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With only pen and paper, is it possible to calculate the ideal Q-factor for e.g. a dipole antenna, loop etc, given a frequency?

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  • $\begingroup$ Do you specifically want the Q-factor for some reason or are you willing to accept more useful information about such antennas? Often a Q-factor is used with an antenna to provide a measure (or, indication) of the bandwidth of the antenna centered around the point of resonance at a given frequency. But, bandwidth for antennas is usually measured as the SWR being some acceptable value, say under 2:1. Often, the SWR which is measurable easily is a better parameter than Q-Factor and gives equivalent information. $\endgroup$ – K7PEH Oct 24 '15 at 2:44
  • $\begingroup$ Follow up to above comment. I went looking for my notes on my own 80-meter dipole because I measured the 2:1 SWR window bandwidth at a number of spots on the antenna. Down in the low end, 3.5 to 3.560 MHz, the bandwidth of my dipole (under 2:1 SWR) window is about 60 KHz centered on 3540 KHz (a frequency I use a lot for a particular CW traffic net). Thus, outside of that 60 KHz window, I usually have to retune the antenna with my antenna tuner (no big thing, it is actually auto-tuned). $\endgroup$ – K7PEH Oct 24 '15 at 2:49
  • $\begingroup$ So you have the data, roughly, already. The Q is something like 3540 / 60, the center frequency divided by the bandwidth. The actual antenna "Q" depends on a bunch of factors, like the wire resistance, wire diameter (bigger=broader), ground proximity, etc. $\endgroup$ – Martin Ewing AA6E Oct 25 '15 at 12:33
  • $\begingroup$ @MartinEwingAA6E -- I may have the data but I don't care about the Q-factor. For my use and analysis of antennas in "ham radio", the q-factor is totally useless compared to an easily directly measured SWR. Now, the OP may be interested in Q-factor because it is some kind of homework problem to compute it. But, if he is interested in a particular antenna to build or use, q-factor is not a big help. $\endgroup$ – K7PEH Oct 26 '15 at 18:14
  • $\begingroup$ The Q tells you the bandwidth of the antenna resonance. If you have a reasonable match at resonance, it is also a measure of "SWR bandwidth". So it's just a different way of looking at an antenna design. $\endgroup$ – Martin Ewing AA6E Oct 26 '15 at 22:59
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There are at least two definitions of Q factor. One is the reciprocial of fractional bandwidth:

$$ Q = {f_r \over \Delta f} $$

Where $ f_r $ is the resonant frequency (or center of the antenna's bandwidth) and $ \Delta f $ is the difference between the upper and lower frequencies of the bandwidth.

By this definition, it's easy to calculate with pen and paper if you are allowed some empirical data. You might define "bandwidth" as the range of frequencies where the VSWR is better than 2:1 and empirically measure this bandwidth. Then calculating the Q factor is some simple arithmetic.

The other definition of Q is the ratio of energy stored to energy dissipated per cycle:

$$ Q = 2 \pi {\text{energy stored} \over \text{energy dissipated per cycle}} $$

This is in theory calculable, but it's complex enough I don't think any normal person would attempt the calculation with pen and paper.

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This is all about definition.

First, define what is acceptable as maximum VSWR for your particular situation, at a given impedance for a given antenna.

Once you have that you can measure the lowest frequency (call this f1) where the maximum VSWR is measured on that antenna, then you measure the highest frequency (call this f2) that maximum VSWR is reached.

Now your Q can be calculated as

$$ f_c = \frac{f_1+f_2}{2} \\ Q = \frac{f_c}{f_2-f_1} $$

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