Telecommunication Networks
Distributed Parameter Networks
Distributed Parameter Networks 1.
d
Lecher wire
koaxial cable
EM waves theory, but if d<<, we
can examine the topic with quasistationary method.
The electromagnetic wave passes on the wire.
The electric and magnetic power
distribute homogeneously along
the wire, but the current changes
in time.
Consumer
Generator
F
C   ;
m
H
L 
m

R   ;
m
S
G  
m
Distributed Parameter Networks 2.
Induction law for ABCD loop:

L E ds   t A B dA Loop equation for dx :
R
u
R


L E ds  i 2 dx  ux, t   x dx  i 2 dx  ux, t    t A B dA   t
The flux is proportional to the current: =Ldxi
u
di

 Ri  L
x
dt
Distributed Parameter Networks 3.
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:
q 
u
 Cdx u  x, t   Cdx
t t
t
Continuity equation :
i
u
i  x, t   dx  u  x, t Gdx  i  x, t   Cdx
x
t
where u(x,t)Gdx is the leakage current
Rearranging:
i
u
  Gu  C
x
t
is the displacement current
eltolási between the two wires
Distributed Parameter Networks 4.
The equvalent circuit of dx on power transmission line:
Two equations on pure sine signal:
u

 Ri  j Li  R  j L i
x
i
  Gu  j Cu  G  j C u
x
Distributed Parameter Networks 5.
u
i
1. 
 Ri  L
x
t
i
u
2.   Gu  C
x
t

/
x
Telegram equations
 2u
i
 2i
i
 i
 2 R L
R L
x
x
xt
x
t x
 2u
u
u
 2u
 RGu  RC
 LG  LC 2
2
x
t
t
t
 2u
 2u
u



LC

RC

LG
 RGu
2
2
x
t
t
The 2. equation differentiated by x and we put du/dx from 1.
equation into 2. equation we get:
 2i
 2i
i
 LC 2  RC  LG   RGi
2
x
t
t
Distributed Parameter Networks 6.
u  U 0  e j t x
2
u
 2u

u

u
2
2
 j u ;



u
;




u
;


u
2
2
t
t
x
x
We try to get a formula like this:
substituting back :
 2u   2 LC  u  j RC  LG u  RGu
 2  R  jL G  jC 
   R  jL G  jC  – propagation coefficient
    j
Considering a pozitive , we get:
Attenuation factor
U   U 0 e jt   j  x  U 0 e x  e j  t   x    csillapítá si tényező
  Phase
fázisténye
ző
factor
Distributed Parameter Networks 7.
If a
 
is considered
u   U 0  e t  e
 x
j t 
 v
This means a wave passing toward negative x direction on rate v
The length of period is:
e
 x

j    2 
 v

e
j
x
v

v
Wave passing on positive x is:

2 v


2

 2
Formula for
wavelength and phase
factor
Distributed Parameter Networks 8.
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
u   U 0  e j t  x voltage wave into
u

 R  j L i equation, we get:

j  t  x


U

e
 R  j L i
0
x
Distributed Parameter Networks 9.
From the previous picture we suppose that the current formula
is:
i   x, t   I 0  e j t  x
Thus:
  U 0  e j t  x  R  j L I 0  e j t  x
U 0 R  j L
R  j L


 Z0

I0

G  j C
Wave impedance
Distributed Parameter Networks 10.
U   U 0  e j t  x Substituting the negative direction wave we get:
U 0
 Z 0

I0
The general solution of the voltage wave is:
u  x, t   u   u   U 0  e j t x  U 0  e j t x
The general solution of the current wave is:
i  x, t   i   i   I 0  e j t x  I 0  e j t x
or:
U 0 j t x U 0 j t x
i  x, t  
e

e
Z0
Z0
Distributed Parameter Networks 11.
R  0 ; G  0 thus  
Ideal wire:
    2 LC
    j
H és
L  
m
 0
F
C   
m
Thomson formula:


v  
R  jLG  jC 
  j LC
   LC

1
v 

m
LC
Phase factor
rate
sec
1
where
LC
1
  
sec
[L]=Henry and [C]=Farad
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.
Distributed Parameter Networks 12.
Two wires
Capacity
Induction

 d
L  ln
 r
Z
 rk
L
ln
2 rb
1
Z
2
C
d
ln
r
2
C
rk
ln
rb
Wave resistance
1

 d
ln
 r
 rk
ln
 rb
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.
Distributed Parameter Networks 13.
For idle wire:
R  j L
L
Z0 

Z0
G  j C
C
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, 30, 50, …
etc. and we would get different phase delays on different
frequencies.
On the cables with big attenuation the big delay time causes big
problems:

d  

d
Distributed Parameter Networks 14.
We examine the transmission line with Z
at the end:


u  x, t   u   u   U 0 e j t  x  U 0 e j t  x  U 0 e  x  U 0 e x e j t





 j t
U
U
U
U


j t  x
j t  x
 x
0
0
0
0  x
i  x, t   i  i 
e

e
 
e 
e e
Z0
Z0
Z0
 Z0

Let’s say: U0+ =A and U0- =B and x=-l. We do not consider the
time dependence case, thus:
l
 l
U l  Ae  Be
I l  Z 0  Ae l  Be  l
Uh
Ur

A  l B  l 
 I l 
e  e 
Z0
Z0


Ih
Ir
Distributed Parameter Networks 15.
Let’s count the value of A and B with UZ and IZ:
If =0, it means U =UZ and I  =IZ
UZ  A  B
Z0 I Z  A  B
2 A  U Z  Z0 I Z
2B  U Z  Z0 I Z
Adding and substracting:
Z  Z0
A  IZ
2
Z  Z0
B  IZ
2
Distributed Parameter Networks 16.
U r  Be  l
U h  Ae l
U r Be  l B  2 l
 2 l
Voltage reflection factor:



e


e
0
U h Ae l
A
Z
Z  Z0
1
IZ
Z  Z0 Z0
B
Z  1
2
0  



Z
 1
A I Z  Z0 Z  Z0
Z
1
Z
Z0
2
at Z helyén
0  aReflection
reflexió factor
a lezárás
B  l

e
0  1
Ir
Current reflection factor is:  

Ih
Z0
B  2 l
 e
 0 e  2 l
A l
A
e
Z0
Distributed Parameter Networks 17.

U l  Ae l  Be  l  e l Ae j l  Be  j l
  0  nál
U l  Ae j l  Be  j l
A and B are complex numbers:
A  A e j és B  B e j
thus:
U l  A e j   Bl   B e j   Bl 
The direct and reflected
waves on complex plain:

    j
Distributed Parameter Networks 18.
Where the two vectors are in phase  voltage-maximum
Where the phase difference is 180o  voltage-minimum
If ZZ0 
 0
along the line standing waves are self created
If Z=Z0 ,
Z  Z0

 0 there are not standing waves
Z  Z0
Distributed Parameter Networks 19.
U max  A  B
U min  A  B
Voltage standing wave:
U max
r
U min
1 r  
B
1
AB
A 1  0



B 1  0
AB
1
A
or:
From the figure:


r 1
0 
 0
r 1
Distance between two minimum or maximum values is /2
Distance between maximum and minimum values is /4
 0