a new time-domain model of the impedance of lossy soil in mtl model

7/17/2019 A New Time-domain Model of the Impedance of Lossy Soil in MTL Model

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Asia-Pacific Conference on Environmental Electromagnetics

CEEM 2000 May 3-7,2000 Shanghai, China

A

New Time-domain Model

of

the Impedance of

Lossy

Soil in MTL M odel

Tiebing Lu Xiang Cui Weidong Zhang

North China Electric Power University, Baoding, Hebei, 071

003,

China

E-mail:

hi

bdcuix@,publ c.bdptt.he.cn

Abstract:

In order to research the problems of

electromagnetic compatibility (EMC) at

electric power substations or electroma gnetic

interference generated by carrier channels on

multi-conductor power transmission lines,

finite-difference time-domain(FDTD ) m ethod

and multi-conductor transmission lines(MTLs)

model can be used. But the im pedance due to

lossy soil is dependent on the frequency, so

the problem is very complex. One new m odel

to deal with it in time-domain is proposed

using Pade's approximation. Contrasted with

other methods, the model is accurate and can

be used practically.

Introduction

The electromagnetic interference

produced by power transmission lines is a

relevant problem in the power community,

and it can be analyzed with Multi-conductor

Transmission Lines (MTLs) model. If the soil

is lossy, it can be easily dealt with in

frequency domain and be transferred to time

domain by IFFT. But it is only available

within the scope of linear condition. For non-

linear problems, methods in time domain

must be used. One of them is Finite-

Difference in Time-Domain (FDTD) method.

Thus the impedance of lossy soil must be

determined in time domain.

There are some to calculate

the impedance of lossy soil, but under the

condition

og

>

WE , ,

one approximation

formula has been proposed by Carson and has

been used in power systemi3]. But the

expressions for ground impedance in

Carson's formula are not convenient for

numerical calculations, since they involve an

integral over an infinite interval.

So

complex

ground return plane method is proposed to

replace Carson's

This paper sets up a time-domain model

to deal with impedance of soil for the

overhead lines using Pade's approximati~n[~I.

Compared with other formulas, the model

is

accurate. Finally a simple example is

analyzed using FDTD method.

Ground Impedance in Complex

Frequency Domain

For Fig.1, assuming P =

l / d z

he

impedance matrix of the lines due to the soil

is

expressed as Z( ), and each element of

the impedance matrix is:

Here, z&( ) is the self ground impedance of

line

k

and zkl( is the mutua l ground

impedance between line

k

and line

1.

Because of the impedance of lossy

ground much less than external inductance in

high freque ncy[*], his m ethod can be used in

a wide frequency range to calculate the per-

unit-length(PUL) parameters of M TLs.

In order to get the model in time domain,

Laplac inverse transformation is used.

So

replace the term j with complex frequency

s, then the following expressions for the

ground impedance are derived:

142

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fro m thLe following e qu at ions :

=E . o a E

Fig.

1

Sketch

of

MTLs

section abo ve

lossy

soil

1

k k s )= In(1

+

Y k k

p= 'g[(hk

+h/ ) z

+dl/ l

(10)

All these terms are determined by the

geometrical parameter

of

transmission lines

and the electromagnetic parameter

of

the

media.

Time-Domain Model of Ground

Impedance

It is clear that in equations (5) and ( 6 )

there is the term ln(1

+ y ,

so once this term

is transformed into time domain, a time-

domain formula for ground impedance can be

gained.

A. the Pad e s Approximation o ln(1

+

y )

Maclaurin series ofy around y=O:

ln(l+ y)c an be expanded into a

i = I

the term U

=w , o

according to Pade's

approximation, the following expression is

derived:

y

P l y r

2 4 , Y '

In(

1

.

y

=

(12)

, = O

the coefficient, pi and

qi

can be calculated

In them , there is

Assumes x=I/y, then

= O7 ~ N I

i = l

=I

Assuming the numerator is P(x) and the

denominator is Q(x),

so

with the help of

Hevisitle theorem, the expression can be

changed to:

Where,, ci is one

of

N I roots

of

Q(x)=O, and

So

it is clear that

b, ,c,

are constants which

are independent of x or y, the error of the

result

is

dependent on

the

value

of

N .

