Electronic – Intuition for resonant, natural, and oscillatory frequencies of RLC circuits


I can't seem to get these concepts straight.

To start, what are the official definitions of each?

My current understanding is that:
\$ w_0 = \sqrt{\frac{1}{LC}} \$ is the natural (and resonant?) frequency that an undriven LC circuit (or a RLC circuit with no damping) oscillates at. When we add nonzero damping though, we then get that the natural frequency that the system tends to oscillate at is \$ s = \frac{R}{2L} \pm \sqrt{\frac{R}{2L}^2 – w_0^2} \$.

(Then there is the damped natural frequency \$ w_d = \sqrt{w_0^2-\frac{R}{2L}^2} \$ for an underdamped system — where does this come in?)

Why are there two natural frequencies — in real life which one does the circuit actually oscillate at? Are these the same natural frequencies for ANY configuration of R, L, C (series, parallel, more complex setups, etc)? Is yes, why does that intuitively make sense? If not, is there any intuition behind the different expressions for natural frequency for series/parallel? (Is the freq greater or less, and how can we roughly judge from a circuit schematic whether its natural frequency will be high/low?)

In the nonzero damping case, does \$ w_0 \$ just become an abstract quantity? Why is it that when we have RLC filters, the peak frequency (where we have the greatest response) is still \$ w_0 \$ and not the more complex expression we have above?

Where does quality factor come into this? Is it always \$ \frac{w_0L}{R} \$ regardless of the circuit setup? (again, what's the intuition to this answer?)

Correct any misconceptions I may have — I'd love to understand this topic more deeply. Thanks so much in advance!

Best Answer

There are quite a few subtle differences between band-pass and low/high filters but, for a simple LCR band-pass filter, damped and un-damped resonant frequencies are the same numerically and formulaically, \$\dfrac{1}{2\pi\sqrt{LC}}\$.

This is also the (un-damped) natural resonant frequency for high/low pass filters.

When damping is added, the natural frequency stays the same but it can (eg for a low pass filter) rotate anticlockwise in the pole zero plane and this leaves (in the jw axis) what is known as the damped resonant frequency, \$\omega_d = \omega_n\sqrt{1 - \zeta^2}\$. See lower part of 1st set of pictures below (it's basically Pythagoras and right-angle triangles).

And for low pass filters, there is the frequency at which peaking occurs and this is slightly different to damped resonant frequency, \$\omega_p = \omega_n\sqrt{1 - 2\zeta^2}\$.

The amplitude that this peaks at is \$\dfrac{1}{2\zeta\sqrt{1-\zeta^2}}\$

enter image description here

The proof of these three different frequencies for a low-pass filter is relatively straightforward but a little long winded. Below is an extract of a design paper for a 2nd order low pass filter where it is shown that a varying Q-factor moved the "peaking" frequency away from the natural frequency (100 Hz in this example) but, as always Q is the value of the peak at the natural resonant frequency: -

enter image description here

Maybe a close up view will be more exciting: -

enter image description here