it

comes

( ) ( ) ( )

0 0 x E x z z E x h j B x z z z

s h

x

s

y

s h

, , , d d 0

E E Es i

( ) ( ) ( ) ( )

0 x E x z z E x h j B x z z E x h z

s h

x y

s h

x

i , , , , d d

0

In the second member of (6) there is only the incident component of the

electric field, which is the source term.

Now we shall rewrite the first member, where are present the electric and

magnetic fields, in terms of line voltages and currents.


Conducting Ground


This is possible for the two main assumptions of the transmission line

theory:

line response to the the incident electromagnetic field is quasi-TEM;

the transverse dimensions (cross sectional dimensions) of the line are

small compared to the minimum wavelength lmin.

From these hypotheses is then possible to define the line scattered voltage as

follows:

( ) ( ) U x E x z z s z s

h

, d 0

(7)

The electric field on the surface of the overhead conductor, , is

linked to the line current I(x) through the conductor surface impedance per

unit of length

( )E x hx ,

Zw

'

( ) ( )E x h Z I xx w, ' (8)

that we shall neglect in what follows.


Conducting Ground


By substituting (7) and (8) in (6) we obtain

From the TL hypotheses is possible to link the scattered magnetic fields with

the line currents by means of a self-inductance coefficient

(9) ( )

( ) ( ) d

d d

U x

x j B x z z E x h

s

y s

h

x i , ,

0

( ) L

B x z z

I x

y

s h

' ,

( )

d 0

Equation (9) can be rewritten, by considering (10), obtaining the first field-

to-transmission line coupling equation:

( ) ( ) d

d

U x

x jL I x E x h

s

x

i ' ( , )

(11)

(10)

(I)


Conducting Ground


(I) (see page before)

For L’ , the following expression applies

1

2 2'

2cosh ln

o oC for h ah h

a a

(II) (see page after)

For C’ , the following expression applies

ahfora

h

a

hL oo

2

ln2

2cosh

2' 1


To obtain the second equation of the field-to-transmission line coupling

model, we shall start from the equation that links the current I(x) and the

charge along the line (continuity equation)

d

d

I x

xj q x

( )( )' 0 (12)

Quantity can be related with the scattered voltage using the

hypotheses previously mentioned through a per unit of length capacitance C' ,

obtaining the second field-to-transmission line coupling equation:

j q x '( )

d

d

I x

x jC U x s ( )

( ) ' 0 (13)

where C’=q’(x)/Us(x)

(II)

(14)


Conducting Ground


In summary: Agrawal Model

( )' ( ) ( , )

si

x

dU xj L I x E x h

dx

0

( )

h

i

zE x dz) ( ) ( ) ( x u x u x u i s t + = Total

voltage:

( )' ( ) 0

dI xj C U x

dx

0

(0) (0) (0, )

h

s i

A zU Z I E z dz

0

( ) ( ) ( , )

h

s i

B zU L Z I L E L z dz Boundary

conditions:


Conducting Ground


In summary: Agrawal Model

( )' ( ) ( , )

si

x

dU xj L I x E x h

dx

) ( ) ( ) ( x u x u x u i s t + = Total

voltage:

( )' ( ) 0

dI xj C U x

dx

( , )i

xE x h

0

0

(0) (0) (0, )

h

s i

zU Z I E z dz

0

( ) ( ) ( , )

h

s i

L zU L Z I L E L z dz Boundary

conditions:

0

( )ih

i

zE x dz

0

(0, )

h

i

zE z dz

0

( , )

h

i

zE L z dz


Conducting Ground


0

+

- -u i (0)

R 0

+

-

L

-u i (L)

R L

u (x)

i(x) L'dx

x x+dx

+ -

i(x+dx)

u s (x+dx)

u i (x)

i E x dx

s C'dx


Conducting Ground


FAQ:

Which component of the LEMP does affect most the induced

voltages?

Vertical E component?

Horizontal E component?

Other components?

Let us keep assuming, for simplicity, a lossless line


Conducting Ground


hiz

ix

hiy dztzxE

xthxEdztzxB

t00

),,(),,(),,(

z

x y

E i, B i


Conducting Ground


Taylor et al.

0

u (x)

i(x) L'dx

x x+dx

+ -

i(x+dx)

+

-

+

-

L

u s (x+dx)

-u i (0,t)

R 0

-u i (L)

R L

u i (x)

i E x dx

s C'dx

Agrawal et al.

0

u (x)

i(x) L'dx

x x+dx

+ -

i(x+dx)

L

u (x+dx)

R 0

R L

i (B

y (x,z) dz)dx

C'dx

d

dt

i (E

z (x,z) dz)dx

d

dt -C’

0

( )' ( ) ( , )

h

i

y

dU xj L I x j B x z dz

dx

0

( )' ( ) ' ( , )

h

i

z

dI xj C U x j C E x z dz

dx

( )' ( ) ( , )i

x

dU xj L I x E x h

dx

( )' ( ) 0sdI x

j C U xdx

dzzxExUxUxUxU

h

i

z

sis ),()()()()(0

Taylor, Satterwhite, and Harrison model

s


Conducting Ground


-20

0

20

40

60

80

0 2 4 6 8

Time (µs)

E contribution

E contribution

Total

X

Z

Agrawal et al. -60

-40

-20

0

20

40

60

80

0 2 4 6 8

Time (µs)

B contribution

E contribution

Total

z

y

Taylor et al.


