Telecommunication Networks

Distributed Parameter Networks

Distributed Parameter Networks 1.

The electric and magnetic power distribute homogeneously along the wire, but the current changes in time.

d

Lecher wire koaxial cable

The electromagnetic wave passes on the wire.

EM waves theory, but if d<<, we can examine the topic

with quasi-stationary method.

m

HL

m

FC ;

m

SG

mR

;

Generator

Consumer

Induction law for ABCD loop:

AL

AdBt

sdE

t

AdBt

txudxR

idxx

utxudx

RisdE

AL

,2

,2

The flux is proportional to the current: =Ldxi

dt

diLRi

x

u

Loop equation for dx :

Distributed Parameter Networks 2.

Rearranging:

On dx section charches accumulate, or accumulated charges disappear. This increases the difference between input and output current. On dx the change of charges in time:

t

uCdxtxuCdx

tt

q

,

t

uCGu

x

i

is the displacement current eltolási between the two wires

Continuity equation : t

uCdxtxiGdxtxudx

x

itxi

,,,

where u(x,t)Gdx is the leakage current

Distributed Parameter Networks 3.

The equvalent circuit of dx on power transmission line:

iLjRLijRix

u

uCjGCujGux

i

Two equations on pure sine signal:

Distributed Parameter Networks 4.

The 2. equation differentiated by x and we put du/dx from 1. equation into 2. equation we get:

t

iLRi

x

u

.1Telegram equations

RGit

iLGRC

t

iLC

x

i

2

2

2

2

t

uCGu

x

i

.2

x

/

x

i

tL

x

iR

tx

iL

x

iR

x

u

2

2

2

2

2

2

2

t

uLC

t

uLG

t

uRCRGu

x

u

RGu

t

uLGRC

t

uLC

x

u

2

2

2

2

Distributed Parameter Networks 5.

We try to get a formula like this:

ux

uu

x

uu

t

uuj

t

u 22

22

2

2

; ; ;

substituting back :

xtjeUu 0

RGuuLGRCjuLCu 22

CjGLjR 2

CjGLjR j

xtjxxjtj eeUeUU 00

Considering a pozitive , we get:

tényezősicsillapítá zőfázisténye

– propagation coefficient

Distributed Parameter Networks 6.

Attenuation factor

Phase factor

If a is considered

v

xtj

t eeUu 0

This means a wave passing toward negative x direction on rate v

v

xj

v

xj

ee

2

2

v

The length of period is:

22

v Formula for

wavelength and phase factor

Wave passing on positive x is:

Distributed Parameter Networks 7.

The time dependence of the voltage measured in a given position on the wire is pure sinusoidal.

At dx distance the amplitude decreases and the phase changes. Substituting voltage wave into

equation, we get:

xtjeUu 0

iLjRx

u

iLjReU xtj 0

Distributed Parameter Networks 8.

From the previous picture we suppose that the current formula is:

xtjeItxi 0,

Thus: xtjxtj eILjReU 0

0

00

0

Z

CjG

LjRLjR

I

U

Wave impedance

Distributed Parameter Networks 9.

Substituting the negative direction wave we get:

xtjeUU 0

The general solution of the voltage wave is:

xtjxtj eUeUuutxu 00,

00

0 ZI

U

xtjxtj eIeIiitxi 00,

or:

xtjxtj eZ

Ue

Z

Utxi

0

0

0

0,

Distributed Parameter Networks 10.

The general solution of the current wave is:

thus

0;0 GR

Thomson formula: where

[L]=Henry and [C]=Farad

0 j

LCjLC 2

If we decrease the value of L and C towards 0, v does not reach c (speed of light). On idle wire the waves pass by c rate.

Phase factor

CjGLjR

LC

LCv

1

LC

1

sec

1

m

HL

sec

mv

m

FC

Ideal wire:

rate

és

Distributed Parameter Networks 11.

If we increase capacities with geometrical sizes the inductivity decreases and vice versa. Thus it is impossible to make a construction which can operate on higher rate than c.

rd

Cln

r

dL ln

r

dZ ln

1

b

k

rr

Cln

2

b

k

r

rL ln

2

b

k

r

rZ ln

2

1

Two wiresCapacity Induction Wave

resistance

Distributed Parameter Networks 12.

In idle case the rate and wave impedance do not depend on frequency.

If rate would depend on frequency, distortion would occure for example on pulse signal, because the spectra is: 0, 30, 50, … etc. and we would get different phase delays on different frequencies.

On the cables with big attenuation the big delay time causes big problems:

00

Z

C

L

CjG

LjRZ

For idle wire:

d

d

Distributed Parameter Networks 13.

Let’s say: U0+ =A and U0

- =B and x=-l. We do not consider the time dependence case, thus:

tjxxxtjxtj eeUeUeUeUuutxu 0000,

We examine the transmission line with Z at the end:

tjxxxtjxtj eeZ

Ue

Z

Ue

Z

Ue

Z

Uiitxi

0

0

0

0

0

0

0

0,

lll BeAeU

lll BeAeZI 0

ll

l eZ

Be

Z

AI

00

Ih IrUh Ur

Distributed Parameter Networks 14.

Adding and substracting:BAUZ

Let’s count the value of A and B with UZ and IZ:

If =0, it means U =UZ and I =IZ

BAIZ Z 0

ZZ IZUA 02

ZZ IZUB 02

20ZZ

IA Z

20ZZ

IB Z

Distributed Parameter Networks 15.

Current reflection factor is:

lll

l

h

r eeA

B

Ae

Be

U

U 20

2

Voltage reflection factor:

lr BeU l

h AeU

1

1

1

1

2

2

0

0

0

0

0

0

0

Z

Z

ZZZZ

ZZ

ZZZZ

I

ZZI

A

B

Z

Z

helyénlezárásareflexióa 0

10 ll

l

l

h

r eeA

B

eZA

eZB

I

I 20

2

0

0

Distributed Parameter Networks 16.

Reflection factor at Z

The direct and reflected waves on complex plain:

ljljllll BeAeeBeAeU j

nál0 ljljl BeAeU

A and B are complex numbers:

jj eBBéseAA

thus:

BljBljl eBeAU

Distributed Parameter Networks 17.

Where the two vectors are in phase voltage-maximum

Where the phase difference is 180o voltage-minimum

If ZZ0

along the line standing waves are self created

If Z=Z0 , there are not standing waves

0

00

0

ZZ

ZZ

Distributed Parameter Networks 18.

BAU max

Voltage standing wave:

BAU min

0

0

min

max

1

1

1

1

ABAB

BA

BA

U

Ur

From the figure: Distance between two minimum or maximum

values is /2 Distance between maximum and minimum values

is /4

00 1

1

r

rr1

or:

0

Distributed Parameter Networks 19.