Fig.2

illustrates the results of function ln(1+y for

different N . In this figure, it

is

clear that the

result is more accurate when N

is

23, and the

corresponding values of b,,

c,

are listed in

Tab.1. Because there are many zeroes for t),,

c,,

only some items a re used in the calculation,

which saves lots of time in the following

iteration. In case

of

engineering c alculation,

5

the values of Pade approx imat ion

-

1O?[TL0'

/q

< - __

***

-

Pl0O

10

1OD

10'

10 1o3 1o4

Fig.2 Pade's approximation with different

N

143



for N is viable.

When y has the expressions in the

equations (5) and (6), the logarithmic item

,

so

the error

as the

f o r m x

produced by Pade's approximation can be

calculated. For example, there is a single line

above the ground, w ith its height h= l Om,

radius r=0.0255m, the length 140m,

conductivity

of

the soil

a

,=O.OlS/m and

relative permittivity

of

the

soil

E rg=4. By

comparing the real and imaginary component

with the real value in Fig.3, it is shown that

when N is 23, the result is more acc urate than

other numbers and the frequency of signal

can be limited above 1Hz. So the Pade's

approximation is very helpful to get more

accurate result in time dom ain.

So

ck s) and ck s)

an be expressed in

the forms in complex frequency domain

which can be transferred into the expressions

in time domain with the help of inverse

Laplace transform.

b~ Y k , k

N + l

l=l

-

l

1

Y k , k

B. Expressions in time domain

o

impedance

due to soil

tk ( S ) and ck, S)an be transformed into

following in time-domain.

+mw

1 9 )

So

once

Y k . k , ~ , ~ , ~nd yk, , ,*

which depend on

the geometry of MTLs are determined, the

impedance

of

lossy soil can be expressed in

time domain. Fig.4 shows t h t the-result of

this proposed method is similar as Timotin's

and more accurate than Vance's. Fig.5

is

the

relative error between the results of this

method and Timotin's method, which shows

that the relative error is less than 5 .

So

from equations (l ), (2), (1

8)

and

(1

9),

the frequency scope of Pade approximation

Id,

1

.

U

m

[

0'

-

E

10''

loo

to' loz 10

frequency

Hz)

(a) real component

the frequency scope of Pade approximation

2 ,

1

OD

10

10'

1

(b) imaginary component

Fig.3

the frequency scope of Pade's approximation

one

l ine abwe

lossy

soil

frequency (Hz)

Proposed

Timotin

Vance

P

lo8 10 1o'( 1 lo

l ime s )

Fig.4 Comparison

of pock ( t )

2 ~ )

n different

methods for a single line above lossy ground (the data

of

Timotin's

and

the proposed are similar)

contrast

of two methods

10 ,

1

,

, 1

1o . ~

1 O 1o.6 1 - ~ 1o . ~

Time(s)

Fi g5 Comparison of two methods

expressions in time domain of z,,,(s) and

Z

,

s )

are:

cf

1 b,

+

1

Y k , k

h

k . k

144



So

the time-domain model of impedance for

soil is set up, which can be implemented in

the FDTD method. In (20) and (21), there is

an i tem of Dirac hction, which can be

looked as the e ffect of the soil resistor.

