Antenna Analyzer AIM4170 page 30
Reflection Coefficient:
To find the SWR (standing wave ratio) of an antenna, we first calculate the reflection
coefficient. This is the ratio of the voltage that is reflected at the antenna to the voltage
that arrives at the antenna from the transmitter. If all the power from the transmitter is
radiated into space, there is no reflection, the reflection coefficient is zero and the
SWR=1.0.
The following discussion uses the concept of complex numbers. A tutorial on complex
numbers is available in Appendix 1.
Reflection_Coefficient = Rho = ( Zload – Zo ) / ( Zload + Zo )
Zload = antenna impedance
Zo = transmission line impedance
Note that in general, Zload and Zo are complex numbers of the form:
Zload = Ra + jXa and
Zo=Ro+jXo.
Xo, which is the imaginary part of Zo, is often neglected since it is usually small
compared to the real part, Ro.
Since Zload is a complex number, the reflection coefficient, Rho, is also a complex
number.
The reflection of the incoming power from the transmitter is caused by a mismatch
between the transmission line impedance (Zo) and the impedance of the antenna at
the operating frequency. For example, if the transmission line has an impedance of 50
ohms and the antenna is a dipole with an impedance of around 75 ohms, there is a
mismatch and some of the power is reflected even though the antenna itself may be very
good. If the transmission line is changed to 75 ohms, the match is much better, there is
less reflection and the SWR is closer to 1.0.
In the special case where the transmission line is open at the antenna (due to a broken
wire), all of the power that arrives at this open circuit will be reflected back toward the
transmitter and the reflection coefficient is 1.0 and the SWR= infinity. Another
interesting case is when the transmission line is shorted at the antenna terminals. Again,
all the power will be reflected (none is radiated) but the signal is inverted, so the
reflection coefficient is now –1.0 (minus one). The magnitude is still unity (that is, +1)
and the SWR=infinity.
Thus, we see that the magnitude of the reflection coefficient will be in the range of zero
to 1.0 for any combination of transmission line and antenna.
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