Conducting Ground


0

u (x)

i(x) L'dx

x x+dx

+ -

i(x+dx)

+

-

+

-

L

u s (x+dx)

-u i (0,t)

R 0

-u i (L)

R L

u i (x)

i E x dx

s C'dx

Agrawal et al.

0

i B y (x,0)

1

L’ -

h

dz

0

u (x)

i(x) L'dx

x x+dx

i(x+dx)

L

u (x+dx)

R 0 R L

C'dx

i B y (x,0)

1

L’ -

h

dz

i B x (x,z)

y

1

L’ 0

-

h

dz [ ] dx

Rachidi

( )' ( ) ( , )i

x

dU xj L I x E x h

dx

( )' ( ) 0sdI x

j C U xdx

( )' ( ) 0sdU x

j L I xdx

0

( , )( ) 1' ( )

'

h is

xB x zdI xj C U x dz

dx L y

dzzxBL

xIxIh

ey

s0

),('

1)()(dzzxExUxUxUxU

hez

ses ),()()()()(0

Rachidi model


Conducting Ground


-20

0

20

40

60

80

0 2 4 6 8

Time (µs)

E contribution

E contribution

Total

X

Z

Agrawal et al.

-20

0

20

40

60

80

0 2 4 6 8

Time (µs)

B contribution

B contribution

Total

X

y

Rachidi


Conducting Ground


Taylor et al. 0

u (x)

i(x) L'dx

x x+dx

+ -

i(x+dx)

+

-

+

-

L

u s (x+dx)

-u i (0,t)

R 0

-u i (L)

R L

u i (x)

i E x dx

s C'dx

Agrawal et al. 0

u (x)

i(x) L'dx

x x+dx

+ -

i(x+dx)

L

u (x+dx)

R 0

R L

i (B

y (x,z) dz)dx

C'dx

d

dt

i (E

z (x,z) dz)dx

d

dt -C’

0

i B y (x,0)

1

L’ -

h

dz

0

u (x)

i(x) L'dx

x x+dx

i(x+dx)

L

u (x+dx)

R 0 R L

C'dx

i B y (x,0)

1

L’ -

h

dz

i B x (x,z)

y

1

L’ 0

-

h

dz [ ] dx

Rachidi

Contribution of the different components of the

electromagnetic field in the coupling mechanism


Conducting Ground


The contribution of a given electromagnetic field

component in the coupling mechanism depends

strongly on the used model.

Thus, when speaking about the contribution of a

given electromagnetic field component to the

induced voltages, one has to specify the coupling

model he is using.


Conducting Ground


Other models

See lecture at the blackboard


Conducting Ground


( ) ( ) d

d

U x

x j L I x '

Classical Transmission Line Equations

(no External Field Illumination)

d

d

I x

x j C U x

( ) ( )

' 0

0 ( , ) ( , )

' 0u x t i x t

Lx t

( , ) ( , )' 0

i x t u x tC

x t

Frequency domain Time domain

By setting equal to zero the source terms in the Taylor et al. model:


Conducting Ground


Agrawal, Price, and Gurbaxani model extended to …

We now extend contour C of surface S up to depth d, which is the depth below

ground beyond which the electric and magnetic field can be considered equal zero. (

d >dg , where dg is the skin depth in the ground at the considered frequency

expressed by ) (1)

and repeat the same procedure already seen in § 1.2.2. This time we consider

losses in the ground and in the line conductor.

We apply again Stokes theoreme to the first Maxwell equation and again obtain

2.3. Single Conductor Line above a Lossy

Ground

h

dxy

z C

2ax

d g rg o g 2 / ( )

E l j B S

C S d d (2)


Considering only the scattered electric and magnetic fields, identified by

superscript s, we can write the first and the second member of (2)

respectively as (C clockwise). We now obtain slightly different expressions

(note the lower extreme of integration of the vertical electric field and

transverse magnetic induction field integrals)

(3)

(4)

As done before, we omit for convenience the notation concerning ordinate

y within the parenthesis, and rewrite (2) as follows

( )d ,0, d d

x xh

s s

y

S x

j B S j B x z x zd

( ) ( )[ ] ( )[ ] ( )[ ]

E x x z E x z z E x h x j B x z x zz

s

z

s

h

x

s

x

x x

y

sh

x

x x

, , , ,d d d dd

d 0

(5)

( ) ( )[ ] ( )[ ] xhxEzzxEzxxElE

xx

x

s

x

h

s

z

s

zC

s d,0,d,0,,0,d

d


Ground


By dividing both members of (5) by x and taking the limit for

x0, we obtain

(6)

(7)

( ) ( ) ( )

d dxE x z z E x h j B x z zz

sh

x

s

y

sh

, , ,d d 0

By adding at both members of (6) the horizontal incident electric field at

height h, identified by superscript i, and reminding that , it

comes:

E E Es i

( ) ( ) ( ) ( )

d dxE x z z E x h j B x z z E x hz

sh

x y

sh

x

i, , , ,d d

In the second member of (7) there is only the incident component of the

electric field, which is the source term.

Now we shall rewrite the first member, where are present the electric and

magnetic fields, in terms of line voltages and currents.


