# Why does the $Q$-factor depend on the resonant frequency?

The $$\Q\$$-factor is defined as $$Q=\frac{\omega_{0}}{\Delta \omega},$$

where $$\ \Delta \omega\$$ is the bandwidth around the resonant frequency $$\ \omega _{0}\$$. For a series RLC circuit, $$\Delta \omega = \frac{R}{L}$$

and $$\omega_{0}=\frac{1}{\sqrt{LC}}.$$

$$\Q\$$ is supposed to measure how sharp the frequency response is. And so a higher $$\Q\$$ means a smaller $$\\Delta \omega \$$. But we can clearly see above that for a series RLC circuit, the $$\Q\$$-factor can be increased just by increasing $$\\omega_{0}\$$, alone, by decreasing the value of $$\C\$$, without even changing how narrow the response curve is, i.e. without even changing $$\\Delta \omega \$$.

So why a higher resonant frequency implies a higher $$\Q\$$-factor, which is supposed to measure how sharp the frequency response of our resonant circuit is? In other words, why should the resonant frequency have anything to do with the the $$\Q\$$-factor?

• That's simply the definition of the quality factor. You'll have to just live with it. Also, you already derived that to increase $\omega_0$ withouh changing $\Delta\omega$, you'd have to touch both L and R, so you can clearly see how that RLC circuit is fundamentally a different one with different ohmic losses / dampening. All already in your question! Feb 16 '20 at 16:26
• Is a bandwidth of 100 Hz small or large? You cannot answer because it depends on the mid frequency. Relative to 500 Hz it is large and relative to 500kHz it is rather small. Hence, the Q-factor is defined as the inverse of the RELATIVE bandwidth (bandwidth divided by the midfrequency).
– LvW
Feb 16 '20 at 16:30
• $\Delta \omega$ is frequency -dependent as well as $Q$. Skin depth is one frequency-dependent factor that raises series-R as frequency goes up. Feb 16 '20 at 16:34
• @LvW I don't think it makes much sense to compare a bandwidth with a frequency. Bandwidths can only be compared to bandwidths, the same goes for frequencies. And so a bandwidth of 100 Hz is greater than a bandwidth of 1 Hz and smaller than one of 1 kHz, comparing a bandwidth of 100 Hz to a frequency of 500 Hz or 500 kHz doesn't make much sense. Feb 16 '20 at 16:47
• If you want a measure of bandwidth or linewidth that doesn't depend on center frequency, just use $\Delta\omega$. Feb 16 '20 at 16:52

So why a higher resonant frequency implies a higher Q-factor, which is supposed to measure how sharp the frequency response of our resonant circuit is?

It measures how sharp the resonance is, relative to the resonance frequency.

This might make resonant circuit topologies or designs easier to compare for certain purposes if they're designed at different resonant frequencies.

If you want a measure of how sharp the resonance is that isn't dependent on the center frequency, you can always just use $$\\Delta\omega\$$.

The Q relationship makes spectrum and filter shapes a constant on a log scale.

Q applies to audio filters. And human perception of audio bandwidth (perceivable pitch differences, bass and treble bands, etc.) is more similar to a log scale than a linear scale. Constant ratio rather than a constant delta Hertz.

Same with the physical component tolerances required to create a musical instrument of RF circuit with a given Q.