passive-networks
Imagine a step response of a series RLC circuit oscillates as an under-damped way:
I also came across the following equation:
As far as I understand the ωd above is the frequency of the damping oscillation (?) and ω0 is the resonant frequency which would be the oscillation freq. if the resistor was zero.
Is my way of thinking correct? If so what is ωd called in literature?
The problem is, I feel, in your thinking. The unit impulse does contain all frequencies and therefore, the RMS value of the particular frequency that corresponds with your filter's resonant frequency is infinitely small.
But, you might say that your filter's bandwidth is still quite wide so the energy of the spectrum around your resonance is not infinitely small. However, all the "in-band" frequencies that are "stimulating" your filter are incoherent and you can't expect to see those energies translated to one clear and obvious sinewave.
What you have is a series RLC circuit. One of the most common introductory circuit in EE. For example, look under Series RLC Circuit in this wikipedia page:
en.wikipedia.org/wiki/RLC_circuit
These are equations copied from the same page for the under-damped case. Given R, C along with measured \$\omega_d\$, L can be solved from these equations. (You don't even need to measure the attenuation factor \$\alpha\$.)
If the RLC do want to oscillate, the node where D1 cathode is connected will go negative. Therefore, depending on how D1 is driven, it may conduct during the negative half cycle and kill the oscillation.
Best Answer
It's sometimes\$^{(note1)}\$ called the damped resonant frequency and applies to some 2nd order filters but not all. I regard it as the frequency where the observable resonant peak is maximum on the bode plot. For a 2nd order low pass filter this might help: -
The peaking frequency becomes the same as \$\omega_n\$ when there is no damping (\$\zeta\$).
For a series RLC circuit there is no difference between \$\omega_n\$ and \$\omega_d\$ - it's just the way the math turns out because there is (what is called) a "zero" at 0 Hz and this nullifies the effect of the pole situated at negative complex frequencies. If you find this last paragraph beyond your learning I do apologize.
\$^{(note1)}\$ - Often the damped resonant frequency is called the peaking frequency (as mentioned above i.e. \$\omega_d=\omega_n\sqrt{1-2\zeta^2}\$) but it is also quite often mentioned as the projection of the pole to the jw axis as per below: -
Note the subtle difference in the two formulas.
That projection to the jw axis is the frequency of the decaying oscillation as seen with a step response on (say) an RLC low pass filter like this: -
With values plugged-into the calculator you get a response like this (and please note the damped oscillation frequency in the lower half of the picture): -
On-line calculator source.