1.7b, the transformed antenna impedance (the equivalent impedance of the antenna with the transmission line) becomes [1]
This result is obtained from impedance definition and by relating voltages and currents on the line via its characteristic impedance, Z0.
We have shown in the previous section that adding the lossless transmission line perfectly matched at the generator does not change antenna power. Now, we are about to show that adding a transmission line mismatched at both ends does change antenna power. It may be quite beneficial for impedance matching.
Example 1.9
A quarter wave transmission line transformer is characterized by (from Eq. (1.27))
An antenna with Za = 200 Ω is to be perfectly matched to the generator with Rg = 50 Ω by a proper choice of line characteristic impedance Z0.
Solution: The perfect match means that generator sees Z* = 50 Ω in the reference plane. Then, from Eq. (1.28), one has
(1.29)
It is not difficult to construct such a transmission line using, for example, a somewhat narrower microstrip as compared to the 50 Ω microstrip line.
Emphasize that the quarter wave transformer is relatively narrowband; it cannot be used for antenna matching over a wide band. Also, some reflections and standing waves will occur along the quarter wave line.
1.11 REFLECTION COEFFICIENT EXPRESSED IN DECIBELS AND ANTENNA BANDWIDTH
It follows from the previous discussion that the major dimensionless parameter that characterizes antenna matching is the antenna reflection coefficient, Γ. To highlight the physical meaning of the reflection coefficient and its acceptable threshold, we rewrite Eq. (1.17) for the antenna power one more time:
and let |Γ|2 to have the value of 0.1. According to Eq. (1.30), this means that the power delivered to the antenna is exactly 90% of the maximum available antenna power. Simultaneously, the reflection coefficient in dB (or power reflection coefficient),
(1.31)
will attain the value of −10 dB. Since ideal antenna matching (|Γ|=0 or |Γ|dB = − ∞) is never possible over the entire frequency band, it is a common agreement that, if
uniformly over the band, then the antenna is said to be matched over the band. In other words, at least 90% of available power from the generator will be delivered to the antenna for any frequency within the band. When applied over a frequency band, Eq. (1.32) thus determines the antenna impedance bandwidth or simply the antenna bandwidth. Several such bands may be present for a multiband antenna.
The TX antenna may be treated as a one port (port 1) of an electric linear network, whereas the other ports (if present) are other antennas. An example is given by a transmit/receive antenna pair, which form a two‐port linear network. In that and similar cases,
(1.33)
where S11 is the port 1 complex reflection coefficient or the first diagonal term of a multi‐port scattering matrix
Note:
Along with the reflection coefficient in dB, |S11|dB is always nonpositive. The terminal condition |S11|dB = 0 dB corresponds to a complete reflection of the voltage generator signal from the antenna. Nothing is being radiated.
Note:
Meanings of the complex reflection coefficient Γ itself and the dB measure of its magnitude, 20log10|Γ|, are often interchanged. For example, if an antenna datasheet reports “antenna reflection coefficient as a function of frequency,” it is 20log10|Γ| that is being plotted (cf. Figure 1.8 as an example).
Using MATLAB Antenna Toolbox, determine antenna impedance bandwidth for the blade dipole antenna with lA = 15 cm, w = 8 mm.
Solution: The approximate resonant frequency of the half wave dipole with the length of 15 cm is about 1 GHz. We again choose the frequency band from 200 to 1200 MHz for testing. A simple MATLAB script given below initializes the dipole antenna, plots the antenna geometry, and computes the dipole reflection coefficient Γ = S11. Note that the computations are performed exactly following Eq. (1.17), i.e. without an extra cable. Figure 1.8 shows the resulting dipole geometry and the reflection coefficient comparison with the analytical result obtained from Eq. (1.14). The agreement between two solutions is good while the numerical solution is again expected to be somewhat more accurate