Ground


It useful to remind that this is possible for the main assumptions of the

transmission line theory

The transverse dimensions (cross sectional dimensions) of the line are

small compared to the minimum wavelength lmin;

line response to the the incident electromagnetic field is quasi-TEM

We can then define the line scattered voltage as follows:

(8)

The electric field on the surface of the overhead conductor, , is

linked to the line current I(x) through the conductor surface impedance per

unit of length

( )E x hx ,Zw

'

( ) ( )E x h Z I xx w, ' (9)

We shall not disregard, as in 1.2.2, this surface impedance as we are now

taking losses into account. This will be clearer later, when we shall give the

expression for

( ) ( ) ( )0

, d , dh hs s s

z zU x E x z z E x z zd

Zw

'


Ground


By substituting (8) and (9) in (7) we obtain

We can now decompose the term of (10) containing as follows

(10)

By virtue of the TL theory assumptions we can correlate magnetic induction

fluxes to line currents via impedance (inductance) coefficients. We define

then

( )( ) ( ) ( )'d

, d ,d

sh s i

w y x

U xZ I x j B x z z E x h

x d

s

yB

( ) ( ) ( )j B x z z j B x z z j B x z zy

sh

y

s

y

sh

d d

, , ,d d d

0

0

( )Z

j B x z z

I xsg

y

s

',

( )

d

d0

'

sgZwhere is the surface impedance per unit length of ground

(11)

(12)


Ground


( )L

B x z z

I xTot

y

sh

',

( )

d0

Concerning the second term on the right-hand side of the equation, we can define

(13)

It is convenient to imagine this term L’Tot as formed by two terms: one

relevant to the line inductance calculated assuming the ground as a perfect

conductor (as in 1.2.2),

and one other due to the lossy ground effects

LB x z z

I x

y

sph

'( , )d

( )

0

( )[ ]L

B x z B x z dz

I xg

y

s

y

sph

', ( , )

( )

0

(14)

(15)

L L LTot g

' ' ' where (16)

(*)

(*) clearly, superscript p in (14) denotes a perfectly conducting ground. This means that (14) is equivalent of (10)

of 1.2.2.


Ground


The ground impedance per unit length is then defined by

(17) ( )

( )Z

j B x z z

I xj Lg

y

sh

',

'

dd

Equation (10) can be then rewritten, by considering (17), obtaining the first

field-to-transmission line coupling equation (lossy):

(18) ( )( )'d

( ,0, )d

six

U xZ I x E x h

x

where Z Z j L Zw g

' ' ' '

We shall soon give the expressions of the above line parameters.


Ground


To obtain the second equation of the field-to-transmission line coupling

model, we shall start from the equation that links the current I(x) and the

charge along the line (continuity equation) d

d

I x

xj q x

( )( )' 0 (19)

Quantity can be related to the scattered voltage by virtue of the TL

theory assumptions previously mentioned. As now we are taking into account

losses we cannot a-priori disregard neither the line conductance G’ nor the

ground admittanceY’g.

We then can write the second field-to-transmission line coupling equation

(lossy):

j q x '( )

(20)

where ( )' ' '

'

' ' '

g

g

G j C YY

G j C Y

'd ( )( ) 0

d

sI xY U x

x

We now give the expressions of the various line parameters introduced so far

(21)

(^) (C’ is the same as (14) of 1.2)

(^)


Ground


Line parameters

)()/2ln(2

' ahahL o

( ))(

/2ln

2' ah

ahC o

'' air CGo

If we assume an axial symmetry for the current in the conductor

(supposed cylindrical), the internal impedance of the conductor can

be expressed as wZ '

aa

Z w

w

w d

2

1' a

a

jZ w

www d

d

2

1'

( )( )

ZI a

a I aw

w w

w w

'

0

12

with clear meaning of used

symbols

where w is the propagation constant of the conductor, w e w are the

conductor conductivity and permittivity respectively.

I 0 and I 1 are the modified Bessel functions of order zero and 1 respectively.


Ground impedance and admittance

Carson formula

'

),(

' LjI

dxzxBj

Z

hsy

g

dxxx

jZ

g

hxo

g

022

2e'

( )g o g o rgj j

h

hjZ

g

gog

1ln

2'

'

'

'( ' ')g

g

j L G j CY

Z

' ' 2 ( )g g g o g o rgZ Y j j

Sunde

approximation

where

Vance

approximation

Propagation constant of the ground

g e rg are the ground conductivity and

relative permittivity respectively.


dxxx

jZ

g

hxo

g

022

2e'

h

hjZ

g

gog

1ln

2'

0

0.5

1

1.5

2

1000 104

105

106

107

108

Integral ExpressionSimplified Expression

Mo

d (

Z'g

ii)

(o

hm

s/m

)

Frequency (Hz)

a)

h>l/10 h>l

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1000 104

105

106

107

108

Integral ExpressionSimplified Expression

Arg

(Z'g

ii)

(o

hm

s/m

)

Frequency (Hz)

b)

h>l/10

h>l

Ground impedance


Relative importance of line parameters for an

overhead line

10

20

30

40

50

60

70

80

1000 104

105

106

107

108

Mo

d(Z

'gii/Z

'wi)

Frequency (Hz)

h>l/10 h>l

Conducteur à 10 m du sol (conductivité 0.01 S/m, permittivité 10).


