Practical_Antenna_Handbook_0071639586

142 P a r t I I : F u n d a m e n t a l s
Now we know l and the tangent of bl, so we can find b, the phase constant of the line from
trig tables or our scientific calculator.
More likely, we would like to know v F , the velocity factor of the line. Rearranging Eq.
(4.16), we obtain
2
v = π f
F
cβ
(4.49)
Because a shorted or open length of transmission line always exhibits a pure reactance
at the other end, such lines are often used to match an antenna to its transmission
line. A specific reactance, either capacitive or inductive, can be obtained simply by adjusting
the length of the line, or stub.
In most cases, however, not only do we need to cancel out the reactive part of the
feedpoint impedance, we also want to change the resistive part to match the Z 0 of the
feedline. Simply adjusting the length of the stub cannot compensate for both the resistive
and the reactive mismatches. A second adjustable variable is needed. That turns out
to be the distance of the stub back toward the transmitter from the antenna feedpoint.
For a variety of reasons, both electrical and mechanical, stubs are more often found
connected in parallel, across the feedline, than in series with one side of the feedline. In
such cases, it is far easier to work with the reciprocal of the impedances involved. These
admittances, denoted by Y, consist of a shunt conductance G (the purely resistive part of
the admittance) in parallel with a shunt susceptance B, which is the purely reactive part.
Mathematically, if Z = R + jX, then
1 1
Y = = = G + jB
(4.50)
Z R + jX
1
To put into the right form, we multiply both top and bottom by R – jX, the
R + jX
complex conjugate of R + jX. This gives us
Thus,
and
⎡ 1 ⎤⎡R
− jX ⎤ R − jX
⎢ ⎥⎢
⎥ =
2 2
⎣R
+ jX ⎦⎣R
− jX ⎦ R + X
R
G =
R + X
2 2
B = − jX
R + X
2 2
(4.51)
(4.52)
(4.53)
In a small percentage of stub-matching cases, it turns out to be possible to locate the
stub right at the antenna feed terminals. Most often, however, a stub must be located at
some specified distance from the antenna feedpoint.
Without the use of paper or online Smith charts, stub matching is far more difficult
than it needs to be. Chapter 26 includes a detailed example of stub matching using a
Smith chart.