FDTD Method for Lines above Lossy Soil

A . Telegraph equations in time-domain

for

M T L S

For MTLs as illustrated in Fig.6, the

voltage V(z,t) and current I(z,t) wave

processes can be expressed with telegraph

equations in time domain:

d

d

(2,

)

+

Y(f )V(2,

)

+

c,

z,

)

=I,(

24

dz d f

In them, z is the direction of transmission line,

Le

and C, are inductance and capacitance

matrixes when the transmission lines are

lossless and the ground is infinite perfect

conductive plane; Z g(t) is the time-domain

expression of impedance which has been

calculated abov e; Z,(t) is the time-domain

expression of impedance produced by lossy

lines which is far sm aller than Z,(t) and can

be treated with Prony's method; Y(t) is the

admittance matrix which can be neglected

generally. Only con sidering the effect of soil,

the term Z,(t) can be om itted.

So

FDTD

method can be used to calculate the wave

processes along the lined6].

( 2 3 )

Rsn

n

RS2

I

Fig. 6 Illustration of M T L

Because of the convolution in the

equation 22), the history data must be used

in the calculation, which limits the efficiency

of the algorithm. In order to save time,

recursive convolution must be used in FDTD

iteration.

After a series

of

complex transformation,

the iterative equation or currents on lines has

the following expression.

N l

A2

At 2 n

F = - L ,

+-

Where

U

is a unit column vector, 9; is ai

symmetrical matrix, its eleme nt is:

(27)

1

9L.1

=

p : , , l , l

+

1 J J

)

And 9y,

,,

9;/; ave the same expression

with

59;,,

except for replacing ykk with

ykl I

,

k , respectively.

YJ

And

tzJ,aJ

are calculated by Prony's

method16].

The term y J is:

'-' =

(I;- '

-

1 -3'2)a,(-eaJ

+

eZaJ + e Jy, -'

( 2 8 )

V,It.(l. W ..................................

lo.

....

__ _

I

1

;:::v,d

.................

L

.................

...............

r

f--jsi+ki~

Tlme n) x

1 0 0

O L L ,

(b)

at the end near the load

Fig.7 Voltage wave processes on one line abo ve lossless and

lossy ground

Examples of Wave Processes along

Overhead Line

As an example, a single line over lossy

ground. as m entioned a bove is calculated

using :FDTD method when the load and the

source resistor have the values of

50 Q

.

145



During calculation, the voltage source is a

unit step function. Fig.7 shows that there is

some difference between wave processes on

the line above lossy soil and lossless soil, and

the difference becomes larger as the time

goes. So in order to determine the accurate

wave process in transient analysis along the

lines during a long period, the impedance of

lossy soil must be under consideration. This

can be demonstrated by the results in

frequency domain.

Conclusions

A new time-domain model of multi-

conductor transmission lines above a lossy

ground to consider the im pedance due to soil

is proposed, which. is very easy to be

implemented in FDTD method. W ith its help,

the tedious FFT and IFFT can be waived.

Contrasted with results by other method, this

model is very efficient and accurate.

The method can be extended to the

problem including multi-layer ground, which

will be researched in the future.

References

[l]

M.D.Amore, M.S.Sarto. A new

formulation of lossy ground return

parameters for transient analysis of multi-

conductor dissipative lines.

IEEE on PD.1997-12(1): pp301-314.

[2] F.MTesche, M.V.Ianoz, T.Karlsson. EM C

analysis methods and computation models.

New York: John Wiley &Sons Press,

1996.

[3] J.R.Carson. Wave propagation in

overhead wires with ground return. Bell

System technical journal. 1926(5): pp539-

554.

[4] A.Deri, G.teven, A.Symlyen, etc. The

complex ground return plane: a simplified

model for homogeneous and multi-layer

earth return. IEEE on PAS.1981-lOO(8):

[5] S.Lin, E.Kuh. Transient simulation

of

lossy interconnects based on recursive

convolution formulation. IEEE on CAS.

[6] C.R.Pau1. Analysis of multi-conductor

transmission lines. New York: John Wiley

&Sons Press, 1994.

~ ~ 3 6 8 6 - 3 6 9 3 .

1992-39(1 ):pp879-892.

Tab.1 Values of

b,,c,

N=23)

146

a new time-domain model of the impedance of lossy soil in mtl model

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