10-5

0.0001

0.001

0.01

0.1

1

1000 104

105

106

107

108

Frequency (Hz)

h>l/10 h>l

Z'gii/L'ii

Z'wi/L'ii

Conducteur à 10 m du sol (conductivité 0.01 S/m, permittivité 10).


overhead line


( )Y G j Cg

' ' ' ( )Y G j C' ' '

'd ( )( ) 0

d

sI xj C U x

x

For practical cases we have that

Additionally, term G’ is negligible compared to jC’

Second transmission line coupling equation


overhead line


In summary: Agrawal Model (lossy)

( )' ( ) ( , )

si

x

dU xZ I x E x h

dx

dI (x)

dx+ jwC 'U s(x) = 0


Ground


0

u (x)

i(x) L'dx

x x+dx

+ -

i(x+dx)

+

-

+

-

L

u s (x+dx)

- u i (0)

R 0

- u i (L)

R L

u i (x)

i E x dx

s C' dx

Z' g

Overhead line above a lossy ground

Z' w

Z'


Ground


( , ) ' ( , ) 0si x t C u x tx t

Note that: ground resistivity plays a role in

1) the calculation of the line parameters

Time-domain representation of Agrawal Model (lossy)

'

0

( , )( , ) ' ( , ) ( ) ( , , )

t

s i

g x

i xu x t L i x t t d E x t h

x t

2) the calculation of the electromagnetic field

( , , )i

xE x t h

We here assume negligible Z’w


Ground


0

u (x,t)

i(x,t) L'dx

x x+dx

+ -

i(x+dx,t)

+

-

+

-

L

u s (x+dx,t)

- u i (0,t)

R 0

- u i (L,t)

R L

u i (x,t)

i E x (x,h,t)dx

s C' dx

t

' g ( ) i(x,t - )d 0

Overhead line above a lossy ground

(time domain representation)

We here assume negligible Z’w


Ground


Application of the convolution integral to the case of the ground

resistance Rg=Rg() (transmission line equations).

From Y(s)=G(s)X(s)

we obtain Z’g()I()

( ) ( ) ( ) ( ) ( )0 0

t t

y t x g t d g x t d

( ) ( ) ( ) ( )0 0

' '

t t

g gz i t d z t i d

( )( ) ( ) ( )

,' , ' , , 0 g

dU xj L I x Z x I x

dx

( ) ( )( )

( )

0

, , ,' ' , 0

t

g

u x t i x t i xL x t d

x t

1'g Fj

gZ'


Ground


( )1 1 1 1

' ( ) min , exp / erfc2 4 42

g go og g

o rg g

t th t t

where

• 0 and rg are the air and ground permittivity respectively;

• 0 is the air permeability;

• g=h20g (where g is the ground conductivity);

• erfc is the complementary error function.

Ground impedance

Timotin and Rachidi et al.


Ground


Classical Transmission Line Equations

(no External Field Illumination)

Frequency domain Time domain

By setting equal to zero the source terms in the coupling equations:

( )' ( ) 0

dU xj Z I x

dx

( )' ( ) 0

dI xj C U x

dx

'

0

( , )( , ) ' ( , ) ( ) 0

t

g

i xu x t L i x t t

x t

( , ) ' ( , ) 0i x t C u x tx t


Ground


2.4 Coupling Equations for a Multiconductor Line

(see also Eq. in time domain)

d

dx[U i

s(x)]+ jw[L 'ij][I

i(x)] = [E

x

e(x,hi)]

d

dx[I

i(x)]+ jw[C '

ij][U i

s(x)] = [0]

0

[ (0)] [ ][ (0)] [ (0, ) ]i

i

hs e

A i zU Z I E z dz

0

[ ( )] [ ][ ( )] [ ( , ) ]i

i

hs e

B i zU L Z I L E L z dz

Boundary Conditions


Hypothesis:

multiconductor transmission in an homogeneous medium;

parallel line conductors with a cylindrical cross-section;

infinite line length (border effects are neglected).

Determination of line inductance and capacitance parameters

2.5. Line Parameters for Multi Conductor Line above

a Perfectly Conducting Ground


1 1 2 2 .......j

j j jn n

dUZ I Z I Z I

dx

Inductive coupling:

N-wire line

+

V

Wire j

Wire i

V i High impedance

voltage meter

measuring Uj

x = L

Conductors short-circuited

at x=L

Reference conductor

x =0

+

U Voltage

source V i

L

Conductors open-circuited

at x=0


a Perfectly Conducting Ground Cont.


1 1 2 2 .......j

j j jn n

dUZ I Z I Z I

dx

For l >> L ( )1 1 2 2(0) ( ) .......j j j j jn nU U L Z I Z I Z I L

( ) 0jU L

'ij

LjZ ij

ikIk 0

with '

(0)1 1ij

j

i

UL

L j I

N-wire line

+

V

Wire j

Wire i

V i High impedance

voltage meter

measuring Uj

x = L


at x=L

Reference conductor

x =0

+

U Voltage

source V i

L


at x=0 Inductance

measurement




Magnetic Field Distribution

Analytical inductance evaluation




Multiconductor line (cylindrical thin conductors)









Circuit b

Circuit a

Conductor 3

r 13

r 14 r 24

r 23

Circuit b

Circuit a

r 13

r 14 r 24

r 23

Conductor 4

Conductor 2 Conductor 1

Current Ia Current Ib

Area S1

Area S2

Area S4 Area S3

Filament i

Filament k Filament l

Filament j

Transverse cross-section of a four-conductor system for two circuits





In an elementary current

filament of cross section dS, a

current dI=JdS flows.

J is assumed to be constant

over the cross section S:

J = I/S.

Thus for a filament area dS1 of

conductor 1, a current

IadS1/S1 flows, and in an

element dS2 of conductor 2,

the current is -IadS2/S2.

