Practical_Antenna_Handbook_0071639586

124 P a r t I I : F u n d a m e n t a l s
reflected. But in a mismatched system (where Z L ≠ Z 0 ), a portion of the wave is reflected
back up the line toward the source.
Reflection Coefficient
Mismatches can vary from a perfect short circuit at the load to an open circuit (no load
at all). In between are all the possible combinations of resistance and reactance—an infinite
number! It seems intuitive that the effect of a very slight difference between Z L
and Z 0 should have less effect on the resulting line conditions than a major mismatch,
such as a short or open circuit.
The reflection coefficient G of a circuit containing a transmission line and load impedance
is a measure of how well the load is matched to the transmission line:
where V REF = reflected voltage
V FWD = forward, or incident, voltage
REF
G = V
(4.18)
V
The absolute value of the reflection coefficient varies from –1 to +1, depending upon
the nature of the reflection; G = 0 indicates a perfect match with no reflection, while –1
indicates a short-circuited load (Z L = 0), and +1 indicates an open circuit (Z L = ∞). To see
how this comes about, consider the set of boundary conditions that must be true at the
junction of the transmission line and the load (antenna).
• Just before arriving at the load, the advancing step function knows only that it’s
traveling on a transmission line of characteristic impedance Z 0 . Therefore, at
each and every point on the transmission line prior to the step function first
reaching the load, the voltage is related to the current by V FWD = I FWD Z 0 .
• The voltage across the load (usually an antenna) must always be related to the
current through it by V L = I L Z L .
• At the junction of the line and the load, the current must be continuous; that is,
the current in the transmission line has no place else to go except into the load.
Thus, if I L = I FWD , then V L = V FWD only if Z 0 = Z L . This is the matched condition,
for which G = 0.
• Suppose, however, that we substitute a short circuit for the original load. Now
Z L = 0, so V L = I L Z L = 0. The advancing wave is attempting to apply a voltage
V FWD = I FWD Z 0 to the boundary, yet we measure a voltage of zero at the boundary.
For this to be true, there must be a second voltage present at the boundary such
that the sum of V FWD and this new voltage is zero (by linear superposition). We
call this new voltage V REF , and it has to be exactly equal in magnitude to V FWD ,
but of opposite polarity. In other words, V REF = -V FWD , hence
FWD
VREF
VFWD
Γ = = − = −1
V V
FWD
FWD