Lecture
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 coskxjI Н Нsinkx
ux
Zx . (1.15)
ix I N j
cos kx sin kx
After the transformations (1.15), we obtain the expressions for Zx виде in the form Znjtgkx
(1.16)
Zx
jZ n
tg kx
which can also be written as
jZ ctgkx
(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
Zx jctgkx. (1.18)
For a closed transmission line, get the expression
Zx jtgkx. (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
j2kx
1Г 0 e
Zx j 0 2 kx . (1.20)
0
1Г 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 Zx is always purely active and exceeds the wave resistance of the transmission line
Zx 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
Zx 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
KBBj0,51KBV sin2kx
Zx. (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
No | 2 | 3 | four |
---|---|---|---|
HR | 10 | | 10 |
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
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Microwave Devices and Antennas
Terms: Microwave Devices and Antennas