Circuit b

Circuit a

Conductor 3

r13

r14r24

r23

Circuit b

Circuit a

r13

r14r24

r23

Conductor 4

Conductor 2Conductor 1


Area S1

Area S2

Area S4Area S3

Filament i


Filament j

Circuit b

Circuit a

Conductor 3

r13

r14r24

r23

Circuit b

Circuit a

r13

r14r24

r23

Conductor 4



Area S1

Area S2

Area S4Area S3

Filament i


Filament j




The partial flux in a length x linking the

filament k and l of circuit b due to the

current (IadS1/S1) in element i of

conductor 1 is equal to:

( )131411

11

,22

14

13

rrdSS

Ix

r

drdS

S

Ixd ao

r

r

aoikl lnln

While the contribution of the current element j of

conductor 2 to the elementary flux between k

and l is:

( )232422

22

,22

24

23

rrdSS

Ix

r

drdS

S

Ixd ao

r

r

aojkl lnln

The total flux between the elementary filament k and l due to the complete circuit a

is:

( )

2 21 1

22

232

2

241

1

131

1

14

2 S SS Sa

oakl dS

S

rdS

S

rdS

S

rdS

S

rI

x lnlnlnln

Circuit b

Circuit a

Conductor 3

r13

r14r24

r23

Circuit b

Circuit a

r13

r14r24

r23

Conductor 4



Area S1

Area S2

Area S4Area S3

Filament i


Filament j

Circuit b

Circuit a

Conductor 3

r13

r14r24

r23

Circuit b

Circuit a

r13

r14r24

r23

Conductor 4



Area S1

Area S2

Area S4Area S3

Filament i


Filament j




Total flux between the elementary filament k

and l due to the complete circuit a

( )

2 21 1

22

232

2

241

1

131

1

14

2 S SS Sa

oakl dS

S

rdS

S

rdS

S

rdS

S

rI

x lnlnlnln

The total coupled flux a,b in circuit b arising from all the

current in circuit a is the algebraic mean (or average) and

is given by the integral of the contribution of the current

element j of conductor 2 to the elementary flux between k

and l over cross-sectional areas S3 and S4:

3 14 1

3113

1341

14

14,

2 S SS Sa

oab dSdS

SS

rdSdS

SS

rI

x lnln[

]lnln

3 24 2

3223

2342

24

24

S SS S

dSdSSS

rdSdS

SS

r

Circuit b

Circuit a

Conductor 3

r13

r14r24

r23

Circuit b

Circuit a

r13

r14r24

r23

Conductor 4



Area S1

Area S2

Area S4Area S3

Filament i


Filament j

Circuit b

Circuit a

Conductor 3

r13

r14r24

r23

Circuit b

Circuit a

r13

r14r24

r23

Conductor 4



Area S1

Area S2

Area S4Area S3

Filament i


Filament j




3 14 1

3113

1341

14

14,

2 S SS Sa

oab dSdS

SS

rdSdS

SS

rI

x lnln[

]lnln

3 24 2

3223

2342

24

24

S SS S

dSdSSS

rdSdS

SS

r

The per-unit lenght mutual

inductance between circuits a and b

due to the current Ia is then given as

xIa

ab

,'ba,M

]lnln

3 24 2

3223

2342

24

24

S SS S

dSdSSS

rdSdS

SS

r

3 14 1

3113

1341

14

14

2 S SS S

o dSdSSS

rdSdS

SS

r lnln['

ba,M

Circuit b

Circuit a

Conductor 3

r13

r14r24

r23

Circuit b

Circuit a

r13

r14r24

r23

Conductor 4



Area S1

Area S2

Area S4Area S3

Filament i


Filament j

Circuit b

Circuit a

Conductor 3

r13

r14r24

r23

Circuit b

Circuit a

r13

r14r24

r23

Conductor 4



Area S1

Area S2

Area S4Area S3

Filament i


Filament j




Courant I

Circuit b

Circuit a

Conducteur 3 Conducteur 4

Conducteur1 Conducteur 2

Aire S

Aire SAire S

Aire S12

34

Filament i Filament j


r13

r14r24

r23

a Courant -IaCourant I

Circuit b

Circuit a



Aire S

Aire SAire S

Aire S12

34



r13

r14r24

r23

a Courant -Ia

3 14 1

3113

1341

14

14

2 S SS S

o dSdSSS

rdSdS

SS

r lnln['

ba,M

]lnln

3 24 2

3223

2342

24

24

S SS S

dSdSSS

rdSdS

SS

r

The geometrical mean distance (GMD) of

cross section Si to cross section Sj, is denoted

by gij and is defined as:

i jS Sjiij

jiij dSdSr

SSg lnln

1

1324

23

2 gg

ggo 14'

ba,M ln

Using the geometrical mean above defined the

analytical expression of the mutual inductance

between two circuits a and b becomes:

Geometrical mean distance between two circuits (GMD)




GMD for the most

common conductors

geometries




For da << D and db << D, the DMGs have the following values:

2/)(1414 ba ddDrg

2/)(2323 ba ddDrg

2/)(2424 ba ddDrg

2/)(1313 ba ddDrg

GMD: Example

d b

Circuit b Circuit a

d a

D

All wire radii =a

1 2 3 4




If the two circuits are identical, da = db = d:

dDg 14 dDg 23Dgg 1324

Ma,b'

o

ln

D2 d 2

D

we obtain:

d b

Circuit b Circuit a

d a

D

All wire radii =a

1 2 3 4




If the two circuits have a common ground retur conductor n as in the following

figure

i iS Siiii

i

ii dSdSrS

g lnln2

1

the mutual inductance between circuits i-n and j-n is:

