Naive question on quality factor

Hello,

I am in the very (very) beginning of my training in antennas and I have a naive question on antennas quality factor.
I always thought that quality factor Q was simply a quantification of the sharpness of the resonance peak. So, if two antennas have the same quality factor, their resonance peaks would have the same sharpness.
Yet, it appears that I was wrong.

Since the formula is Q = f/Δf, the sharpness is only defined by the bandwidth Δf. So if two antennas have their resonance frequency at 1 GHz and 2 GHz respectively, and the same resonance sharpness, then the 2 GHz antenna would have a 2 times the Q-factor of the 1 GHz antenna.

Am I right or do I miss something obvious ?

I am sorry for the naive question but I am pretty confused, because I know that researchers that use waveguides to estimate the complex permittivity of materials correlate the variation of the resonance frequency with the real part of the permittivity, and Q-factor variations with the dielectric losses. If Q variations contains frequency variations then it must be more difficult than I thought.

Thank you!

Have a look at the following it may have answers for most of your questions
https://uspas.fnal.gov/materials/09UNM/ResonantCavities.pdf
Actually for me who works on RF cavities the quality factor is the ration of energy stored in cavity to the energy dissipated in cavity walls and apart from frequency which is mostly fixed it is highly material and design dependent.
Have a look at the link

Thank you for this very interesting link.
I am very uncomfortable with some of the technical aspects involved here (I am chemist) but if I understand what I have read in this link and in related documents, I can answer my own questions as follows:

1) For the Q-factors, if we take the - 3dB bandwidth definition, it is correct to say that two resonance peaks with the same bandwidth and amplitude (ie the same sharpness) will have different Q-factors. If one resonance frequency is the double than the other, then there will be a factor of 2 between the two Q-factors.

2) The point 1) doesn't matter when it comes to complex permittivity measurements through cavity perturbation because this method is only valid in the small perturbation theory conditions (volume of the sample very small compared to the cavity volume). In small perturbation conditions, the frequency shift will be relatively small, so it stays negligible in the Q-factor determination.

Is this correct?
Maybe the energy definition of the Q-factor eliminates any frequency shift impact on Q-factor value but I don't fully understand the definition/how it is measured.

Thanks again.

I think you are creating a pseudo problem by specifying resonance sharpness in absolute bandwidth (Δf) units. If you observe resonance effects over a larger frequency range, it's practical to read it as relative bandwidth Δf/f.

Quality factor is defined in various ways depending on the context (examples include fc/BW, X/R, Pstored/Pdiss, etc). Some of these definitions can contradict each other, especially when your system has a response with order higher than 2. Responsible engineers are careful to specify how they define Q factor, and treat it appropriately when making comparisons.

First lets start with what Q really is about.
The Q factor is the amount of energy stored - in a component or resonant circuit - compared to the amount of energy dissipated as heat.
The energy is stored as a magnetic field in an inductor and as an electric field in a capacitor.
In a resonant circuit the energy gets converted to a magnetic field for some time until the magnetic field collapses and converts it to current that ruses in to the capacitor and gets converted to an electric field, this in turn collapses after some time and gets converted in to current that goes in to the inductor and the cycle starts over again.
This conversion is completely loss-free and could go on forever, how ever when the energy is a current some small portion gets converted to heat, how much that gets converted each time depends on the resistance. Low resistance = low conversion rate to heat = High Q

High Q means the circuit (or antenna) "rings" for a long time after being exited with energy.
Think of a tuning fork, when you strike it it rings for a long time, it has a high Q.
If you put a piece of cloth to the tuning fork to dampen it it will only ring for a very short time after being striked, you have lowered it's Q value by introducing loss to the resonator.
The undampened tuning fork has a high "unloaded Q" the dampened tuning fork has a lower "loaded Q".
Loaded Q is always lower.


There are many ways to calculate the Q value but it is always a relationship between to factors.
One way to calculate the Q can be to divide the reactance of an inductor with its resistance.
Say the reactance is 1000ohm at a certain frequency and the resistance is 10ohm the Q value of the inductor at that frequency would be 100.

Another way of deducing the Q value of a resonant circuit (like an antenna) would be to observe the bandwidth and divide the center frequency with this value.
Eg. an antenna is at resonance at 1GHz and has a bandwith of 100MHz.
The antenna has a Q of 10.

The same antenna shortened in half will be resonant at 2 GHz and have a bandwith of 200MHz, giving the same Q value of 10.

I hope this clears up the concept of Q for you.

//Harry