Practical_Antenna_Handbook_0071639586

C h a p t e r 4 : T r a n s m i s s i o n L i n e s a n d I m p e d a n c e M a t c h i n g 141
In general, the simplest ways to determine Z IN as a function of feedline length for a
specific load are to use the graphical Smith chart techniques of Chap. 26 or one of the
many online calculators available. However, when the load on a nearly lossless length
of line is either a short circuit or an open circuit, the math required to find Z IN for a given
length of line is quite a bit simpler and yields interesting and useful results.
For a short circuit at the far end of the line, Z L = 0, and Eq. (4.42) reduces to
Z
IN−SHORT 0
⎡0 + jZ0
tan βl
⎤
= Z ⎢
⎥ = jZ
⎣ Z + j0
⎦
0
( )
0
tan( βl)
(4.43)
Since the only term in the solution is imaginary, Z IN is purely reactive—alternating
between capacitive and inductive reactance as the length of the short-circuited line is
varied and tan (bl) swings negative or positive, respectively.
Similarly, for an open circuit at the far end of the line, Z L = ∞, and Eq. (4.42)
reduces to
⎡ 1 ⎤
ZIN -OPEN
= Z0 ⎢ ⎥ = −jZ0
cot ( β l)
⎣ j tan( β l )
(4.44)
⎦
Note: If you have a network analyzer, VNA, or antenna analyzer that provides R and X for an
unknown impedance attached to its terminals, you can determine Z 0 and the velocity factor
(v F ) for a mystery piece of transmission line. First obtain Z IN-OPEN and Z IN-SHORT for the line.
Now multiply Eq. (4.43) by Eq. (4.44) to obtain Eq. (4.45):
( Z )( Z ) = ⎡⎣ jZ tan( βl) ⎤ ⎦ ⎡⎣ −Z cot( βl)
⎤ (4.45)
⎦
IN-SHORT IN-OPEN 0 0
But tan x = 1/cot x, so Eq. (4.45) reduces to
2 2 2
( Z )( Z ) = − j Z = Z
(4.46)
IN -SHORT
IN -OPEN
0
0
In other words, we can determine the characteristic impedance of our unlabeled transmission
line by taking the square root of the product of the short-circuit and open-circuit impedance
measurements we made.
If now we obtain the ratio of Eqs. (4.43) and (4.44), we have
Z
Z
IN-SHORT
IN -OPEN
jZ0
tan βl
2
=
tan ( l)
−jZ
cot β l
= − β (4.47)
0
( )
( )
Multiplying both sides by -1 and taking the positive square root, we obtain
ZIN-SHORT
tan( β l)
= − (4.48)
Z
IN -OPEN