I

I

Return conductor

Cond. i Cond. j

i

j

- (I + I ) j i

nnij

jnino

gg

ggln

2

'ba,M

where gin, gjn, and gij are the GMD between

conductors i-n, j-n, and i-j respectively, and gnn is the

GMR of the neutral conductor n, which is defined as

the geometrical mean distance of a surface Sn to

itself:

Geometrical mean radius (GMR)




Courant I

Circuit b

Circuit a



Aire S

Aire SAire S

Aire S12

34



r13

r14r24

r23

a Courant -IaCourant I

Circuit b

Circuit a



Aire S

Aire SAire S

Aire S12

34



r13

r14r24

r23

a Courant -Ia

Single current loop

resulting from the

superimposion of

circuit a and b.


Filament l Filament k

Area S4 Area S3

r13 r24

r23 r14

Conductors 1 & 3 Conductors 2 & 4

Current I Current -I

Self-inductance per-unit-length




42

31

2314

SS

SS

rr

113313

12342314 ,

rrr

gggg

4433

234

2 gg

go ln'bb,M



Area S4 Area S3

r13 r24

r23 r14






4433

234

2 gg

go ln'bb,M

More generally per-unit-length self inductance of a loop comprised of

conductors i and a neutral (return) conductor n is:

nnii

ino

gg

g 2

2ln'

iiM



Area S4 Area S3

r13 r24

r23 r14






kiik

kiiko

gg

gg

ln22

1'ikM

i*

i

k

y ik

h i

k*

Images

Ground plane

Conductors

d*

d

2r ii

Mutual and self-inductance of lines with earth as a return conductor

(ideal ground)




kiik

kiiko

gg

gg

ln22

1'ikM

gik* = gki* and gik = gi*k*, the above

equation can be rewritten as:

ik

iko

ik

iko

g

g

g

g

lnln24

2

'ikM

i*

i

k

yik

hi

k*

Images

Ground plane

Conductors

d*

d

2rii

i*

i

k

yik

hi

k*

Images

Ground plane

Conductors

d*

d

2rii




For the self inductance (i=k) we

obtain

iiii

iio

gg

g 2

4ln'

iiM

i*

i

k

y ik

h i

k*

Images

Ground plane

Conductors

d*

d

2r ii




From self inductance

iiii

iio

gg

g 2

4ln'

iiM

since gii* = 2hi and gii = gi*i*=rii we have

ii

io

ii

io

g

h

g

h 2

2

2

4

2

lnln'iiM

i*

i

k

yik

hi

k*

Images

Ground plane

Conductors

d*

d

2rii

i*

i

k

yik

hi

k*

Images

Ground plane

Conductors

d*

d

2rii




GMR for the most

common conductors

cross-section




GMD for the most

common conductors

geometries




1 1 2 2 .......ij j jn n

dIY U Y U Y U

dx Capacitive coupling:

N-wire line

+

Wire j

Wire i

V Low impedance

current meter

x = L


at x=L

Reference conductor

x =0

+

Uj

Voltage

source A

L


at x=0

Capacitance

measurement




For l >> L

0kU k j

'ij

CjYij

0)( LIi

with

1 1 2 2 .......ij j jn n

dIY U Y U Y U

dx

( )1 1 2 2(0) ( ) .......j j j j jn nI I Y U Y U Y U L L

' (0)1 1ij

i

j

IC

L j U

N - wire line

+

Wire j

Wire i

V Low impedance

current meter

x = L

Conductors open - circuited at x=L

Reference conductor

x =0

+

j

Voltage

source A

L

Conductors short - circuited at x=0 N - wire line

+

Wire j

Wire i

V Low impedance

current meter

x = L

Conductors open - circuited at x=L

Reference conductor

x =0

+ Voltage

source A

L

Conductors short - circuited at x=0

U




Neutral (reference)

+

-

11 C'

22 C'

33 C'

Cond. 1

Cond. 2

Cond. 3

12 C'

23 C'

13 C'

U 12

U 23

U 13

+ -

+ -

+

-

+

-

+

- U U U 1 2 3

+

-

11 C'

22 C'

33 C'

Cond. 1

Cond. 2

Cond. 3

12 C'

23 C'

13 C'

12

23

13

+ -

+ -

+

-

+

-

+

- 1 2 3

For a system of many conductors, calculation of the capacitance is more involved.

Consider a system of three charged conductors over a perfecly conducting plane as

illustrated in the above figure. For this system the per-unit-length charges on the

conductors are denoted by q1’, q2’ and q3’ respectively; the potential differences (i.e. the

voltages) between the conductors are u12 , u23 and u31 .

