Month: October 2013

Mostly on 40m band, mode CW, and ANT is dipole, POW 20-30W.

Now, let’s think of a case, for example, in which Z=10+jX.

This time Z can be anywhere on the r=0.2 circle (outer brown circle) and you can see from the figure that Z is again most close to the origin (red dot) when X=0.

See that the two brown circles (outer and inner ones) are mutually tangent to each other, the point of the contact being at X=0.

In general, the feed point impedance of an antenna is Z=R+jX, where R could be either greater than or less than 50 ohm.

First, let’s think of a case, for example, in which Z=100+jX. Now Z can be anywhere on the r=2 circle (blue on the right) and it is obvious that Z is most close to the origin (red dot) when X=0.

This can also be confirmed by looking at the two blue circles which are mutually tangent to each other, the point of the contact being at X=0.

The same can be said for any R > 50, which you can see immediately from the figure if you imagine two such blue circles.

(picture from [1], but the color added by the author.)

So we would like to know if an antenna in resonance gives the minimum VSWR, under the assumption that only the imaginary part of the feed point impedance varies when when we change the element length.

Let’s assume for the moment that the system impedance Z0 is 50 ohm, and if we further assume that the feed point impedance is given by "Z=50+j*X", then Z is always on the r=1 circle (green), and in this case obviously Z is most close to the origin (red dot) when X=0.

Actually in this particular case, Z becomes exactly 50 ohm when X=0, and coincides with Z0, thus the complex reflection coefficient gamma is 0, which means rho is also 0 and VSWR=1.

[1] http://www.arrl.org/files/file/Antenna%20Book%20Supplemental%20Files/22nd%20Edition/Smith%20Chart%20Supplement%20-%20Corrected%20Jan%202012.pdf

An antenna is said to be resonant if its input impedance (or feed point impedance), Z=R+jX, is purely resistive, which means its reactance X is exactly zero.

The input impedance of a half-wave dipole is approximated by the equation:

where We and k are some constants, and delta is the excess length of an antenna over half the wavelength. Since both We and k are positive, we can make X to be zero by shortening the element slightly, usually a few percent depending on the diameter of the element.

The question here is what is the physical and practical meaning of the antenna being in resonant.

References:
[1] “Antenna and Radio propagation (in Japanese)” by Yasuto Mushiake,
[2] MMANA-GAL basic.

The above two figures are from
Mats Gustafsson et al., Forward scattering of loaded and unloaded antennas.

A receiving antenna interacts with an incoming electromagnetic wave. However, its interaction is not limited to absorption of the energy but includes some scattering from the antenna.

As is described in “Antenna and Radio propagation (in Japanese)” written by Yasuto Mushiake, pp 39-42, the scattering cross section of an (receiving) antenna loaded with Zl is given by

where Z=R+jX is the input impedance and Ae is the effective area of the antenna.

The scattering cross section As is a function of Zl, and obviously As(max)=4Ae when Zl=0. And if Zl is selected to be Zconj=R-jX, then As=Ae, which means the antenna emits the same amount of power as it supplies to the receiver.