1.4. The second system of telegraph equations

A characteristic feature of long lines is the possibility of the existence of two waves in them, extending towards each other. One of these waves is formed by an electromagnetic generator *** connected to the line and is called the incident one. Another wave is formed due to the reflection of the incident wave from the load connected to the opposite end of the line, and is called reflected. The reflected wave propagates in the direction opposite to the incident wave. The whole variety of processes occurring in a long line is determined by the amplitude-phase relations between the incident and reflected waves.

Consider a lossless transmission line loaded with complex resistance Z Н - see fig. 1.14. The distribution of voltage and current along the transmission line can be described using the equation system:

uxU xU x;

От pad re

 (1.3)

ix Id Ip .

xx



l

x lx 0

The falling and reflected stress wave is described by the expressions

U xU 0e j kx ;

 pad pad

 (1.4)

jkx

U x U  0 e.

 rejected

Introduce the reflection coefficient of the stress G  x, which is determined by the formula

U x

Гx  р. (1.5)

U pad 

x

The current reflectance is G I  x   x .

From (1.4) and (1.5) we determine the value of Гx

j2kx

ГxГ н e, (1.6)

0

where G  G0  G e; Г 0 ,  - modulus and phase of reflection coefficient at a point

n 00

connecting load. After substituting (1.5) and (1.6) into (1.3), we obtain expressions for determining the distribution of voltage and current along the transmission line:  Ud x1 0 expj 0 kx;

ux   2 

 (1.7)

 Id 1 0 expj  0 2kx

ix x  .

From (1.7) we obtain the expressions for calculating the distribution of the amplitudes of voltage and current along the transmission line:

2



U 1Г 2Г cos 2kx;

ux

pad 00 0



(1.8)



2

I 1Г 2Г cos 2kx.

ix



pad 00 0



Normalized voltage distribution is determined by the formula

the norm

2

x

u  1Г 2Г cos 2kx. (1.9)

00 0

The maximum and minimum voltage value is determined by the formulas: U U (0) 1Г ;

max pad 0

 (1.10)

 U min  U pad (0) 1 G 0 .

Coordinate of the location of the first load maximum voltage x max

is related to the phase of the reflection coefficient in the load  0 by the formula

 0

x max . (1.11)

2k

The coordinate of the location of the first side of the load minimum voltage

x min

determined by the formula



x x . (1.12)

min max

four

The reflection coefficient is associated with the radiation resistance and wave impedance of the transmission line:

Z 

Mr. N ; (1.13)

Z n  1Г n

Z . (1.14)

n 1 G

n

Formulas (1.8) - (1.14) allow calculating the amplitude distribution of voltage and current along the conductors of a two-wire transmission line. Example. For the transmission line with the characteristics l , 75 Ohm, construct the normalized voltage and current distributions for the load Z Н 100 j150 Ohm.

We carry out calculations for the load Z H 100  j150.

The reflection coefficient in the load according to the formula (2.13): G 0  0.66,  0  40  . The location of the highs and lows of the voltage according to the formula (1.11) and (1.12): x  0.055, x  0,305.

max min

The maximum and minimum values ​​of the normalized voltage and current

norms  norms 

to formulas (1.10): U max 1.66, U min 0.34. According to the formulas (1.8) and (1.9) we construct the distribution of voltage and current - see fig. 1.15.

x

1 0.9 0.8 0.7 0.6 0.5 0.4 0.2 0.2 0.1 0 0

u (x) i (x) 

Fig. 1.15 - The distribution of the normalized amplitude of voltage and current along the transmission line

We carry out calculations for the load Z H 100  j150.

The reflection coefficient in the load according to the formula (1.13): G 0  0.66, 0  320  . The location of the highs and lows of the voltage according to the formula (1.11) and (1.12): x max  0.445, x min 0.195.

The maximum and minimum values ​​of the normalized voltage and current

norms  norms 

to formulas (1.10): U max 1.66, U min 0.34. According to the formulas (1.8) and (1.9) we construct the distribution of voltage and current - see fig. 1.16.

u m (x)

im (x)



If the nature of the load is inductive, then the first on the load side will be



voltage, located at a distance of not more than the load. If character

4 reactivity Z N capacitive, then the first from the load will be located at a minimum voltage.

The distance between the minima and the voltage maxima is a quarter of the wavelength.