Analytical capacitance evaluation:

partial capacitances




' ' ' '1 12 12 13 13 1 1

' ' '12 10 20 13 10 30 11 10

' ' ' ' '11 12 13 10 12 20 13 30

( ) ( )

( )

n nq C U C U C U

C U U C U U C U

C C C U C U C U

Neutral (reference)

+

-

11 C'

22 C'

33 C'

Cond . 1

Cond . 2

Cond . 3

12 C' 23

C'

13 C'

U 12

U 23

U 13

+ -

+ -

+

-

+

-

+

- U U 1 2 3

+

-

11 C'

22 C'

33 C'

Cond . 1

Cond . 2

Cond . 3

12 C' 23

C'

13 C'

12

23

13

+ -

+ -

+

-

+

-

+

- 1 2 3

Neutral (reference)

+

-

11 C'

22 C'

33 C'

Cond . 1

Cond . 2

Cond . 3

12 C' 23

C'

13 C'

12

23

13

+ -

+ -

+

-

+

-

+

- 1 2 3

+

-

11 C'

22 C'

33 C'

Cond . 1

Cond . 2

Cond . 3

12 C' 23

C'

13 C'

12

23

13

+ -

+ -

+

-

+

-

+

- 1 2 3

Using the partial capacitance between the conductors and between the

conductors and the ground as defined in the above figure, the relations between

the conductor charges and the voltages are:

U




( )

( )

2 21 20 21 22 23 20 23 30

3 31 20 32 20 31 32 33 30

' ' ' ' ' '

' ' ' ' ' '

q C U C C C U C U

q C U C U C C C U

Neutral (reference)

+

-

11C'

22C'

33C'

Cond. 1

Cond. 2

Cond. 3

12C'

23C'

13C'

V12

V23

V13

+ -

+-

+

-

+

-

+

-V V V1 2 3

+

-

11C'

22C'

33C'

Cond. 1

Cond. 2

Cond. 3

12C'

23C'

13C'

V12

V23

V13

+ -

+-

+

-

+

-

+

-V V V1 2 3

Neutral (reference)

+

-

11C'

22C'

33C'

Cond. 1

Cond. 2

Cond. 3

12C'

23C'

13C'

V12

V23

V13

+ -

+-

+

-

+

-

+

-V V V1 2 3

+

-

11C'

22C'

33C'

Cond. 1

Cond. 2

Cond. 3

12C'

23C'

13C'

V12

V23

V13

+ -

+-

+

-

+

-

+

-V V V1 2 3

In a similar way one finds that:

U1 U2 U3

U23 U12

U13 -




' ' '1 12 1( 1)

1'1 1

' ' ''21 2 2( 1) 22

1

' 11

' ' '( 1)1 ( 1)2 ( 1)

1

.

.

... . . .

.

n

j nj

n

j nj

nn n

n n n jj

q U

Uq

Uq

C -C -C

C C -C

-C -C C

Neutral (reference)

+

-

11C'

22C'

33C'

Cond. 1

Cond. 2

Cond. 3

12C'

23C'

13C'

V12

V23

V13

+ -

+-

+

-

+

-

+

-V V V1 2 3

+

-

11C'

22C'

33C'

Cond. 1

Cond. 2

Cond. 3

12C'

23C'

13C'

V12

V23

V13

+ -

+-

+

-

+

-

+

-V V V1 2 3

Neutral (reference)

+

-

11C'

22C'

33C'

Cond. 1

Cond. 2

Cond. 3

12C'

23C'

13C'

V12

V23

V13

+ -

+-

+

-

+

-

+

-V V V1 2 3

+

-

11C'

22C'

33C'

Cond. 1

Cond. 2

Cond. 3

12C'

23C'

13C'

V12

V23

V13

+ -

+-

+

-

+

-

+

-V V V1 2 3

For a system of n conductors (including the reference conductor), the previous

relations can be generalized in a matrix form as:

Analytical capacitance evaluation:

static capacitances

U1 U2 U2

U23

U12

U13 -




' ' '1 12 1( 1)

1'1 1

' ' ''21 2 2( 1) 22

1

' 11

' ' '( 1)1 ( 1)2 ( 1)

1

.

.

... . . .

.

n

j nj

n

j nj

nn n

n n n jj

q U

Uq

Uq

C -C -C

C C -C

-C -C C

' ' ''11 12 1( 1)1 1' ' ''

221 22 2( 1)2

' ' ' ' 11 ( 1)1 ( 1)2 ( 1)( 1)

.

.

.. . . . .

.

n

n

nn n n n n

C C Cq U

UC C Cq

Uq C C C

The above matrix equation can also be rewritten as:




' ' ''11 12 1( 1)1 1' ' ''

221 22 2( 1)2

' ' ' ' 11 ( 1)1 ( 1)2 ( 1)( 1)

.

.

.. . . . .

.

n

n

nn n n n n

C C Cq U

UC C Cq

Uq C C C

n

j

ijiiC1

'' C ' '

ij ijC Cwhere

[ ] [ ]' 'q C U

Note that all of the C’ii elements are positive and all of the C’ij elements for i j

are negative. The [C ’] matrix is called the static or Maxwellian capacitance

matrix or, in the power network literature, the nodal capacitance matrix. These

capacitance have no physical meaning but can be used to calculate the

physical partial capacitance between the conductors.




