1.5. Analysis of the distribution of resistance along the transmission line

1.3. Analysis of the distribution of resistance along the transmission line

In any section of the transmission line, the equivalent resistance is determined by the formula, which, based on the first system of equations, can be written as U coskxjI Н Нsinkx

ux

Zx . (1.15)

 UN

ix I N j  

cos kx sin kx

After the transformations (1.15), we obtain the expressions for Zx виде in the form Znjtgkx

 (1.16)

Zx

jZ n 

tg kx

which can also be written as

jZ ctgkx

 (1.17)

Zx n .

Z n j 

c tg kx

Example. We construct the distribution of resistance along an open transmission line (Z N  - idling mode) and a closed transmission line (Z Н 0 - short-circuit mode). From (1.16) for the open line transmission, we obtain the expression

Zx jctgkx. (1.18)

For a closed transmission line, get the expression

Zx jtgkx. (1.19) On the basis of (1.18) and (1.19), we construct the distributions of equivalent resistance along the transmission line, which is open and closed at short in the load - see fig. 1.17.

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

Z H  Z H  0

short circuit

15

From the second system of telegraph equations (1.7) we obtain the expression for the distribution of resistance in the transmission line

j2kx

1Г 0 e

Zx  j   0  2 kx  . (1.20)

0

1Г 0 e

Let us analyze (1.20) in maxima and minima. In the place of the maximum amplitude of the voltage, with arbitrary loads, the resistance Zx is always purely active and exceeds the wave resistance of the transmission line

Zx R   . (1.21)

max max

KBV At the point of minima of the voltage amplitude, the resistance is also active and less than the wave impedance

Zx R  KBV. (1.22)

min min

For an arbitrary load it is necessary to get rid of 0 . To do this, we introduce a new

coordinate system of the location of the first minimum of the voltage amplitude x'xx x 0

- see fig. 1.18.

Then you can get an expression to calculate the distribution of resistance along a uniform transmission line without any loss

2

KBBj0,51KBV sin2kx

Zx. (1.23)

2 22

cos KBV sin 

kx kx

Example. Consider the active load Z R, X H 0, for which the values ​​of R

NN N

are given in table. 1.2. Consider the transmission line in impedance 50, Ohm. Tab. 1.2 - Load resistance values

The results of calculations of the distribution of resistance, made by the formula (1.16) are presented by the active component of the resistance - in Fig. 1.19, and; on the reactive component resistance - in fig. 1.19 b.

Z Н  Z Н  Z Н  10

10 b) Pic. 1.19 - Resistance distribution along the transmission line