There is a problem with evaluating the matrix [C ’] directly. Unlike the

inductance matrix discussed in the previuos section, where M ’ij was a

function only of the ith and jth conductor geometry, the terms C ’ij

depend on the geometry of the entire collection of conductors. By

defining a potential coefficient matrix [K ’] as:

[U] = [K'] [q']

[C'] = [K']-1

The advantage of using the potential coefficient matrix is that they may

be calculated analytically for a number of particular configurations.

the static capacitance matrix can be calculated by inversion from the

matrix of the potential coefficients:

Calculation of the static capacitances




Pj

Pj

Pj

j

ojP

jP

jP

j

ot

r

r

r

q

r

r

r

qEEE

''

2

1

2

1

1

2

n

1*

n*

2* Image s

Point P

Charge q' j

j Charge - q'

r j*p

r jp

E _

_ E*

E-fields

Integration path P

Neutral (reference)

1

2

n

1*

n*

2* Image s

Charge q' j

j Charge - q'

r j*p

r jp

n Conductors

E _

_ E*

j

Conductors in a

homogeneous medium

over the ideal ground

h




' ' '

ln ln ln2 2 2

j j j j P j j PP

o jP o j o jP

r rq h q qU

r h r

'

1

1ln

2

nj k

kn jo jkj

rU q

r

For a configuratin of n arbitrary

conductors, the potential of the k-th

conductor with respect to the ground

due to the charges on all the other

conductors ( j =1 to n ) is expressed

as:

For j = k, rk*k = 2hk represents the distance

between conductor k and its image

1

2

n

1*

n*

2* Image s

Point P

Charge q' j

j Charge - q'

r j*p

r jp

E

_

_

E* E - fields h j

Integration path P

Neutral (reference)

1

2

n

1*

n*

2* Image s

Charge q' j

j Charge - q'

r j*p

r jp

n Conductors

E

_

_

E*

h j

1

2

n

1*

n*

2* Image s

Point P

Charge q' j

j Charge - q'

r j*p

r jp

E

_

_

E* E - fields h j

Integration path P

Neutral (reference)

1

2

n

1*

n*

2* Image s

Charge q' j

j Charge - q'

r j*p

r jp

n Conductors

E

_

_

E*




'

1

1ln

2

nj k

kn jo jkj

rU q

r

Comparing the obtainet relation with

[V] = [K'] [q']

we obtain that

jk

kj

okj

r

r

K ln2

1'

1

2

n

1*

n*

2* Image s

Point P

Charge q' j

j Charge - q'

r j*p

r jp

E

_

_

E* E - fields h j

Integration path P

Neutral (reference)

1

2

n

1*

n*

2* Image s

Charge q' j

j Charge - q'

r j*p

r jp

n Conductors

E

_

_

E*

h j

1

2

n

1*

n*

2* Image s

Point P

Charge q' j

j Charge - q'

r j*p

r jp

E

_

_

E* E - fields h j

Integration path P

Neutral (reference)

1

2

n

1*

n*

2* Image s

Charge q' j

j Charge - q'

r j*p

r jp

n Conductors

E

_

_

E*

h j




2.6. Line Parameters for Multi Conductor

Line above a Lossy Ground

Analytical approximations*

( )

4

1erfc/exp

4

1

2

1,

2

1min)('

tt

tht ii

ii

ii

ii

ii

gg

g

g

o

rgo

o

ig

( ) ( )

4

)cos(

2

12cos

2

1sincose

4

12/cos

2

1,

ˆ2

1min)('

2

12

0

/)cos( ijij

n

n

ijnijij

ijtTij

o

rgo

og

n

t

Ta

t

T

t

T

Tht ijijij

ij

ij

*F. Rachidi, S. Loyka, C.A. Nucci, M. Ianoz, "A New Expression For the Ground Transient Resistance Matrix Elements of

Multiconductor Overhead Transmission Lines", To be published in Electric Power System Research Journal, 2003.


dx)xrcos(xx

ej'Z ij

g

x)hh(

og

ji

ij

022

22

22

22

221

4ij

g

ji

g

ij

g

ji

g

og

rhh

r)

hh(

lnj

'Zij




Dependence of mutual ground impedance of two 10 m

high wires as a function of their horizontal distance.

(Ground conductivity: 0.01 S/m, rel. permittivity 10)




Coupling equations for a multiconductor line

Time Domain (Agrawal)

[ ] [ ]( , ) ' ( , ) ' ( , ) ( , , )i

s e

ij i g i x iijv x t L i x t i x t E x h t

x t t

[ ] [ ] [ ] 0),('),('),(

txv

tCtxvGtxi

x

sij

siji ii

1'

[ ' ] ijg

gij

Z

j

F

where denotes convolution integral

is the transient ground resistance matrix and

[ ]

i

ii

hezA dztzEtiRtv

0

),,0(),0(),0(

[ ]

i

ii

hezB dztzLEiRLv

0

),,()0()(

Boundary Conditions:


Further Reading and Acknowledgements

F.M. Tesche, M. Ianoz, Karlsson,” Emc Analysis Methods and Computational Models”, Wiley-Interscience,

1997

C.A. Nucci, F. Rachidi, “Interaction of electromagnetic fields with electrical networks generated by lightning”,

Chapter 8 of "The Lightning Flash: Physical and Engineering Aspects", IEE Power and Energy series 34,

IEE Press, London, 2003.

C.A. Nucci, F. Rachidi, M. Rubinstein, “Derivation of telegrapher’s equations and field-to-transmission line

interaction”, Chapter 1 of “Electromagnetic field interaction with transmission lines. From classic theory to HF

radiation effects”, WIT Press, Southampton, Boston, 2008.

C.A. Nucci, F. Rachidi, “Lightning protection of medium voltage lines”, Chapter 13 of "Lightning Protection",

IEE Power and Energy series 58, IEE Press, London, 2010.

A. Borghetti, S. Guerrieri, M. Ianoz, C. Mazzetti, M. Paolone, F. Rachidi, M. Rubinstein are gratefully

acknowledged.


Documents

Transmission Line Electromagnetic Transients with …catedraendesa.us.es/documentos/sem_Nucci/Course Sevilla...Transmission Line Electromagnetic Transients with special Reference to